Advanced Fixed Point Theory for
Economics
Andrew McLennan
April 8, 2014
Preface
Over two decades ago now I wrote a rather long survey of the mathematical
theory of fixed points entitled Selected Topics in the Theory of Fixed Points. It
had no content that could not be found elsewhere in the mathematical literature,
but nonetheless some economists found it useful. Almost as long ago, I began
work on the project of turning it into a proper book, and finally that project is
coming to fruition. Various events over the years have reinforced my belief that the
mathematics presented here will continue to influence the development of theoretical
economics, and have intensified my regret about not having completed it sooner.
There is a vast literature on this topic, which has influenced me in many ways,
and which cannot be described in any useful way here. Even so, I should say
something about how the present work stands in relation to three other books on
fixed points. Fixed Point Theorems with Applications to Economics and Game
Theory by Kim Border (1985) is a complement, not a substitute, explaining various
forms of the fixed point principle such as the KKMS theorem and some of the
many theorems of Ky Fan, along with the concrete details of how they are actually
applied in economic theory. Fixed Point Theory by Dugundji and Granas (2003) is,
even more than this book, a comprehensive treatment of the topic. Its fundamental
point of view (applications to nonlinear functional analysis) audience (professional
mathematicians) and technical base (there is extensive use of algebraic topology)
are quite different, but it is still a work with much to offer to economics. Particularly
notable is the extensive and meticulous information concerning the literature and
history of the subject, which is full of affection for the theory and its creators. The
book that was, by far, the most useful to me, is The Lefschetz Fixed Point Theorem
by Robert Brown (1971). Again, his approach and mine have differences rooted in
the nature of our audiences, and the overall objectives, but at their cores the two
books are quite similar, in large part because I borrowed a great deal.
I would like to thank the many people who, over the years, have commented
favorably on Selected Topics. It is a particular pleasure to acknowledge some very
detailed and generous written comments by Klaus Ritzberger. This work would not
have been possible without the support and affection of my families, both present
and past, for which I am forever grateful.
i
Contents
1 Introduction and Summary
1.1 The First Fixed Point Theorems
1.2 “Fixing” Kakutani’s Theorem .
1.3 Essential Sets of Fixed Points .
1.4 Index and Degree . . . . . . . .
1.4.1 Manifolds . . . . . . . .
1.4.2 The Degree . . . . . . .
1.4.3 The Fixed Point Index .
1.5 Topological Consequences . . .
1.6 Dynamical Systems . . . . . . .
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Topological Methods
22
2 Planes, Polyhedra, and Polytopes
2.1 Affine Subspaces . . . . . . . . .
2.2 Convex Sets and Cones . . . . . .
2.3 Polyhedra . . . . . . . . . . . . .
2.4 Polytopes . . . . . . . . . . . . .
2.5 Polyhedral Complexes . . . . . .
2.6 Graphs . . . . . . . . . . . . . . .
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3 Computing Fixed Points
3.1 The Lemke-Howson Algorithm . . . . . . . .
3.2 Implementation and Degeneracy Resolution
3.3 Using Games to Find Fixed Points . . . . .
3.4 Sperner’s Lemma . . . . . . . . . . . . . . .
3.5 The Scarf Algorithm . . . . . . . . . . . . .
3.6 Homotopy . . . . . . . . . . . . . . . . . . .
3.7 Remarks on Computation . . . . . . . . . .
4 Topologies on Spaces of Sets
4.1 Topological Terminology . . .
4.2 Spaces of Closed and Compact
4.3 Vietoris’ Theorem . . . . . . .
4.4 Hausdorff Distance . . . . . .
4.5 Basic Operations on Subsets .
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Sets
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23
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iii
CONTENTS
4.5.1
4.5.2
4.5.3
4.5.4
4.5.5
4.5.6
Continuity of Union . . . . . . . . .
Continuity of Intersection . . . . . .
Singletons . . . . . . . . . . . . . . .
Continuity of the Cartesian Product
The Action of a Function . . . . . . .
The Union of the Elements . . . . . .
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5 Topologies on Functions and Correspondences
5.1 Upper and Lower Semicontinuity . . . . . . . .
5.2 The Strong Upper Topology . . . . . . . . . . .
5.3 The Weak Upper Topology . . . . . . . . . . . .
5.4 The Homotopy Principle . . . . . . . . . . . . .
5.5 Continuous Functions . . . . . . . . . . . . . . .
6 Metric Space Theory
6.1 Paracompactness . . . . .
6.2 Partitions of Unity . . . .
6.3 Topological Vector Spaces
6.4 Banach and Hilbert Spaces
6.5 EmbeddingTheorems . . .
6.6 Dugundji’s Theorem . . .
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7 Retracts
7.1 Kinoshita’s Example . . . . . . .
7.2 Retracts . . . . . . . . . . . . . .
7.3 Euclidean Neighborhood Retracts
7.4 Absolute Neighborhood Retracts
7.5 Absolute Retracts . . . . . . . . .
7.6 Domination . . . . . . . . . . . .
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71
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104
8 Essential Sets of Fixed Points
107
8.1 The Fan-Glicksberg Theorem . . . . . . . . . . . . . . . . . . . . . 108
8.2 Convex Valued Correspondences . . . . . . . . . . . . . . . . . . . . 110
8.3 Kinoshita’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 112
9 Approximation of Correspondences
9.1 The Approximation Result . . . . . . . . .
9.2 Extending from the Boundary of a Simplex
9.3 Extending to All of a Simplicial Complex .
9.4 Completing the Argument . . . . . . . . .
II
Smooth Methods
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115
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124
10 Differentiable Manifolds
125
10.1 Review of Multivariate Calculus . . . . . . . . . . . . . . . . . . . . 126
10.2 Smooth Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . 128
1
CONTENTS
10.3
10.4
10.5
10.6
10.7
10.8
10.9
Manifolds . . . . . . . . . . . . . . . .
Smooth Maps . . . . . . . . . . . . . .
Tangent Vectors and Derivatives . . . .
Submanifolds . . . . . . . . . . . . . .
Tubular Neighborhoods . . . . . . . . .
Manifolds with Boundary . . . . . . .
Classification of Compact 1-Manifolds .
11 Sard’s Theorem
11.1 Sets of Measure Zero . . . . . . . .
11.2 A Weak Fubini Theorem . . . . . .
11.3 Sard’s Theorem . . . . . . . . . . .
11.4 Measure Zero Subsets of Manifolds
11.5 Genericity of Transversality . . . .
12 Degree Theory
12.1 Orientation . . . . . . . . . . . . .
12.2 Induced Orientation . . . . . . . .
12.3 The Degree . . . . . . . . . . . . .
12.4 Composition and Cartesian Product
13 The
13.1
13.2
13.3
13.4
13.5
III
Fixed Point Index
Axioms for an Index on a Single
Multiple Spaces . . . . . . . . .
The Index for Euclidean Spaces
Extension by Commutativity . .
Extension by Continuity . . . .
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Space
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131
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189
Applications and Extensions
193
14 Topological Consequences
14.1 Euler, Lefschetz, and Eilenberg-Montgomery
14.2 The Hopf Theorem . . . . . . . . . . . . . .
14.3 More on Maps Between Spheres . . . . . . .
14.4 Invariance of Domain . . . . . . . . . . . . .
14.5 Essential Sets Revisited . . . . . . . . . . . .
15 Vector Fields and their Equilibria
15.1 Euclidean Dynamical Systems . . .
15.2 Dynamics on a Manifold . . . . . .
15.3 The Vector Field Index . . . . . . .
15.4 Dynamic Stability . . . . . . . . . .
15.5 The Converse Lyapunov Problem .
15.6 A Necessary Condition for Stability
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194
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207
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211
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226
Chapter 1
Introduction and Summary
The Brouwer fixed point theorem states that if C is a nonempty compact convex
subset of a Euclidean space and f : C → C is continuous, then f has a fixed point,
which is to say that there is an x∗ ∈ C such that f (x∗ ) = x∗ . The proof of
this by Brouwer (1912) was one of the major events in the history of topology.
Since then the study of such results, and the methods used to prove them, has
flourished, undergoing radical transformations, becoming increasingly general and
sophisticated, and extending its influence to diverse areas of mathematics.
Around 1950, most notably through the work of Nash (1950, 1951) on noncooperative games, and the work of Arrow and Debreu (1954) on general equilibrium
theory, it emerged that in economists’ most basic and general models, equilibria
are fixed points. The most obvious consequence of this is that fixed point theorems provide proofs that these models are not vacuous. But fixed point theory also
informs our understanding of many other issues such as comparative statics, robustness under perturbations, stability of equilibria with respect to dynamic adjustment
processes, and the algorithmics and complexity of equilibrium computation. In particular, since the mid 1970’s the theory of games has been strongly influenced by
refinement concepts defined largely in terms of robustness with respect to certain
types of perturbations.
As the range and sophistication of economic modelling has increased, more advanced mathematical tools have become relevant. Unfortunately, the mathematical
literature on fixed points is largely inaccessible to economists, because it relies heavily on homology. This subject is part of the standard graduate school curriculum
for mathematicians, but for outsiders it is difficult to penetrate, due to its abstract
nature and the amount of material that must be absorbed at the beginning before
the structure, nature, and goals of the theory begin to come into view. Many researchers in economics learn advanced topics in mathematics as a side product of
their research, but unlike infinite dimensional analysis or continuous time stochastic
processes, algebraic topology will not gradually achieve popularity among economic
theorists through slow diffusion. Consequently economists have been, in effect,
shielded from some of the mathematics that is most relevant to their discipline.
This monograph presents an exposition of advanced material from the theory of
fixed points that is, in several ways, suitable for graduate students and researchers
in mathematical economics and related fields. In part the “fit” with the intended
2
1.1. THE FIRST FIXED POINT THEOREMS
3
audience is a matter of coverage. Economic models always involve domains that
are convex, or at least contractible, so there is little coverage here of topics that
only become interesting when the underlying space is more complicated. For the
settings of interest, the treatment is comprehensive and maximally general, with
issues related to correspondences always in the foreground. The project was originally motivated by a desire to understand the existence proofs in the literature
on refinements of Nash equilibrium as applications of preexisting mathematics, and
the continuing influence of this will be evident.
The mathematical prerequisites are within the common background of advanced
students and researchers in theoretical economics. Specifically, in addition to multivariate calculus and linear algebra, we assume that the reader is familiar with basic
aspects of point-set topology. What we need from topics that may be less familiar
to some (e.g., simplicial complexes, infinite dimensional linear spaces, the theory
of retracts) will be explained in a self-contained manner. There will be no use of
homological methods.
The avoidance of homology is a practical necessity, but it can also be seen as
a feature rather than a bug. In general, mathematical understanding is enhanced
when brute calculations are replaced by logical reasoning based on conceptually
meaningful definitions. To say that homology is a calculational machine is a bit
simplistic, but it does have that potential in certain contexts. Avoiding it commits
us to work with notions that have more direct and intuitive geometric content.
(Admittedly there is a slight loss of generality, because there are acyclic—that is,
homologically trivial—spaces that are not contractible, but this is unimportant
because such spaces are not found “in nature.”) Thus our treatment of fixed point
theory can be seen as a mature exposition that presents the theory in a natural and
logical manner.
In the remainder of this chapter we give a broad overview of the contents of
the book. Unlike many subjects in mathematics, it is possible to understand the
statements of many of the main results with much less preparation than is required
to understand the proofs. Needless to say, as usual, not bothering to study the
proofs has many dangers. In addition, the material in this book is, of course,
closely related to various topics in theoretical economics, and in many ways quite
useful preparation for further study and research.
1.1
The First Fixed Point Theorems
A fixed point of a function f : X → X is an element x∗ ∈ X such that
f (x∗ ) = x∗ . If X is a topological space, it is said to have the fixed point
property if every continuous function from X to itself has a fixed point. The first
and most famous result in our subject is Brouwer’s fixed point theorem:
Theorem 1.1.1 (Brouwer (1912)). If C ⊂ Rm is nonempty, compact, and convex,
then it has the fixed point property.
Chapter 3 presents various proofs of this result. Although some are fairly brief,
none of them can be described as truly elementary. In general, proofs of Brouwer’s
4
CHAPTER 1. INTRODUCTION AND SUMMARY
theorem are closely related to algorithmic procedures for finding approximate fixed
points. Chapter 3 discusses the best known general algorithm due to Scarf, a new
algorithm due to the author and Rabee Tourky, and homotopy methods, which are
the most popular in practice, but require differentiability. The last decade has seen
major breakthroughs in computer science concerning the computational complexity
of computing fixed points, with particular reference to (seemingly) simple games
and general equilibrium models. These developments are sketched briefly in Section
3.7.
In economics and game theory fixed point theorems are most commonly used to
prove that a model has at least one equilibrium, where an equilibrium is a vector of
“endogenous” variable for the model with the property that each individual agent’s
predicted behavior is rational, or “utility maximizing,” if that agent regards all
the other endogenous variables as fixed. In economics it is natural, and in game
theory unavoidable, to consider models in which an agent might have more than
one rational choice. Our first generalization of Brouwer’s theorem addresses this
concern.
If X and Y are sets, a correspondence F : X → Y is a function from X to the
nonempty subsets of Y . (On the rare occasions when they arise, we use the term set
valued mapping for a function from X to all the subsets of Y , including the empty
set.) We will tend to regard a function as a special type of correspondence, both
intuitively and in the technical sense that we will frequently blur the distinction
between a function f : X → Y and the associated correspondence x 7→ {f (x)}.
If Y is a topological space, F is compact valued if, for all x ∈ X, F (x) is
compact. Similarly, if Y is a subset of a vector space, then F is convex valued if
each F (x) is convex.
The extension of Brouwer’s theorem to correspondences requires a notion of
continuity for correspondences. If X and Y are topological spaces, a correspondence
F : X → Y is upper semicontinuous if it is compact valued and, for each x0 ∈ X
and each neighborhood V ⊂ Y of F (x0 ), there is a neighborhood U ⊂ X of x0 such
that F (x) ⊂ V for all x ∈ U. It turns out that if X and Y are metric spaces and Y
is compact, then F is upper semicontinuous if and only if its graph
Gr(F ) := { (x, y) ∈ X × Y : y ∈ F (x) }
is closed. (Proving this is a suitable exercise, if you are so inclined.) Thinking of
upper semicontinuity as a matter of the graph being closed is quite natural, and in
economics this condition is commonly taken as definition, as in Debreu (1959). In
Chapter 5 we will develop a topology on the space of nonempty compact subsets
of Y such that F is upper semicontinuous if and only if it is a continuous function
relative to this topology.
A fixed point of a correspondence F : X → X is a point x∗ ∈ X such that
∗
x ∈ F (x∗ ). Kakutani (1941) was motivated to prove the following theorem by the
desire to provide a simple approach to the von Neumann (1928) minimax theorem,
which is a fundamental result of game theory. This is the fixed point theorem that
is most commonly applied in economic analysis.
1.2. “FIXING” KAKUTANI’S THEOREM
5
Theorem 1.1.2 (Kakutani’s Fixed Point Theorem). If C ⊂ Rm is nonempty,
compact, and convex, and F : C → C is an upper semicontinuous convex valued
correspondence, then F has a fixed point.
1.2
“Fixing” Kakutani’s Theorem
Mathematicians strive to craft theorems that maximize the strength of the conclusions while minimizing the strength of the assumptions. One reason for this is
obvious: a stronger theorem is a more useful theorem. More important, however, is
the desire to attain a proper understanding of the principle the theorem expresses,
and to achieve an expression of this principle that is unencumbered by useless clutter. When a theorem that is “too weak” is proved using methods that “happen
to work” there is a strong suspicion that attempts to improve the theorem will
uncover important new concepts. In the case of Brouwer’s theorem the conclusion,
that the space has the fixed point property, is a purely topological assertion. The
assumption that the space is convex, and in Kakutani’s theorem the assumption
that the correspondence’s values are convex, are geometric conditions that seems
out of character and altogether too strong. Suitable generalizations were developed
after World War II.
A homotopy is a continuous function h : X × [0, 1] → Y where X and Y are
topological spaces. It is psychologically natural to think of the second variable in the
domain as representing time, and we let ht := h(·, t) : X → Y denote “the function
at time t,” so that h is a process that continuously deforms a function h0 into h1 .
Another intuitive picture is that h is a continuous path in the space C(X, Y ) of
continuous function from X to Y . As we will see in Chapter 5, this intuition can
be made completely precise: when X and Y are metric spaces and X is compact,
there is a topology on C(X, Y ) such that a continuous path h : [0, 1] → C(X, Y ) is
the same thing as a homotopy.
We say that two functions f, g : X → Y are homotopic if there is a homotopy
h with h0 = f and h1 = g. This is easily seen to be an equivalence relation:
symmetry and reflexivity are obvious, and to establish transitivity we observe that
if e is homotopic to f and f is homotopic to g, then there is a homotopy between
e and g that follows a homotopy between e and f at twice its original speed, then
follows a homotopy between f and g at double the pace. The equivalence classes
are called homotopy classes.
A space X is contractible if the identity function IdX is homotopic to a constant
function. That is, there is a homotopy c : X × [0, 1] → X such that c0 = IdX and
c1 (X) is a singleton; such a homotopy is called a contraction. Convex sets are
contractible. More generally, a subset X of a vector space is star-shaped if there
is x∗ ∈ X (the star) such that X contains the line segment
{ (1 − t)x + tx∗ : 0 ≤ t ≤ 1 }
between each x ∈ X and x∗ . If X is star-shaped, there is a contraction
(x, t) 7→ (1 − t)x + tx∗ .
6
CHAPTER 1. INTRODUCTION AND SUMMARY
It seems natural to guess that a nonempty compact contractible space has the
fixed point property. Whether this is the case was an open problem for several years,
but it turns out to be false. In Chapter 7 we will see an example due to Kinoshita
(1953) of a nonempty compact contractible subset of R3 that does not have the
fixed point property. Fixed point theory requires some additional ingredient.
If X is a topological space, a subset A ⊂ X is a retract if there is a continuous
function r : X → A with r(a) = a for all a ∈ A. Here we tend to think of X as a
“simple” space, and the hope is that although A might seem to be more complex,
or perhaps “crumpled up,” it nonetheless inherits enough of the simplicity of X. A
particularly important manifestation of this is that if r : X → A is a retraction and
X has the fixed point property, then so does A, because if f : A → A is continuous,
then so is f ◦ r : X → A ⊂ X, so f ◦ r has a fixed point, and this fixed point
necessarily lies in A and is consequently a fixed point of f . Also, a retract of a
contractible space is contractible because if c : X × [0, 1] → X is a contraction of
X and r : X → A ⊂ X is a retraction, then
(a, t) 7→ r(c(a, t))
is a contraction of A.
A set A ⊂ Rm is a Euclidean neighborhood retract (ENR) if there is an
open superset U ⊂ Rm of A and a retraction r : U → A. If X and Y are metric
spaces, an embedding of X in Y is a function e : X → Y that is a homeomorphism
between X and e(X). That is, e is a continuous injection1 whose inverse is also
continuous when e(X) has the subspace topology inherited from Y . An absolute
neighborhood retract (ANR) is a separable2 metric space X such that whenever
Y is a separable metric space and e : X → Y is an embedding, there is an open
superset U ⊂ Y of e(X) and a retraction r : U → e(X). This definition probably
seems completely unexpected, and it’s difficult to get any feeling for it right away.
In Chapter 7 we’ll see that ANR’s have a simple characterization, and that many
of the types of spaces that come up most naturally are ANR’s, so this condition
is quite a bit less demanding than one might guess at first sight. In particular, it
will turn out that every ENR is an ANR, so that being an ENR is an “intrinsic”
property insofar as it depends on the topology of the space and not on how the
space is embedded in a Euclidean space.
An absolute retract (AR) is a separable metric space X such that whenever
Y is a separable metric space and e : X → Y is an embedding, there is a retraction
r : Y → e(X). In Chapter 7 we will prove that an ANR is an AR if and only if it
is contractible.
Theorem 1.2.1. If C is a nonempty compact AR and F : C → C is an upper
semicontinuous contractible valued correspondence, then F has a fixed point.
An important point is that the values of F are not required to be ANR’s.
1
We will usually use the terms “injective” rather than “one-to-one,” “ surjective” rather than
“onto,” and “bijective” to indicate that a function is both injective and surjective. An injection is
an injective function, a surjection is a surjective function, and a bijection is a bijective function.
2
A metric space is separable if it has a countable dense subset.
7
1.3. ESSENTIAL SETS OF FIXED POINTS
For “practical purposes” this is the maximally general topological fixed point
theorem, but for mathematicians there is an additional refinement. There is a concept called acyclicity that is defined in terms of the concepts of algebraic topology.
A contractible set is necessarily acyclic, but there are acyclic spaces (including compact ones) that are not contractible. The famous Eilenberg-Montgomery fixed point
theorem is:
Theorem 1.2.2 (Eilenberg and Montgomery (1946)). If C is a nonempty compact
AR and F : C → C is an upper semicontinuous acyclic valued correspondence, then
F has a fixed point.
1.3
Essential Sets of Fixed Points
It might seem like we have already reached a satisfactory and fitting resolution
of “The Fixed Point Problem,” but actually (both in pure mathematics and in
economics) this is just the beginning. You see, fixed points come in different flavors.
1
b
b
0
0
s
t
1
Figure 1.1
The figure above shows a function f : [0, 1] → [0, 1] with two fixed points, s
and t. If we perturb the function slightly by adding a small positive constant, s
“disappears” in the sense that the perturbed function does not have a fixed point
anywhere near s, but a function close to f has a fixed point near t. More precisely,
if X is a topological space and f : X → X is continuous, a fixed point x∗ of f is
essential if, for any neighborhood U of x∗ , there is a neighborhood V of the graph
of f such that any continuous f ′ : X → X whose graph is contained in V has a
fixed point in U. If a fixed point is not essential, then we say that it is inessential.
These concepts were introduced by Fort (1950).
There need not be an essential fixed point. The function shown in Figure 1.2
8
CHAPTER 1. INTRODUCTION AND SUMMARY
has an interval of fixed points. If we shift the function down, there will be a fixed
point near the lower endpoint of this interval, and if we shift the function up there
will be a fixed point near the upper endpoint.
This example suggests that we might do better to work with sets of fixed points.
A set S of fixed points of a function f : X → X is essential if it is closed, it has a
neighborhood that contains no other fixed points, and for any neighborhood U of S,
there is a neighborhood V of the graph of f such that any continuous f ′ : X → X
whose graph is contained in V has a fixed point in U. The problem with this
concept is that “large” connected sets are not of much use. For example, if X is
compact and has the fixed point property, then the set of all fixed points of f is
essential. It seems that we should really be interested in sets of fixed points that are
either essential and connected3 or essential and minimal in the sense of not having
a proper subset that is also essential.
1
0
0
1
Figure 1.2
In Chapter 8 we will show that any essential set of fixed points contains a minimal essential set, and that minimal essential sets are connected. The theory of
refinements of Nash equilibrium (e.g., Selten (1975); Myerson (1978); Kreps and
Wilson (1982); Kohlberg and Mertens (1986); Mertens (1989, 1991); Govindan and
Wilson (2008)) has many concepts that amount to a weakening of the notion of
essential set, insofar as the set is required to be robust with respect to only certain types of perturbations of the function or correspondence. In particular, Jiang
(1963) pioneered the application of the concept to game theory, defining an essential!Nash equilibrium and an essential set of Nash equilibria in terms
of robustness with respect to perturbations of the best response correspondence
induced by perturbations of the payoffs. The mathematical foundations of such
3
We recall that a subset S of a topological space X is connected if there do not exist two
disjoint open sets U1 and U2 with S ∩ U1 6= ∅ 6= S ∩ U2 and S ⊂ U1 ∪ U2 .
9
1.4. INDEX AND DEGREE
concepts are treated in Section 8.3.
1.4
Index and Degree
There are different types of essential fixed points. Figure 1.3 shows a function
with three fixed points. At two of them the function starts above the diagonal and
goes below it as one goes from left to right, and at the third it is the other way
around. For any k it is easy to imagine a function with k fixed points of the first
type and k − 1 fixed points of the second type.
This phenomenon generalizes to higher dimensions. Let
D m = { x ∈ Rm : kxk ≤ 1 } and S m−1 = { x ∈ Rm : kxk = 1 }
be the m-dimensional unit disk and the (m − 1)-dimensional unit sphere, and suppose that f : D m → D m is a C ∞ function. In the best behaved case each fixed
point x∗ is in the interior D m \ S m−1 of the disk and regular, which means that
IdRm −Df (x∗ ) is nonsingular, where Df (x∗ ) : Rm → Rm is the derivative of f at x∗ .
We define the index of x∗ to be 1 if the determinant of IdRm − Df (x∗ ) is positive
and −1 if this determinant is negative. We will see that there is always one more
fixed point of index 1 than there are fixed points of index −1, which is to say that
the sum of the indices is 1.
What about fixed points on the boundary of the disk, or fixed points that aren’t
regular, or nontrivial connected sets of fixed points? What about correspondences?
What happens if the domain is a possibly infinite dimensional ANR? The most challenging and significant aspect of our work will be the development of an axiomatic
theory of the index that is general enough to encompass all these possibilities. The
work proceeds through several stages, and we describe them in some detail now.
1
b
b
b
0
0
1
Figure 1.3
10
1.4.1
CHAPTER 1. INTRODUCTION AND SUMMARY
Manifolds
First of all, it makes sense to expand our perspective a bit. An m-dimensional
manifold is a topological space that resembles Rm in a neighborhood of each of its
points. More precisely, for each p ∈ M there is an open U ⊂ Rm and an embedding
ϕ : U → M whose image is open and contains p. Such a ϕ is a parameterization
and its inverse is a coordinate chart. The most obvious examples are Rm itself
and S m . If, in addition, N is an n-dimensional manifold, then M × N is an (m + n)dimensional manifold. Thus the torus S 1 × S 1 is a manifold, and this is just the
most easily visualized member of a large class of examples. An open subset of an
m-dimensional manifold is an m-dimensional manifold. A 0-dimensional manifold is
just a set with the discrete topology. The empty set is a manifold of any dimension,
including negative dimensions. Of course these special cases are trivial, but they
come up in important contexts.
A collection {ϕi : Ui → M}i∈I of parameterizations is an atlas if its images
cover M. The composition ϕ−1
j ◦ ϕi (with the obvious domain of definition) is called
a transition function. If, for some 1 ≤ r ≤ ∞, all the transition functions are
C r functions, then the atlas is a C r atlas. An m-dimensional C r manifold is an
m-dimensional manifold together with a C r atlas. The basic concepts of differential
and integral calculus extend to this setting, leading to a vast range of mathematics.
In our formalities we will always assume that M is a subset of a Euclidean
space Rk called the ambient space, and that the parameterizations ϕi and the
coordinate charts ϕ−1
are C r functions. This is a bit unprincipled—for example,
i
physicists see only the universe, and their discourse is more disciplined if it does
not refer to some hypothetical ambient space—but this maneuver is justified by
embedding theorems due to Whitney that show that it does not entail any serious
loss of generality. The advantages for us are that this approach bypasses certain
technical pathologies while allowing for simplified definitions, and in many settings
the ambient space will prove quite handy. For example, a function f : M → N
(where N is now contained in some Rℓ ) is C r for our purposes if it is C r in the
standard sense: for any S ⊂ Rk a function h : S → Rℓ is C r , by definition, if there
is an open W ⊂ Rk containing S and a C r function H : W → Rℓ such that h = H|S .
Having an ambient space around makes it relatively easy to establish the basic
objects and facts of differential calculus. Suppose that ϕi : Ui → M is a C r
parameterization. If x ∈ Ui and ϕi (x) = p, the tangent space of M at p, which we
denote by Tp M, is the image of Dϕi (x). This is an m-dimensional linear subspace
of Rk . If f : M → N is C r , the derivative
Df (p) : Tp M → Tf (p) N
of f at p is the restriction to Tp M of the derivative DF (p) of any C r function
F : W → Rℓ defined on an open W ⊂ Rk containing M whose restriction to M is
f . (In Chapter 10 we will show that the choice of F doesn’t matter.) The chain
rule holds: if, in addition, P is a p-dimensional C r manifold and g : N → P is a C r
function, then g ◦ f is C r and
D(g ◦ f )(p) = Dg(f (p)) ◦ Df (p) : Tp M → Tg(f (p)) P.
1.4. INDEX AND DEGREE
11
The inverse and implicit function theorems have important generalizations. The
point p is a regular point of f if the image of Df (p) is all of Tf (p) N. We say that
f : M → N is a C r diffeomorphism if m = n, f is a bijection, and both f and f −1
are C r . The generalized inverse function theorem asserts that if m = n, f : M → N
is C r , and p is a regular point of f , then there is an open U ⊂ M containing p such
that f (U) is an open subset of N and f |U : U → f (U) is a C r diffeomorphism.
If 0 ≤ s ≤ m, a set S ⊂ Rk is an s-dimensional C r submanifold of M if it
is an s-dimensional C r submanifold that happens to be contained in M. We say
that q ∈ N is a regular value of f if every p ∈ f −1 (q) is a regular point. The
generalized implicit function theorem, which is known as the regular value theorem,
asserts that if q is a regular value of f , then f −1 (q) is an (m − n)-dimensional C r
submanifold of M.
1.4.2
The Degree
The degree is closely related to the fixed point index, but it has its own theory,
which has independent interest and significance. The approach we take here is to
work with the degree up to the point where its theory is more or less complete, then
translate what we have learned into the language of the fixed point index.
We now need to introduce the concept of orientation. Two ordered bases
v1 , . . . , vm and w1 , . . . , wm of an m-dimensional vector space have the same orientation if the determinant of the linear transformation taking each vi to wi is
positive. It is easy to see that this is an equivalence relation with two equivalence
classes. An oriented vector space is a finite dimensional vector space with a designated orientation whose elements are said to be positively oriented. If V and
W are m-dimensional oriented vector spaces, a nonsingular linear transformation
L : V → W is orientation preserving if it maps positively oriented ordered bases
of V to positively oriented ordered bases of W , and otherwise it is orientation
reversing. For an intuitive appreciation of this concept just look in a mirror: the
linear map taking each point in the actual world to its position as seen in the mirror
is orientation reversing, with right shoes turning into left shoes and such.
In our discussion of degree theory nothing is lost by working with C ∞ objects
rather than C r objects for general r, and smooth will be a synonym for C ∞ . An
orientation for a smooth manifold M is a “continuous” specification of an orientation
of each of the tangent spaces Tp M. We say that M is orientable if it has an
orientation; the most famous examples of unorientable manifolds are the Möbius
strip and the Klein bottle. (From a mathematical point of view 2-dimensional
projective space is perhaps more fundamental, but it is difficult to visualize.) An
oriented manifold is a manifold together with a designated orientation.
If M and N are oriented smooth manifolds of the same dimension, f : M → N
is a smooth map, and p is a regular point of f , we say that f is orientation
preserving at p if Df (p) : Tp M → Tf (p) N is orientation preserving, and otherwise
f is orientation reversing at p. If q is a regular value of f and f −1 (q) is finite,
then the degree of f over q, denoted by deg∞
q (f ), is the number of points in
−1
f (q) at which f is orientation preserving minus the number of points in f −1 (q)
at which f is orientation reversing.
12
CHAPTER 1. INTRODUCTION AND SUMMARY
We need to extend the degree to situations in which the target point q is not a
regular value of f , and to functions that are merely continuous. Instead of being
able to define the degree directly, as we did above, we will need to proceed indirectly,
showing that the generalized degree is determined by certain of its properties, which
we treat as axioms.
The first step is to extend the concept, giving it a “local” character. For a
compact C ⊂ M let ∂C = C ∩ (M \ C) be the topological boundary of C, and let
int C = C \ ∂C be its interior. A smooth function f : C → N with compact domain
C ⊂ M is said to be smoothly degree admissible over q ∈ N if f −1 (q) ∩ ∂C = ∅
and q is a regular value of f . As above, for such a pair (f, q) we define deg∞
q (f ) to
be the number of p ∈ f −1 (q) at which f is orientation preserving minus the number
∞
of p ∈ f −1 (q) at which f is orientation reversing. Note that deg∞
q (f ) = deg q (f |C ′ )
′
−1
whenever C is a compact subset of C and f (q) has an empty intersection with
the closure of C \ C ′ . Also, if C = C1 ∪ C2 where C1 and C2 are compact and
disjoint, then
∞
∞
deg∞
q (f ) = deg q (f |C1 ) + deg q (f |C2 ).
From the point of view of topology, what makes the degree important is its
invariance under homotopy. If C ⊂ M is compact, a smooth homotopy h : C ×
[0, 1] → N is smoothly degree admissible over q if h−1 (q) ∩ (∂C × [0, 1]) = ∅
and q is a regular value of h0 and h1 . In this circumstance
∞
deg∞
q (h0 ) = deg q (h1 ).
(∗)
Figure 1.4 illustrates the intuitive character of the proof.
b
b
−1
b
b
−1
+1
b
b
+1
b
+1
−1
−1
b
+1
b
b
b
b
t=0
t=1
Figure 1.4
1.4. INDEX AND DEGREE
13
The notion of an m-dimensional manifold with boundary is a generalization
of the manifold concept in which each point in the space has a neighborhood that is
homeomorphic to an open subset of the closed half space { x ∈ Rm : x1 ≥ 0 }. Aside
from the half space itself, the closed disk D m = { x ∈ Rm : kxk ≤ 1 } is perhaps
the most obvious example, but for us the most important example is M × [0, 1]
where M is an (m − 1)-dimensional manifold without boundary. Note that any mdimensional manifold without boundary is (automatically and trivially) a manifold
with boundary. All elements of our discussion of manifolds generalize to this setting.
In particular, the generalization of the regular value theorem states that if M is
an m-dimensional smooth manifold with boundary, N is an n-dimensional (boundaryless) manifold, f : M → N is smooth, and q ∈ N is a regular value of both f and
the restriction of f to the boundary of M, then f −1 (q) is an (m − n)-dimensional
manifold with boundary, its boundary is its intersection with the boundary of M,
and at each point in this intersection the tangent space of f −1 (q) is not contained in
the tangent space of the boundary of M. In particular, if the dimension of M is the
dimension of N plus one, then f −1 (q) is a 1-dimensional manifold with boundary.
If, in addition, f −1 (q) is compact, then it has finitely many connected components.
Suppose now that h : C × [0, 1] → N is smoothly degree admissible over q,
and that q is a regular value of h. The consequences of applying the regular value
theorem to the restriction of h to int C × [0, 1] are as shown in Figure 1.4: h−1 (q)
is a 1-dimensional manifold with boundary, its boundary is its intersection with
C × {0, 1}, and h−1 (q) is not tangent to C × {0, 1} at any point in this intersection.
In addition h−1 (q) is compact, so it has finitely many connected components, each
of which is compact. A connected compact 1-dimensional manifold with boundary
is either a circle or a line segment. (It will turn out that this obvious fact is
surprisingly difficult to prove!) Thus each component of h−1 (q) is either a circle or
a line segment connecting two points in its boundary. If a line segment connects
two points in C × {0}, say (p, 0) and (p′ , 0), then it turns out that h0 is orientation
preserving at p if and only if it is orientation reversing at p′ . Similarly, if a line
segment connects two points (p, 1) and (p′ , 1) in C × {1}, then h1 is orientation
preserving at p if and only if it is orientation reversing at p′ . On the other hand,
if a line segment connects a point (p0 , 0) in C × {0} to a point (p1 , 1) in C × {1},
then h0 is orientation preserving at p0 if and only if h1 is orientation preserving at
p1 . Equation (∗) is obtained by summing these facts over the various components
of h−1 (q).
This completes our discussion of the proof of (∗) except for one detail: if h :
C → [0, 1] → N is a smooth homotopy that is smoothly degree admissible over q,
q is not necessarily a regular value of h. Nevertheless, Sard’s theorem (which is
the subject of Chapter 11, and a crucial ingredient of our entire approach) implies
that h has regular values in any neighborhood of q, and it is also the case that
∞
∞
∞
′
deg∞
q ′ (h0 ) = deg q (h0 ) and deg q ′ (h1 ) = deg q (h1 ) when q is sufficiently close to q.
It turns out that the smooth degree is completely characterized by the properties
we have seen. That is, if D ∞ (M, N) is the set of pairs (f, q) in which f : C → N
is smoothly degree admissible over q, then (f, q) 7→ deg∞
q (f ) is the unique function
∞
from D (M, N) to Z satisfying:
∞
−1
(q) is a singleton {p}
(∆1) deg∞
q (f ) = 1 for all (f, q) ∈ D (M, N) such that f
14
CHAPTER 1. INTRODUCTION AND SUMMARY
and f is orientation preserving at p.
Pr
∞
∞
(∆2) deg∞
q (f ) =
i=1 deg q (f |Ci ) whenever (f, q) ∈ D (M, N), the domain of f is
C, and C1 , . . . , Cr are pairwise disjoint compact subsets of C such that
f −1 (q) ⊂ int C1 ∪ . . . ∪ int Cr .
∞
(∆3) deg∞
q (h0 ) = deg q (h1 ) whenever C ⊂ M is compact and the homotopy h :
C × [0, 1] → N is smoothly degree admissible over q.
We note two additional properties of the smooth degree. The first is that if, in
addition to M and N, M ′ and N ′ are m′ -dimensional smooth functions, (f, q) ∈
D ∞ (M, N), and (f ′ , q ′ ) ∈ D ∞ (M ′ , N ′ ), then
(f ×f ′ , (q, q ′)) ∈ D ∞ (M ×M ′ , N ×N ′ ) and
∞
∞
′
′
deg∞
(q,q ′ ) (f ×f ) = degq (f )·degq ′ (f ).
Since (f × f ′ )−1 (q, q ′) = f −1 (q) × f ′ −1 (q ′ ), this boils down to a consequence of
elementary facts about determinants: if (p, p′ ) ∈ (f × f ′ )−1 (q, q ′ ), then f × f ′ is
orientation preserving at (p, p′) if and only if f and f ′ are either both orientation
preserving or both orientation reversing at p and p′ respectively.
The second property is a strong form of continuity. A continuous function
f : C → N with compact domain C ⊂ M is degree admissible over q ∈ N if
f −1 (q) ∩ ∂C = ∅. If this is the case, then there is a neighborhood U ⊂ C × N of
the graph of f and a neighborhood V ⊂ N \ f (∂C) of q such that
′′
′
deg∞
q ′ (f ) = deg q ′′ (f )
whenever f ′ , f ′′ : C → N are smooth functions whose graphs are contained in U,
q ′ , q ′′ ∈ V , q ′ is a regular value of f ′ , and q ′′ is a regular value of q ′′ .
′
We can now define degq (f ) to be the common value of deg∞
q ′ (f ) for such pairs
(f ′ , q ′ ). Let D(M, N) be the set of pairs (f, q) in which f : C → N is a continuous
function with compact domain C ⊂ M that is degree admissible over q ∈ N. The
fully general form of degree theory asserts that (f, q) 7→ degq (f ) is the unique
function from D(M, N) to Z such that:
(D1) degq (f ) = 1 for all (f, q) ∈ D(M, N) such that f is smooth, f −1 (q) is a
singleton {p}, and f is orientation preserving at p.
P
(D2) degq (f ) = ri=1 degq (f |Ci ) whenever (f, q) ∈ D(M, N), the domain of f is C,
and C1 , . . . , Cr are pairwise disjoint compact subsets of U such that
f −1 (q) ⊂ C1 ∪ . . . ∪ Cr \ (∂C1 ∪ . . . ∪ ∂Cr ).
(D3) If (f, q) ∈ D(M, N) and C is the domain of f , then there is a neighborhood
U ⊂ C × N of the graph of f and a neighborhood V ⊂ N \ f (∂C) of q such
that
degq′ (f ′ ) = degq′′ (f ′′ )
whenever f ′ , f ′′ : C → N are continuous functions whose graphs are contained
in U and q ′ , q ′′ ∈ V .
15
1.4. INDEX AND DEGREE
1.4.3
The Fixed Point Index
Although the degree can be applied to continuous functions, and even to convex
valued correspondences, it is restricted to finite dimensional manifolds. For such
spaces the fixed point index is merely a reformulation of the degree. Its application
to general equilibrium theory was initiated by Dierker (1972), and it figures in the
analysis of the Lemke-Howson algorithm of Shapley (1974). There is also a third
variant of the underlying principle, for vector fields, that is developed in Chapter 15,
and which is related to the theory of dynamical systems. Hofbauer (1990) applied
the vector field index to dynamic issues in evolutionary stability, and Ritzberger
(1994) applies it systematically to normal form game theory.
However, it turns out that the fixed point index can be generalized much further,
due to the fact that, when we are discussing fixed points, the domain and the
range are the same. The general index is developed in three main stages. In
order to encompass these stages in a single system of terminology and notation we
take a rather abstract approach. Fix a metric space X. An index admissible
correspondence for X is an upper semicontinuous correspondence F : C → X,
where C ⊂ X is compact, that has no fixed points in ∂C. An index base for X is
a set I of index admissible correspondences such that:
(a) f ∈ I whenever C ⊂ X is compact and f : C → X is an index admissible
continuous function;
(b) F |D ∈ I whenever F : C → X is an element of I, D ⊂ C is compact, and
F |D is index admissible.
Definition 1.4.1. Let I be an index base for X. An index for I is a function
ΛX : I → Z satisfying:
(I1) (Normalization) If c : C → X is a constant function whose value is an element
of int C, then ΛX (c) = 1.
(I2) (Additivity) If F : C → X is an element of I, C1 , . . . , Cr are pairwise disjoint
compact subsets of C, and F P(F ) ⊂ int C1 ∪ . . . ∪ int Cr , then
X
ΛX (F ) =
ΛX (F |Ci ).
i
(I3) (Continuity) For each element F : C → X of I there is a neighborhood
U ⊂ C × X of the graph of F such that ΛX (F̂ ) = ΛX (F ) for every F̂ ∈ I
whose graph is contained in U.
For each m = 0, 1, 2, . . . an index base for Rm is given by letting I m be the set
of index admissible continuous functions f : C → Rm . Of course (I1)-(I3) parallel
(D1)-(D3), and it is not hard to show that there is a unique index ΛRm for I m given
by
ΛRm (f ) = deg0 (IdC − f ).
We now extend our framework to encompass multiple spaces. An index scope
S consists of a class of metric spaces SS and an index base IS (X) for each X ∈ SS
such that
16
CHAPTER 1. INTRODUCTION AND SUMMARY
(a) SS contains X × X ′ whenever X, X ′ ∈ SS ;
(b) F × F ′ ∈ IS (X × X ′ ) whenever X, X ′ ∈ SS , F ∈ IS (X), and F ′ ∈ IS (X ′ ).
These conditions are imposed in order to express a property of the index that is
inherited from the multiplicative property of the degree for cartesian products.
The index also has an additional property that has no analogue in degree theory.
Suppose that C ⊂ Rm and C̃ ⊂ Rm̃ are compact, g : C → C̃ and g̃ : C̃ → C are
continuous, and g̃ ◦ g and g ◦ g̃ are index admissible. Then
ΛRm (g̃ ◦ g) = ΛRm̃ (g ◦ g̃).
When g and g̃ are smooth and the fixed points in question are regular, this boils
down to a highly nontrivial fact of linear algebra (Proposition 13.3.2) that was
unknown prior to the development of this aspect of index theory.
This property turns out to be the key to moving the index up to a much higher
level of generality, but before we can explain this we need to extend the setup a
bit, allowing for the possibility that the images of g and g̃ are not contained in C̃
and C, but that there are compact sets D ⊂ C and D̃ ⊂ C̃ with g(D) ⊂ C̃ and
g̃(D̃) ⊂ C that contain the relevant sets of fixed points.
Definition 1.4.2. A commutativity configuration is a tuple
(X, C, D, g, X̂, Ĉ, D̂, ĝ)
where X and X̂ are metric spaces and:
(a) D ⊂ C ⊂ X, D̂ ⊂ Ĉ ⊂ X̂, and C, Ĉ, D, and D̂ are compact;
(b) g ∈ C(C, X̂) and ĝ ∈ C(Ĉ, X) with g(D) ⊂ int Ĉ and ĝ(D̂) ⊂ int C;
(c) ĝ ◦ g|D and g ◦ ĝ|D̂ are index admissible;
(d) g(F P(ĝ ◦ g|D )) = F P(g ◦ ĝ|D̂ ).
After all these preparations we can finally describe the heart of the matter.
Definition 1.4.3. An index for an index scope S is a specification of an index ΛX
for each X ∈ SS such that:
(I4) (Commutativity) If (X, C, D, g, X̂, Ĉ, D̂, ĝ) is a commutativity configuration
with X, X̂ ∈ SS , (D, ĝ ◦ g|D ) ∈ IS (X), and (D̂, g ◦ ĝ|D̂ ) ∈ IS (X̂), then
ΛX (ĝ ◦ g|D ) = ΛX̂ (g ◦ ĝ|D̂ ).
The index is said to be multiplicative if:
(M) (Multiplication) If X, X ′ ∈ SS , F ∈ IS (X), and F ′ ∈ IS (X ′ ), then
ΛX×X ′ (F × F ′ ) = ΛX (F ) · Λ′X (F ′ ).
1.5. TOPOLOGICAL CONSEQUENCES
17
Let SS Ctr be the class of ANRs, and for each X ∈ SS Ctr let IS Ctr (X) be the union over
compact C ⊂ X of the sets of index admissible upper semicontinuous contractible
valued correspondences F : C → X. The central goal of this book is:
Theorem 1.4.4. There is a unique index ΛCtr for S Ctr , which is multiplicative.
The passage from the indices ΛRm to ΛCtr has two stages. The first exploits
Commutativity to extend from Euclidean spaces and continuous functions to ANR’s
and continuous functions. There is a significant result that is the technical basis for
this. Let X be a metric space with metric d. If Y is a topological space and ε > 0,
a homotopy η : Y × [0, 1] → X is an ε-homotopy if
d η(y, s), η(y, t) < ε
for all y ∈ Y and all 0 ≤ s, t ≤ 1. We say that h0 and h1 are ε-homotopic. For
ε > 0, a topological space D ε-dominates C ⊂ X if there are continuous functions
ϕ : C → D and ψ : D → X such that ψ ◦ ϕ : C → X is ε-homotopic to IdC . In
Section 7.6 we show that:
Theorem 1.4.5. If X is a separable ANR, C ⊂ X is compact, and ε > 0, then
there is an open U ⊂ Rm , for some m, such that U is compact and ε-dominates C.
The second stage passes from continuous function to contractible valued correspondences. As in the passage from the smooth degree to the continuous degree,
the idea is to use approximation by functions to define the extension. The basis
of this is a result of Mas-Colell (1974) that was extended to ANR’s by the author
(McLennan (1991)) and is the topic of Chapter 9.
Theorem 1.4.6 (Approximation Theorem). Suppose that X is a separable ANR
and C and D are compact subsets of X with C ⊂ int D. Let F : D → Y be an upper
semicontinuous contractible valued correspondence. Then for any neighborhood U
of Gr(F |C ) there are:
(a) a continuous f : C → Z with Gr(f ) ⊂ U;
(b) a neighborhood U ′ of Gr(F ) such that, for any two continuous functions f0 , f1 :
D → Y with Gr(f0 ), Gr(f1 ) ⊂ U ′ , there is a homotopy h : C × [0, 1] → Y with
h0 = f0 |C , h1 = f1 |C , and Gr(ht ) ⊂ U for all 0 ≤ t ≤ 1.
1.5
Topological Consequences
The final section of the book develops applications of the index. Chapter 14
presents a number of classical concepts and results from topology that are usually
proved homologically. Let X be a compact ANR. The Euler characteristic of X
is the index of IdX . If F : X → X is an upper semicontinuous contractible valued
correspondence, the index of F is called the Lefschetz number of F . Of course
Additivity implies that F has a fixed point if its Lefschetz number is not zero.
The celebrated Lefschetz fixed point theorem is this assertion (usually restricted to
18
CHAPTER 1. INTRODUCTION AND SUMMARY
compact manifolds and continuous functions) together with a homological characterization of the Lefschetz number. If X is contractible, then the Lefschetz number
of any F : X → X is equal to the Euler characteristic of F , which is one. Thus
we arrive at our version of the Eilenberg-Montgomery theorem: if X is a compact
AR and F : X → X is a upper semicontinuous contractible valued correspondence,
then F has a fixed point.
Chapter 14 also develops many of the classical theorems concerning maps between spheres. The most basic of these is Hopf’s theorem: two continuous functions
f, f ′ : S m → S m are homotopic if and only if they have the same degree, so that the
degree is a “complete” homotopy invariant for maps between spheres of the same
dimension. There are many other theorems concerning maps between spheres of
the same dimension. Of these, one in particular has greater depth: if f : S m → S m
is continuous and f (−p) = −f (p) for all p ∈ S m , then the degree of f is odd. This
and its many corollaries constitute the Borsuk-Ulam theorem.
Using these results, we prove the frequently useful theorem known as invariance
of domain: if U ⊂ Rm is open and f : U → Rm is continuous and injective, then
f (U) is open and f is a homeomorphism onto its image.
If a connected set of fixed points has nonzero index, then it is essential, by virtue
of Continuity. The result in Section 14.5 shows that the converse holds for convex
valued correspondences with convex domains, so for the settings most commonly
considered in economics the notion of essentiality does not have independent significance. But it is important to understand that this result does not imply that
a component of the set of Nash equilibria of a normal form game of index zero is
inessential in the sense of Jiang (1963). In fact Hauk and Hurkens (2002) provide
a concrete example of an essential component of index zero.
1.6
Dynamical Systems
Dynamic stability is a problematic issue for economic theory. On the one hand,
particularly in complex settings, it seems that an equilibrium cannot a plausible
prediction unless it can be understood as the end state of a dynamic adjustment
process for which it is dynamically stable. In physics and chemistry there are explicit dynamical systems, and with respect to those stability is a well accepted principle. But in economics, explicit models of dynamic adjustment are systematically
inconsistent with the principle of rational expectations: if a model of continuous
adjustment of prices, or of mixed strategies, is understood and anticipated by the
agents in the model, their behavior will exploit the process, not conform to it.
Early work in general equilibrium theory (e.g., Arrow and Hurwicz (1958); Arrow et al. (1959)) found special cases, such as a single agent or two goods, in which
at least one equilibrium is necessarily stable with respect to natural price adjustment processes. But Scarf (1960) produced examples showing that one could not
hope for more general positive results, in the sense that “naive” dynamic adjustment
processes, such as Walrasian tatonnement, can easily fail to have stable dynamics,
even when there is a unique equilibrium and as few as three goods. A later stream
of research (Saari and Simon (1978); Saari (1985); Williams (1985); Jordan (1987))
1.6. DYNAMICAL SYSTEMS
19
showed that stability is informationally demanding, in the sense that an adjustment process that is guaranteed to return to equilibrium after a small perturbation
requires essentially all the information in the matrix of partial derivatives of the
aggregate excess demand function. On the whole there seems to be little hope of
finding a theoretical basis for an assertion that some equilibrium is stable, or that
a stable equilibrium exists.
In his Foundations of Economic Analysis Samuelson (1947) Samuelson describes
a correspondence principle, according to which the stability of an equilibrium
has implications for the qualitative properties of its comparative statics. In this
style of reasoning the stability of a given equilibrium is a hypothesis rather than a
conclusion, so the problematic state of the existence issue is less relevant. That is,
instead of claiming that some dynamical process should result in a stable equilibrium, one argues that equilibria with certain properties are not stable, so if what
we observe is an equilibrium, it cannot have these properties.
Proponents of such reasoning still need to wrestle with the fact that there is no
canonical dynamical process. (The conceptual foundations of economic dynamics,
and in particular the principle of rational expectations, were not well understood in
Samuelson’s time, and his discussion would be judged today to have various weaknesses.) Here there is the possibility of arguing that although any one dynamical
process might be ad hoc, the instability is common to all “reasonable” or “natural”
dynamics, for example those in which price adjustment is positively related to excess
demand, or that each agent’s mixed strategy adjusts in a direction that would improve her expected utility if other mixed strategies were not also adjusting. From a
strictly logical point of view, such reasoning might seem suspect, but it seems quite
likely that most economists find it intuitively and practically compelling.
In Chapter 15 we present a necessary condition for stability of a component
of the set of equilibria that was introduced into game theory by Demichelis and
Ritzberger (2003). (See also Demichelis and Germano (2000).) We now give an
informal description of this result, with the relevant background, and relate it to
Samuelson’s correspondence principle.
Let M be an m-dimensional C 2 manifold, where r ≥ 2. A vector field ζ on a
set S ⊂ M is a continuous (in the obvious sense) assignment of a tangent vector
ζp ∈ Tp M to each p ∈ S. Vector fields have many applications, but by far the most
important is that if ζ is defined on an open U ⊂ M and satisfies a mild technical
condition, then it determines an autonomous dynamical system: there is an open
W ⊂ U × R such that for each p ∈ U, { t ∈ R : (p, t) ∈ W } is an interval containing
0, and a unique function Φ : W → U such that Φ(p, 0) = p for all p and, for each
(p, t) ∈ W , the time derivative of Φ at (p, t) is ζΦ(p,t) . If W is the maximal domain
admitting such a function, then Φ is the flow of ζ. A point p where ζp = 0 is an
equilibrium of ζ.
A set A ⊂ M is invariant if Φ(p, t) ∈ A for all p ∈ A and t ≥ 0. The ω-limit
set of p ∈ M is
\
{ Φ(p, t) : t ≥ t0 }.
t0 ≥0
The domain of attraction of A is
D(A) = { p ∈ M : the ω-limit set of p is nonempty and contained in A }.
20
CHAPTER 1. INTRODUCTION AND SUMMARY
A set A ⊂ M is asymptotically stable if:
(a) A is compact;
(b) A is invariant;
(c) D(A) is a neighborhood of A;
(d) for every neighborhood Ũ of A there is a neighborhood U such that Φ(p, t) ∈ Ũ
for all p ∈ U and t ≥ 0.
There is a well known sufficient condition for asymptotic stability. A function
f : M → R is ζ-differentiable if the ζ-derivative
ζf (p) =
d
f (Φ(p, t))|t=0
dt
is defined for every p ∈ M. A continuous function L : M → [0, ∞) is a Lyapunov
function for A ⊂ M if:
(a) L−1 (0) = A;
(b) L is ζ-differentiable with ζL(p) < 0 for all p ∈ M \ A;
(c) for every neighborhood U of A there is an ε > 0 such that L−1 ([0, ε]) ⊂ U.
One of the oldest results in the theory of dynamical systems (Theorem 15.4.1) due
to Lyapunov, is that if there is a Lyapunov function for A, then A is asymptotically
stable.
A converse Lyapunov theorem is a result asserting that if A is asymptotically
stable, then there is a Lyapunov function for A. Roughly speaking, this is true, but
there is in addition the question of what sort of smoothness conditions one may
require of the Lyapunov function. The history of converse Lyapunov theorems is
rather involved, and the issue was not fully resolved until the 1960’s. We present
one such theorem (Theorem 15.5.1) that is sufficient for our purposes.
There is a well established definition of the index of an isolated equilibrium of
a vector field. We show that this extends to an axiomatically defined vector field
index. The theory of the vector field index is exactly analogous to the theories of
the degree and the fixed point index, and it can be characterized in terms of the
fixed point index. Specifically, a vector field ζ defined on a compact C ⊂ M is
index admissible if it does not have any equilibria in the boundary of C. It turns
out that if ζ is defined on a neighborhood of C, and satisfies the technical condition
guaranteeing the existence and uniqueness of the flow, then the vector field index
of ζ is the fixed point index of Φ(·, t)|C for small negative t. (The characterization
is in terms of negative time due to an unfortunate normalization axiom for the
vector field index that is now traditional.) One may define the vector field index
of a compact connected component of the set of equilibria to be the index of the
restriction of the vector field to a small compact neighborhood of the component.
The definition of asymptotic stability, and in particular condition (d), should
make us suspect that there is a connection with the Euler characteristic, because
1.6. DYNAMICAL SYSTEMS
21
for small positive t the flow Φ(·, t) will map neighborhoods of A into themselves.
The Lyapunov function given by the converse Lyapunov theorem is used in Section
15.6 to show that if A is dynamically stable and an ANR (otherwise the Euler characteristic is undefined) then the vector field index of A is (−1)m χ(A). In particular,
if A is a singleton, then A can only be stable when the vector field index of A is
(−1)m . This is the result of Demichelis and Ritzberger. The special case when
A = {p0 } is a singleton is a prominent result in the theory of dynamical systems
due to Krasnosel’ski and Zabreiko (1984).
We now describe the relationship between this result and qualitative properties
of an equilibrium’s comparative statics. Consider the following stylized example.
Let U be an open subset of Rm ; an element of U is thought of as a vector of
endogenous variables. Let P be an open subset of Rn ; an element of P is thought of
as a vector of exogenous parameters. Let z : U × P → Rm be a C 1 function, and let
∂x z(x, α) and ∂α z(x, α) denote the matrices of partial derivatives of the components
of z with respect to the components of x and α.
We think of z as a parameterized vector field on U. An equilibrium for a
parameter α ∈ P is an x ∈ U such that z(x, α) = 0. Suppose that x0 is an
equilibrium for α0 , and ∂x z(x0 , α0 ) is nonsingular. The implicit function theorem
gives a neighborhood V of α and a C 1 function σ : V → U with σ(α0 ) = x0 and
z(σ(α), α) = 0 for all α ∈ V . The method of comparative statics is to differentiate
this equation with respect to α at α0 , then rearrange, obtaining the equation
dσ
(α0 ) = −∂x z(x0 , α0 )−1 · ∂α (x0 , α0 )
dα
describing how the endogenous variables adjust, in equilibrium, to changes in the
vector of parameters. The Krasnosel’ski-Zabreiko theorem implies that if {x0 } is an
asymptotically stable set for the dynamical system determined by the vector field
z(·, α0 ), then the determinant of −∂x z(x0 , α0 )−1 is positive. This is a precise and
general statement of the correspondence principle.
Part I
Topological Methods
22
Chapter 2
Planes, Polyhedra, and Polytopes
This chapter studies basic geometric objects defined by linear equations and
inequalities. This serves two purposes, the first of which is simply to introduce
basic vocabulary. Beginning with affine subspaces and half spaces, we will proceed to (closed) cones, polyhedra, and polytopes, which are polyhedra that are
bounded. A rich class of well behaved spaces is obtained by combining polyhedra
to form polyhedral complexes. Although this is foundational, there are nonetheless
several interesting and very useful results and techniques, notably the separating
hyperplane theorem, Farkas’ lemma, and barycentric subdivision.
2.1
Affine Subspaces
Throughout the rest of this chapter we work with a fixed d-dimensional real inner
product space V . (Of course we are really talking about Rd , but a more abstract
setting emphasizes the geometric nature of the constructions and arguments.) We
assume familiarity with the concepts and results of basic linear algebra.
An affine combination of y0 , . . . , yr ∈ V is a point of the form
α0 y 0 + · · · + αr y r
where α = (α0 , . . . , αr ) is a vector of real numbers whose components sum to 1. We
say that y0 , . . . , yr are affinely dependent if it is possible to represent a point as
an affine combination of these points in two different ways: that is, if
X
X
X
X
αj = 1 =
αj′ and
αj y j =
αj′ yj ,
j
j
j
j
then α = α′ . If y0 , . . . , yr are not affinely dependent, then they are affinely independent.
Lemma 2.1.1. For any y0 , . . . , yr ∈ V the following are equivalent:
(a) y0 , . . . , yr are affinely independent;
(b) y1 − y0 , . . . , yr − y0 are linearly independent;
23
24
CHAPTER 2. PLANES, POLYHEDRA, AND POLYTOPES
(c) therePdo not exist β0 , . . . , βr ∈ R, not all of which are zero, with
and j βj yj = 0.
P
j
βj = 0
Proof. Suppose that y0 , . . . , P
yr are affinely dependent,
and let αj and αj′ be as above.
P
′
If P
we set βj = αj −
Pαj , then j βj = 0 and j βj yj = 0, so (c) implies (a). In turn,
if j βj = 0 and j βj yj = 0, then
β1 (y1 − y0 ) + · · · + βr (yr − y0 ) = −(β1 + · · · + βr )y0 + β1 y1 + · · · + βr yr = 0,
so y1 − y0 , . . . , yr − y0 are linearly dependent. Thus (b) implies (c). If β1 (y1 − y0 ) +
· · · + βr (yr − y0 ) = 0, then for any α0 , . . . , αr with α0 + · · · + αr = 1 we can set
β0 = −(β1 + · · · + βr ) and αj′ = αj + βj for j = 0, . . . , r, thereby showing that
y0 , . . . , yr are affinely dependent. Thus (a) implies (b).
The affine hull aff(S) of a set S ⊂ V is the set of all affine combinations of
elements of S. The affine hull of S contains S as a subset, and we say that S is
an affine subspace if the two sets are equal. That is, S is an affine subspace if it
contains all affine combinations of its elements. Note that the intersection of two
affine subspaces is an affine subspace. If A ⊂ V is an affine subspace and a0 ∈ A,
then { a − a0 : a ∈ A } is a linear subspace, and the dimension dim A of A is,
by definition, the dimension of this linear subspace. The codimension of A is
d − dim A. A hyperplane is an affine subspace of codimension one.
A (closed) half-space is a set of the form
H = { v ∈ V : hv, ni ≤ β }
where n is a nonzero element of V , called the normal vector of H, and β ∈ R.
Of course H determines n and β only up to multiplication by a positive scalar. We
say that
I = { v ∈ V : hv, ni = β }
is the bounding hyperplane of H. Any hyperplane is the intersection of the two
half-spaces that it bounds.
2.2
Convex Sets and Cones
A convex combination of y0 , . . . , yr ∈ V is a point of the form α0 y0 +· · ·+αr yr
where α = (α0 , . . . , αr ) is a vector of nonnegative numbers whose components sum
to 1. A set C ⊂ V is convex if it contains all convex combinations of its elements,
so that (1 − t)x0 + tx1 ∈ C for all x0 , x1 ∈ C and 0 ≤ t ≤ 1. For any set S ⊂ V the
convex hull conv(S) of S is the smallest convex containing S. Equivalently, it is
the set of all convex combinations of elements of S.
The following fact is a basic tool of geometric analysis.
Theorem 2.2.1 (Separating Hyperplane Theorem). If C is a closed convex subset
of V and z ∈ V \ C, then there is a half space H with C ⊂ H and z ∈
/ H.
2.2. CONVEX SETS AND CONES
25
Proof. The case C = ∅ is trivial. Assuming C 6= ∅, the intersection of C with a
closed ball centered at z is compact, and it is nonempty if the ball is large enough,
in which case it must contain a point x0 that minimizes the distance to z over the
points in this intersection. By construction this point is as close to z as any other
point in C. Let n = z − x0 and β = h(x0 + z)/2, ni. Checking that hn, zi > β is a
simple calculation.
We claim that hx, ni ≤ hx0 , ni for all x ∈ C, which is enough to imply the
desired result because hx0 , ni = β − 21 hn, ni. Aiming at a contradiction, suppose
that x ∈ C and hx, ni > hx0 , ni, so that hx − x0 , z − x0 i > 0. For t ∈ R we have
k(1 − t)x0 + tx − zk2 = kx0 − zk2 + 2thx0 − z, x − x0 i + t2 kx − x0 k2 ,
and for small positive t this is less than kx0 −zk2 , contradicting the choice of x0 .
A convex cone is convex set C that is nonempty and closed under multiplication
by nonnegative scalars, so that αx ∈ C for all x ∈ C and α ≥ 0. Such a cone is
closed under addition: if x, y ∈ C, then x + y = 2( 21 x + 12 y) is a positive scalar
multiple of a convex combination of x and y. Conversely, if a set is closed under
addition and multiplication by positive scalars, then it is a cone.
The dual of a convex set C is
C ∗ = { n ∈ V : hx, ni ≥ 0 for all x ∈ C }.
Clearly C ∗ is a convex cone, and it is closed, regardless of whether C is closed,
because C ∗ is the intersection of the closed half spaces { n ∈ V : hx, ni ≥ 0 }.
An intersection of closed half spaces is a closed convex cone. Farkas’ lemma is the
converse of this: a closed convex cone is an intersection of closed half spaces. From a
technical point of view, the theory of systems of linear inequalities is dominated by
this result because a large fraction of the results about systems of linear inequalities
can easily be reduced to applications of it.
Theorem 2.2.2 (Farkas’ Lemma). If C is a closed convex cone, then for any b ∈
V \ C there is n ∈ C ∗ such that hn, bi < 0.
Proof. The separating hyperplane theorem gives n ∈ V and β ∈ R such that
hn, bi < β and hn, xi > β for all x ∈ C. Since 0 ∈ C, β < 0. There cannot be x ∈ C
with hn, xi < 0 because we would have hn, αxi < β for sufficiently large α > 0, so
n ∈ C∗.
The recession cone of a convex set C is
RC = { y ∈ V : x + αy ∈ C for all x ∈ C and α ≥ 0 }.
Clearly RC is, in fact, a convex cone.
Lemma 2.2.3. Suppose C is nonempty, closed, and convex. Then RC is the set
of y ∈ V such that hy, ni ≤ 0 whenever H = { v ∈ V : hv, ni ≤ β } is a half space
containing C, so RC is closed because it is an intersection of closed half spaces. In
addition, C is bounded if and only if RC = {0}.
26
CHAPTER 2. PLANES, POLYHEDRA, AND POLYTOPES
Proof. Since C 6= ∅, if y ∈ RC , then hy, ni ≤ 0 whenever H = { v ∈ V : hv, ni ≤ β }
is a half space containing C. Suppose that y satisfies the latter condition and x ∈ C.
Then for all α ≥ 0, x + αy is contained in every half space containing C, and the
separating hyperplane theorem implies that the intersection of all such half spaces
is C itself. Thus y is in RC .
If RC has a nonzero element, then of course C is unbounded. Suppose that C
is unbounded. Fix a point x ∈ C, and let y1 , y2, . . . be a divergent sequence in C.
y −x
Passing to a subsequence if need be, we can assume that kyjj −xk converges to a unit
vector w. To show that w ∈ RC it suffices to observe that if H = { v : hv, ni ≤ β }
is a half space containing C, then hw, ni ≤ 0 because
yj − x
β − hx, ni
,n ≤
→ 0.
kyj − xk
kyj − xk
The lineality space of a convex set C is
LC = RC ∩ −RC = { y ∈ V : x + αy ∈ C for all x ∈ C and α ∈ R }.
The lineality space is closed under addition and scalar multiplication, so it is a
linear subspace of V , and in fact it is the largest linear subspace of V contained in
RC . Let L⊥
C be the orthogonal complement of LC . Clearly C + LC = C, so
C = (C ∩ L⊥
C ) + LC .
A convex cone is said to be pointed if its lineality space is {0}.
Lemma 2.2.4. If C =
6 V is a closed convex cone, then there is n ∈ C ∗ with hn, xi > 0
for all x ∈ C \ LC .
Proof. For n ∈ C ∗ let Zn = { x ∈ C : hx, ni = 0 }. Let n be a point in C ∗ that
minimizes the dimension of the span of Zn . Aiming at a contradiction, suppose
that 0 6= x ∈ Zn \ LC . Then −x ∈
/ C because x ∈
/ LC , and Farkas Lemma gives
an n′ ∈ C ∗ with hx, n′ i < 0. Then Zn+n′ ⊂ Zn ∩ Zn′ (this inclusion holds for all
n, n′ ∈ C ∗ ) and the span of Zn+n′ does not contain x, so it is a proper subspace of
the span of Zn .
2.3
Polyhedra
A polyhedron in V is an intersection of finitely many closed half spaces. We
adopt the convention that V itself is a polyhedron by virtue of being “the intersection of zero half-spaces.” Any hyperplane is the intersection of the two half-spaces
it bounds, and any affine subspace is an intersection of hyperplanes, so any affine
subspace is a polyhedron. The dimension of a polyhedron is the dimension of its
affine hull. Fix a polyhedron P .
A face of P is either the empty set, P itself, or the intersection of P with the
bounding hyperplane of some half-space that contains P . Evidently any face of P
27
2.3. POLYHEDRA
is itself a polyhedron. If F and F ′ are faces of P with F ′ ⊂ F , then F ′ is a face
of F , because if F ′ = P ∩ I ′ where I ′ is the bounding hyperplane of a half space
containing P , then that half space contains F and F ′ = F ∩ I ′ . A face is proper if
it is not P itself. A facet of P is a proper face that is not a proper subset of any
other proper face. An edge of P is a one dimensional face, and a vertex of P is a
zero dimensional face. Properly speaking, a vertex is a singleton, but we will often
blur the distinction between such a singleton and its unique element, so when we
refer to the vertices of P , usually we will mean the points themselves.
We say that x ∈ P is an initial point of P if there does not exist x′ ∈ P and
a nonzero y ∈ RP such that x = x′ + y. If the lineality subspace of P has positive
dimension, so that RP is not pointed, then there are no initial points.
Proposition 2.3.1. The set of initial points of P is the union of the bounded faces
of P .
Proof. Let F be a face of P , so that F = P ∩ I where I is the bounding hyperplane
of a half plane H containing P . Let x be a point in F .
We first show that if x is noninitial, then F is unbounded. Let Let x = x′ + y
for some x′ ∈ P and nonzero y ∈ RP . Since x − y and x + y are both in H, they
must both be in I, so F contains the ray { x + αy : α ≥ 0, and this ray is contained
in P because y ∈ RP , so F is unbounded.
We now know that the union of the bounded faces is contained in the set of
initial points, and we must show that if x is not contained in a bounded face, it
is noninitial. We may assume that F is the smallest face containing x. Since F
is unbounded there is a nonzero y ∈ RF . The ray { x − αy : α ≥ 0 } leaves P at
some α ≥ 0. (Otherwise the lineality of RP has positive dimension and there are
no initial points.) If α > 0, then x is noninitial, and α = 0 is impossible because it
would imply that x belonged to a proper face of F .
Proposition 2.3.2. If RP is pointed, then every point in P is the sum of an initial
point and an element of RP .
Proof. Lemma 2.2.4 gives an n ∈ V such that hy, ni > 0 for all nonzero y ∈ RP .
Fix x ∈ P . Clearly K = (x − RP ) ∩ P is convex, and it is bounded because its
recession cone is contained in −Rp ∩ RP = {0}. Lemma 2.2.3 implies that K is
closed, hence compact. Let x′ be a point in K that minimizes hx′ , ni. Then x is a
sum of x′ and a point in RP , and if x′ was not initial, so that x′ = x′′ + y where
x′′ ∈ P and 0 6= y ∈ RP , then hx′′ , ni < hx′ , ni, which is impossible.
Any polyhedron has a standard representation, which is a representation of
the form
k
\
P =G∩
Hi
i=1
where G is the affine hull of P and H1 , . . . , Hk are half-spaces. T
This representation
of P is minimal if it is irredundant, so that for each j, G ∩ i6=j Hi is a proper
superset. Starting with any standard representation of P , we can reduce it to a
minimal representation by repeatedly eliminating redundant half spaces. We now
fix a minimal representation, with Hi = { v ∈ V : hv, ni i ≤ αi } and Ii the bounding
hyperplane of Hi .
28
CHAPTER 2. PLANES, POLYHEDRA, AND POLYTOPES
Lemma 2.3.3. P has a nonempty interior in the relative topology of G.
Proof. For each i we cannot have P ⊂ Ii because that would imply that G ⊂ Ii ,
making Hi redundant. Therefore P must contain some xi in the interior of each
Hi . If x0 is a convex combination of x1 , . . . , xk with positive weights, then x0 is
contained in the interior of each Hi .
T
Proposition 2.3.4. For J ⊂ {1, . . . , k} let FJ = P ∩ j∈J Ij . Then FJ is a face
of P , and every nonempty face of P has this form.
Proof. If we choose numbers βj > 0 for all j ∈ J, then
X
X
x,
βj nj ≤
βj αj
j∈J
j∈J
for all x ∈ P , with equality if and only if x ∈ FJ . We have displayed FJ as a face.
Now let F = P ∩ H where H = { v ∈ V : hv, ni ≤ α } is a half-space containing
P , and let J = { j : F ⊂ Ij }. Of course F ⊂ FJ . Aiming at a contradiction,
suppose there is a point x ∈ FJ \ F . Then hx, ni i ≤ αi for all i ∈
/ J and hx, nj i = αj
for all j ∈ J. For each i ∈
/ J there is a yi ∈ F with hyi , ni i < αi ; let y be a strict
convex combination of these. Then hy, ni i < αi for all i ∈
/ J and hy, nj i ≤ αj for
all j ∈ J. Since x ∈
/ H and y ∈ H, the ray emanating from x and passing through
y leaves H at y, and consequently it must leave P at y, but continuing along this
ray from y does not immediately violate any of the inequalities defining P , so this
is a contradiction.
This result has many worthwhile corollaries.
Corollary 2.3.5. P has finitely many faces, and the intersection of any two faces
is a face.
Corollary 2.3.6. If F is a face of P and F ′ is a face of F , then F ′ is a face of P .
T
Proof. If G0 is the affine hull of F , then F =TG0 ∩ i Hi is a standard representation
T
′
of F . The proposition implies that
F
=
P
∩
I
for
some
J,
that
F
=
F
∩
i
i∈J
i∈J ′ Ii
T
for some J ′ , and that F ′ = P ∩ i∈J∪J ′ Ii is a face of P .
Corollary 2.3.7. The facets of P are F{1} , . . . , F{k} . The dimension of each F{i}
is one less than the dimension of P , The facets are the only faces of P with this
dimension.
Proof. Minimality implies that each F{i} is a proper face, and the result above
implies that F{i} cannot be a proper subset of another proper face. Thus each F{i}
is a facet.
For each i minimality implies that for each j 6= i there is some xj ∈ F{i} \ F{j} .
Let x be a convex combination of these with positive weights, then F{i} contains a
neighborhood of x in Ii , so the dimension of F{i} is the dimension of G ∩ Ii , which
is one less than the dimension of P .
A face F that is not a facet is a proper face of some facet, so its dimension is
not greater than two less than the dimension of P .
29
2.4. POLYTOPES
Now suppose that P is bounded. Any point in P that is not a vertex can be
written as a convex combination of points in proper faces of P . Induction on the
dimension of P proves that:
Proposition 2.3.8. If P is bounded, then it is the convex hull of its set of vertices.
An extreme point of a convex set is a point that is not a convex combination
of other points in the set. This result immediately implies that only vertices of P
can be extreme. In fact any vertex v is extreme: if {v} = P ∩ I where I is the
bounding hyperplane of a half space H containing P , then v cannot be a convex
combination of elements of P \ I.
2.4
Polytopes
A polytope in V is the convex hull of a finite set of points. Polytopes were
already studied in antiquity, but the subject continues to be an active area of
research; Ziegler (1995) is a very accessible introduction. We have just seen that
a bounded polyhedron is a polytope. The most important fact about polytopes is
the converse:
Theorem 2.4.1. A polytope is a polyhedron.
Proof. Fix P = conv{q1 , . . . , qℓ }. The property of being a polyhedron is invariant
under translations: for any x ∈ V , P is a polyhedron if and only if x + P is also a
polyhedron. It is also invariant under passage to subspaces: P is a polyhedron in V
if and only if it is a polyhedron in the span of P , and in any intermediate subspace.
The two invariances imply that we may reduce to a situation where the dimension
of P is the same as the dimension of V , and from there we may translate to make
the origin of V an interior point of P . Assume this is the case.
Let
P ∗ = { v ∈ V : hv, pi ≤ 1 for all p ∈ P }
and
P ∗∗ = { u ∈ V : hu, vi ≤ 1 for all v ∈ P ∗ }.
Since P is bounded and has the origin as T
an interior point, P ∗ is bounded with the
origin in its interior. The formula P ∗ = j { v ∈ V : hv, qj i ≤ 1 } displays P ∗ as a
polyhedron, hence a polytope. This argument with P ∗ in place of P implies that
P ∗∗ is a bounded polyhedron, so it suffices to show that P ∗∗ = P . The definitions
immediately imply that P ⊂ P ∗∗ .
Suppose that z ∈
/ P . The separating hyperplane theorem gives w ∈ V and
β ∈ R such that hw, zi < β and hw, pi > β for all p ∈ P . Since the origin is in P ,
β < 0. Therefore −w/β ∈ P ∗ , and consequently z ∈
/ P ∗∗ .
Wrapping things up, there is the following elegant decomposition result:
Proposition 2.4.2. Any polyhedron P is the sum of a linear subspace, a pointed
cone, and a polytope.
30
CHAPTER 2. PLANES, POLYHEDRA, AND POLYTOPES
Proof. Let L be its lineality, and let K be a linear subspace of V that is complementary to L in the sense that K ∩ L = {0} and K + L = V . Let Q = P ∩ K. Then
P = Q + L, and the lineality of Q is {0}, so RQ is pointed. Let S be the convex
hull of the set of initial points of Q. Above we saw that this is the convex hull of
the set of vertices of Q, so S is a polytope. Now Proposition 2.3.2 gives
P = L + RQ + S.
2.5
Polyhedral Complexes
A wide variety of spaces can be created by taking the union of a finite collection
of polyhedra.
Definition 2.5.1. A polyhedral complex is a finite set P = {P1 , . . . , Pk } of
polyhedra in V such that:
(a) F ∈ P whenever P ∈ P and F is a nonempty face of P ;
(b) for any 1 ≤ i, j ≤ k, Pi ∩ Pj is a common (possibly empty) face of Pi and Pj .
The underlying space of the complex is
|P| :=
[
P,
P ∈P
and we say that P is a polyhedral subdivision of |P|. The dimension of P is the
maximum dimension of any of its elements.
To illustrate this concept we mention a structure that was first studied by
Descartes, and that has accumulated a huge literature over the centuries . Let
x1 , . . . , xn be distinct points in V . The Voronoi diagram determined by these
points is
P = { PJ : ∅ =
6 J ⊂ {1, . . . , n} } ∪ {∅}
where
PJ = { y ∈ V : ky − xj k ≤ ky − xi k for all j ∈ J and i = 1, . . . , n }
is the set of points such that the xj for j ∈ J are as close to y as any of the points
x1 , . . . , xn . From Euclidean geometry we know that the condition ky−xj k ≤ ky−xi k
determines a half space in V (a quick calculation shows that ky −xj k2 ≤ ky −xi k2 if
and only if hy, xj −xi i ≥ 21 (kxj k2 −kxi k)) so each PJ is a polyhedron, and conditions
(a) and (b) are easy consequences of Proposition 2.3.4.
Fix a polyhedral complex P. A subcomplex of P is a subset Q ⊂ P that
contains all the faces of its elements, so that Q is also a polyhedral complex. If this
is the case, then |Q| is a closed (because it is a finite union of closed subsets) subset
of |P|. We say that P is a polytopal complex if each Pj is a polytope, in which
case P is said to be a polytopal subdivision of |P|. Note that |P| is necessarily
31
2.5. POLYHEDRAL COMPLEXES
compact because it is a finite union of compact sets. A k-dimensional simplex is
the convex hull of an affinely independent collection of points x0 , . . . , xk . We say
that P is a simplicial complex, and that P is a simplicial subdivision of |P|,
or a triangulation, if each Pj is a simplex.
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
We now describe a general method of subdividing a polytopal complex P into a
simplicial complex Q. For each P ∈ P choose wP in the relative interior of P . Let
Q be the collection of sets of the form
σQ = conv({ wP : P ∈ Q })
where Q is a subset of P that is completely ordered by inclusion. We claim that Q
is a simplicial complex, and that |Q| = |P|.
Suppose that Q = {P0 , . . . , Pk } where Pi−1 is a proper subset of Pi for 1 ≤ i ≤ k.
For each i, wP0 , . . . , wPi−1 are contained in Pi−1 , and wPi is not contained in the
affine hull of Pi−1 , so wPi − wP0 is not spanned by wP1 − wP0 , . . . , wPi−1 − wP0 . By
induction, wP1 − wP0 , . . . , wPk − wP0 are linearly independent. Now Lemma 2.1.1
implies that wP0 , . . . , wPk are affinely independent, so σQ is a simplex.
′
In addition to Q, suppose that Q′ = {P0′ , . . . , Pk′ ′ } where Pj−1
is a proper subset
′
′
of Pj for 1 ≤ j ≤ k . Clearly σQ∩Q′ ⊂ σQ ∩ σQ′ , and we claim that it is also the
case that the σQ ∩ σQ′ ⊂ σQ∩Q′ . Consider an arbitrary x ∈ σQ ∩ σQ′ . It suffices to
show the desired inclusion with Q and Q′ replaced by the smallest sets Q̃ ⊂ Q and
Q̃′ ⊂ Q′ such that x ∈ σQ̃ ∩σQ̃′ , so we may assume that x is in the interior of Pk and
in the interior of Pk′ ′ , and it follows that Pk = Pk′ ′ . In addition, the ray emanating
from wPk and passing through x leaves Pk at a point y ∈ σ{P0 ,...,Pk−1 } ∩ σ{P0′ ,...,Pk′ ′ −1 } ,
and the claim follows by induction on max{k, k ′ }. We have shown that Q is a
simplicial complex.
Evidently |Q| ⊂ |P|. Choosing x ∈ |P| arbitrarily, let P be the smallest element
of P that contains x. If x = wP , then x ∈ σ{P } , and if P is 0-dimensional then this
is the only possibility. Otherwise the ray emanating from wP and passing through
x intersects the boundary of P at a point y, and if y ∈ σQ , then x ∈ σQ∪{P } . By
induction on the dimension of P we see that x is contained in some element of Q,
so |Q| = |P|.
32
CHAPTER 2. PLANES, POLYHEDRA, AND POLYTOPES
This construction shows that the underlying space of a polytopal complex is also
the underlying space of a simplicial complex. In addition, repeating this process
can give a triangulation with small simplices. The diameter of a polytope is the
maximum distance between any two of its points. The mesh of a polytopal complex
is the maximum of the diameters of its polytopes.
Consider an ℓ-dimensional simplex P whose vertices are v0 , . . . , vℓ . The barycenter of P is
1
(v0 + · · · + vℓ ).
β(P ) :=
ℓ+1
In the construction above, suppose that P is a simplicial complex, and that we
chose wP = βP for all P . We would like to bound the diameter of the simplices in
the subdivision of |P|, which amounts to giving a bound on the maximum distance
between the barycenters of any two nested faces. After reindexing, these can be
taken to be the faces spanned by v0 , . . . , vk and v0 , . . . , vℓ where 0 ≤ k < ℓ ≤ m and
m is the dimension of P. The following rather crude inequality is sufficient for our
purposes.
1
1
(v0 + · · · + vk ) −
(v0 + · · · + vℓ )
k+1
ℓ+1
X X
1
vi − vj =
(k + 1)(ℓ + 1) 0≤i≤k 0≤j≤ℓ
X
X
1
≤
kvi − vj k
(k + 1)(ℓ + 1)
0≤i≤k 0≤j≤ℓ,j6=i
1
m
≤
(k + 1)ℓD ≤
D.
(k + 1)(ℓ + 1)
m+1
It follows from this that the mesh of the subdivision of |P| is not greater than
m/(m + 1) times the mesh of P. Since we can subdivide repeatedly:
Proposition 2.5.2. The underlying space of a polytopal complex has triangulations
of arbitrarily small mesh.
Simplicial complexes can be understood in purely combinatoric terms. An abstract simplicial complex is a pair (V, Σ) where V is a finite set of vertices and
Σ is a collection of subsets of V with the property that τ ∈ Σ whenever σ ∈ Σ and
τ ⊂ σ. The geometric interpretation is as follows. Let { ev : v ∈ V } be the standard unit basis vectors of RV : the v-component of ev is 1 and all other coordinates
are 0. (Probably most authors would work with R|V | , but our approach is simpler
and formally correct insofar as Y X is the set of functions from X to Y .) For each
nonempty σ ∈ Σ let Pσ be the convex hull of { ev : v ∈ σ }, and let P∅ = ∅. The
simplicial complex
P(V,Σ) = { Pσ : σ ∈ Σ }
is called the canonical realization of (V, Σ).
Let P be a simplicial complex, and let V be the set of vertices of P. For each
P ∈ P let σP = P ∩ V be the set of vertices of P , and let Σ = { σP : P ∈ P }.
It is easy to see that extending the map v 7→ e affinely on each simplex induces
2.6. GRAPHS
33
a homeomorphism between |P| and |P(V,Σ) |. Thus the homeomorphism type of a
simplicial complex is entirely determined by its combinatorics, i.e., the “is a face
of” relation between the various simplices. Geometric simplicial complexes and
abstract simplicial complexes encompass the same class of homeomorphism types
of topological spaces.
Simplicial complexes are very important in topology. On the one hand a wide
variety of important spaces have simplicial subdivisions, and certain limiting processes can be expressed using repeated barycentric subdivision. On the other hand,
the purely combinatoric nature of an abstract simplicial complex allows combinatoric and algebraic methods to be applied. In addition the requirement that a
simplicial subdivision exists rules out spaces exhibiting various sorts of pathologies
and infinite complexities. A nice example of a space that does not have a simplicial
subdivision is the Hawaiian earring, which is the union over all n = 1, 2, 3, . . . of
the circle of radius 1/n centered at (1/n, 0) ∈ R2 .
2.6
Graphs
A graph is a one dimensional polytopal complex. That is, it consists of finitely
many zero and one dimensional polytopes, with the one dimensional polytopes intersecting at common endpoints, if they intersect at all. A one dimensional polytope
is just a line segment, which is a one dimensional simplex, so a graph is necessarily
a simplicial complex.
Relative to general simplicial complexes, graphs sound pretty simple, and from
the perspective of our work here this is indeed the case, but the reader should be
aware that there is much more to graph theory than this. The formal study of
graphs in mathematics began around the middle of the 20th century and quickly
became an extremely active area of research, with numerous subfields, deep results,
and various applications such as the theory of networks in economic theory. Among
the numerous excellent texts in this area, Bollobás (1979) can be recommended to
the beginner.
This book will use no deep or advanced results about graphs. In fact, almost
everything we need to know about them is given in Lemma 2.6.1 below. The main
purpose of this section is simply to introduce the basic terminology of the subject,
which will be used extensively.
Formally, a graph1 is a triple G = (V, E) consisting of a finite set V of vertices
and a set E of two element subsets of V . An element of e = {v, w} of E is called an
edge, and v and w are its endpoints. Sometimes one writes vw in place of {v, w}.
Two vertices are neighbors if they are the endpoints of an edge. The degree of a
vertex is the cardinality of its set of neighbors.
A walk in G is a sequence v0 v1 · · · vr of vertices such that vj−1 and vj are
neighbors for each j = 1, . . . , r. It is a path if v0 , . . . , vr are all distinct. A path is
1
In the context of graph theory the sorts of graphs we describe here are said to be “simple,”
to distinguish them from a more complicated class of graphs in which there can be loops (that is,
edges whose two endpoints are the same) and multiple edges connecting a single pair of vertices.
They are also said to be “undirected” to distinguish them from so-called directed graphs in which
each edge is oriented, with a “source” and “target.”
34
CHAPTER 2. PLANES, POLYHEDRA, AND POLYTOPES
maximal if it not contained (in the obvious sense) in a longer path. Two vertices
are connected if they are the endpoints of a path. This is an equivalence relation,
and a component of G is one of the graphs consisting of an equivalence class and
the edges in G joining its vertices. We say that G is connected if it has only one
component, so that any two vertices are connected. A walk v0 v1 · · · vr is a cycle if
r ≥ 3, v0 , . . . , vr−1 are distinct, and vr = v0 . If G has no cycles, then it is said to
be acyclic. A connected acyclic graph is a tree.
The following simple fact is the only “result” from graph theory applied in this
book. It is sufficiently obvious that there would be little point in including a proof.
Lemma 2.6.1. If the degree of each of the vertices of G is at most two, then the
components of G are maximal paths, cycles, and vertices with no neighbors.
This simple principle underlies all the algorithms described in Chapter 3. There
are an even number of endpoints of paths in G. If it is known that an odd number
represent or embody a situation that is not what we are looking for, then the rest
do embody what we are looking for, and in particular the number of “solutions” is
odd, hence positive. If it is known that exactly one endpoint embodies what we are
not looking for, and that endpoint is easily computed, then we can find a solution
by beginning at that point and following the path to its other endpoint.
Chapter 3
Computing Fixed Points
When it was originally proved, Brouwer’s fixed point theorem was a major breakthrough, providing a resolution of several outstanding problems in topology. Since
that time the development of mathematical infrastructure has provided access to
various useful techniques, and a number of easier demonstrations have emerged, but
there are no proofs that are truly simple.
There is an important reason for this. The most common method of proving
that some mathematical object exists is to provide an algorithm that constructs it,
or some proxy such as an arbitrarily accurate approximation, but for fixed points
this is problematic. Naively, one might imagine a computational strategy that
tried to find an approximate fixed point by examining the value of the function at
various points, eventually halting with a declaration that a certain point was a good
approximation of a fixed point. For a function f : [0, 1] → [0, 1] such a strategy
is feasible because if f (x) > x and f (x′ ) < x′ (as is the case if x = 0 and x′ = 1
unless one of these is a fixed point) then the intermediate value function implies
that there is a fixed point between x and x′ . According to the sign of f (x′′ ) − x′′ ,
where x′′ = (x+ x′ )/2, we can replace x or x′ with x′′ , obtaining an interval with the
same property and half the length. Iterating this procedure provides an arbitrarily
fine approximation of a fixed point.
In higher dimensions such a computational strategy can never provide a guarantee that the output is actually near a fixed point. To say precisely what we mean
by this we need to be a bit more precise. Suppose you set out in search of a fixed
point of a continuous function f : X → X (where X is nonempty, compact, and
convex subset of a Euclidean space) armed with nothing more than an “oracle” that
evaluates f . That is, the only computational resources you can access are the theoretical knowledge that f is continuous, and a “black box” that tells you the value of
f at any point in its domain that you submit to it. An algorithm is, by definition,
a computational procedure that is guaranteed to halt eventually, so our supposed
algorithm for computing a fixed point necessarily halts after sampling the oracle
finitely many times, say at x1 , . . . , xn , with some declaration that such-and-such is
at least an approximation of a fixed point. Provided that the dimension of X is
at least two, the Devil could now change the function to one that agrees with the
original function at every point that was sampled, is continuous, and has no fixed
points anywhere near the point designated by the algorithm. (One way to do this is
35
36
CHAPTER 3. COMPUTING FIXED POINTS
to replace f with h−1 ◦ f ◦ h where h : X → X is a suitable homeomorphism satisfying h(xi ) = xi and h(f (xi )) = f (xi ) for all i = 1, . . . , n.) The algorithm necessarily
processes the new function in the same way, arriving at the same conclusion, but
for the new function that conclusion is erroneous.
Our strategy for proving Brouwer’s fixed point theorem will, of necessity, be a
bit indirect. We will prove the existence of objects that we will describe as “points
that are approximately fixed.” (The exact nature of such objects will vary from
one proof to the next.) An infinite sequence of such points, with the “error” of
the approximation converging to zero, will have the property that each of its limit
points is a fixed point.
The proof that any sequence in a compact space has an accumulation point uses
the axiom of choice, and in fact Brouwer’s fixed point theorem cannot be proved
without it. The axiom of choice was rather controversial when it emerged, with
constructivists (Brouwer himself became one late in life) arguing that mathematics
should only consider objects whose definitions are, in effect, algorithms for computing the object in question, or at least a succession of finer and finer approximations.
It turns out that this is quite restrictive, so the ‘should’ of the last sentence becomes quite puritanical, at least in comparison with the rich mathematics allowed
by a broader set of allowed definitions and accepted axioms, and constructivism has
almost completely faded out in recent decades.
This chapter studies two algorithmic ideas for computing points that are approximate fixed. One of these uses an algorithm for computing a Nash equilibrium
of a two person game. The second may be viewed as a matter of approximating the
given function or correspondence with an approximation that is piecewise linear in
the sense that its graph is a polyhedral complex. In both cases the algorithm traverses a path of edges in a polyhedral complex, and in the final section we explain
recent advances in computer science concerning such algorithms and the problems
they solve.
3.1
The Lemke-Howson Algorithm
In a two person game each of the two players is required to choose an element
from a set of strategies, without being informed of the other player’s choice, and each
player’s payoff depends jointly on the pair of strategies chosen. A pair consisting
of a strategy for each agent is a Nash equilibrium if neither agent can do better by
switching to some other strategy. The “mixed extension” is the derived two person
game with the same two players in which each player’s set of strategies is the set of
probability measures on that player’s set of strategies in the original game. Payoffs
in the mixed extension are computed by taking expectations.
In a sense, our primary concern in this section and the next is to show that when
the sets of strategies in the given game are finite, the mixed extension necessarily has
a Nash equilibrium. But we will actually do something quite a bit more interesting
and significant, by providing an algorithm that computes a Nash equilibrium. We
will soon see that the existence result is a special case of the Kakutani fixed point
theorem. But actually this case is not so “special” because we will eventually
37
3.1. THE LEMKE-HOWSON ALGORITHM
see that two person games can be used to approximate quite general fixed point
problems.
Formally, a finite two person game consists of:
(a) nonempty finite sets S = {s1 , . . . , sm } and T = {t1 , . . . , tn } of pure strategies for the two agents, who will be called agent 1 and agent 2;
(b) payoff functions u, v : S × T → R.
Elements of S × T are called pure strategy profiles. A pure Nash equilibrium
is a pure strategy profile (s, t) such that u(s′ , t) ≤ u(s, t) for all s′ ∈ S and v(s, t′ ) ≤
v(s, t) for all t′ ∈ T .
To define the mixed extension we need notational conventions for probability
measures on finite sets. For each k = 0, 1, 2, . . . let
∆k−1 = { ρ ∈ Rk+ : ρ1 + · · · + ρk = 1 }
be the k − 1 dimensional simplex. We will typically think of this as the set of
probability measures on a set with k elements indexed by the integers 1, . . . , k. In
particular, let S = ∆m−1 and T = ∆n−1 ; elements of these sets are called mixed
strategies for agents 1 and 2 respectively. Abusing notation, we will frequently
identify pure strategies si ∈ S and tj ∈ T with the mixed strategies in S and T
that assign all probability to i and j.
An element of S × T is called a mixed strategy profile. We let u and v
also denote the bilinear extensions of the given payoff functions to S × T , so the
expected payoffs resulting from a mixed strategy profile (σ, τ ) ∈ S × T are
u(σ, τ ) =
m X
n
X
u(si , tj )σi τj
and
i=1 j=1
v(σ, τ ) =
m X
n
X
v(si , tj )σi τj
i=1 j=1
respectively. A (mixed) Nash equilibrium is a mixed strategy profile (σ, τ ) ∈
S × T such that each agent is maximizing her expected payoff, taking the other
agent’s mixed strategy as given, so that u(σ ′ , τ ) ≤ u(σ, τ ) for all σ ′ ∈ S and
v(σ, τ ′ ) ≤ v(σ, τ ) for all τ ′ ∈ T .
The algebraic expressions for expected payoffs given above are rather bulky.
There is a way to “lighten” our notation that also allows linear algebra to be applied.
Let A and B be the m × n matrices with entries aij = u(si , tj ) and bij = v(si , tj ).
Treating mixed strategies as column vectors, we have
u(σ, τ ) = σ T Aτ
and v(σ, τ ) = σ T Bτ,
so that (σ, τ ) is a Nash equilibrium if σ ′ T Aτ ≤ σ T Aτ for all σ ′ ∈ S and σ T Bτ ′ ≤
σ T Bτ for all τ ′ ∈ T . The set of Nash equilibria can be viewed as the set of fixed
points of an upper semicontinuous convex valued correspondence β : S ×T → S ×T
where β(σ, τ ) = β1 (τ ) × β2 (σ) is given by
T
β1 (τ ) = argmax σ ′ Aτ
σ′ ∈S
and
β2 (σ) = argmax σ T Bτ ′ .
τ ′ ∈T
38
CHAPTER 3. COMPUTING FIXED POINTS
A concrete example may help to fix ideas. Suppose that



3 3 4
4 5



A= 4 3 3
and
B= 4 2
3 4 3
5 4
m = n = 3, with

2
5 .
2
These payoffs determine the divisions of S and T , according to best responses,
shown in Figure 3.1 below.
s2
t2
b
b
t3
S
T
s3
t1
s2
t2
s1
b
b
s3
s1
b
t1
Figure 3.1
Specifically, for any σ ∈ S, β2 (σ) is the set of probability measures that assign
all probability to pure strategies whose associated regions in S contain σ in their
closure, and similarly for β1 (τ ). With a little bit of work you should have no
difficulty verifying that the divisions of S and T are as pictured, but the discussion
uses only the qualitative information shown in the figure, so you can skip this chore
if you like.
Because the number of pure strategies is quite small, we can use exhaustive
search to find all Nash equilibria. For games in which each pure strategy has a
unique best response a relatively quick way to find all pure Nash equilibria is to
start with an arbitrary pure strategy and follow the sequence of pure best responses
until it visits a pure strategy a second time. The last two strategies on the path
constitute a Nash equilibrium if they are best responses to each other, and none of
the preceeding strategies is part of a pure Nash equilibrium. If there are any pure
strategies that were not reached, we can repeat the process starting at one of them,
continuing until all pure strategies have been examined. For this example, starting
at s1 gives the cycle
s1 −→ t2 −→ s3 −→ t1 −→ s2 −→ t3 −→ s1 ,
so there are no pure Nash equilibria.
b
t3
39
3.1. THE LEMKE-HOWSON ALGORITHM
A similar procedure can be used to find Nash equilibria in which each agent
mixes over two pure strategies. If we consider s1 and s2 , we see that there are two
mixtures that allow agent 2 to mix over two pure strategies, and we will need to
consider both of them, so things are a bit more complicated than they were for pure
strategies because the process “branches.” Suppose that agent 1 mixes over s1 and
s2 in the proportion that makes t1 and t2 best responses. Agent 2 has a mixture of
t1 and t2 that makes s2 and s3 best responses. There is a mixture of s2 and s3 that
makes t1 and t3 best responses, and a certain mixture τ ∗ of t1 and t3 makes s1 and
s2 best responses. The only hope for continuing this path in a way that might lead
to a Nash equilibrium is to now consider the mixture σ ∗ of s1 and s2 that makes t1
and t3 best responses, and indeed, (σ ∗ , τ ∗ ) is a Nash equilibrium.
We haven’t yet considered the possibility that agent 1 might mix over s1 and s3 ,
nor have we examined what might happen if agent 2 mixes over t2 and t3 . There is
a mixture of s1 and s3 that allow agent 2 to mix over t1 and t2 , which is a possibility
we have already considered and there is a mixture of t2 and t3 that allows agent
1 to mix over s1 and s3 , which we also analyzed above. Therefore there are no
additional Nash equilibria in which both agents mix over two pure strategies.
Could there be a Nash equilibrium in which one of the agents mixes over all
three pure strategies? Agent 2 does have one mixed strategy that allows agent 1 to
mix freely, but this mixed strategy assigns positive probability to all pure strategies
(such a mixed strategy is said to be totally mixed) so it is not a best response
to any of agent 1’s mixed strategies, and we can conclude that there is no Nash
equilibrium of this sort. Thus (σ ∗ , τ ∗ ) is the only Nash equilibrium.
This sort of analysis quickly becomes extremely tedious as the game becomes
larger. In addition, the fact that we are able to find all Nash equilibria in this way
does not prove that there is always something to find.
Before continuing we reformulate Nash equilibrium using a simple principle with
numerous repercussions, namely that a mixed strategy maximizes expected utility if
and only if it assigns all probability to pure strategies that maximize expected utility.
To understand this formally it suffices to note that agent 1’s problem is to maximize
T
ui (σ, τ ) = σ Aτ =
m
X
i=1
subject to the constraints σi ≥ 0 for all i and
this it follows that:
σi
n
X
j=1
Pm
i=1
aij τj
σi = 1, taking τ as given. From
Lemma 3.1.1. A mixed strategy profile (σ, τ ) is a Nash equilibrium if and only if:
Pn
Pn
(a) for each i = 1, . . . , m, either σi = 0 or
j=1 ai′ j τj for all
j=1 aij τj ≥
′
i = 1, . . . , m;
P
Pm
′
(b) for each j = 1, . . . , n, either τj = 0 or m
b
σ
≥
ij
i
i=1
i=1 bij ′ σi for all j =
1, . . . , n.
For each m + n conditions there are two possibilities, so there are 2m+n cases. For
each of these cases the intuition derived from counting equations and unknowns
40
CHAPTER 3. COMPUTING FIXED POINTS
suggests that the set of solutions of the conditions given in Lemma 3.1.1 will typically be zero dimensional, which is to say that it is a finite set of points. Thus we
expect that the set of Nash equilibria will typically be finite.
The Lemke-Howson algorithm is based on the hope that if we relax one of the
conditions above, say the one saying that either σ1 = 0 or agent 1’s first pure
strategy is a best response, then we may expect that the resulting set will be one
dimensional. Specifically, we let M be the set of pairs (σ, τ ) ∈ S × T satisfying:
P
P
(a) for each i = 2, . . . , m, either σi = 0 or nj=1 aij τj ≥ nj=1 ai′ j τj for all i′ =
1, . . . , m;
Pm
P
′
b
σ
≥
(b) for each j = 1, . . . , n, either τj = 0 or m
ij
i
i=1 bij ′ σi for all j =
i=1
1, . . . , n.
For the rest of the section we will assume that M is 1-dimensional, and that it does
not contain any point satisfying more than m + n of the 2(m + n − 1) conditions
“σi = 0,” “strategy i is optimal,” “τj = 0,” and “strategy j is optimal,” for 2 ≤ i ≤
m and 1 ≤ j ≤ n.
For our example there is a path in M that follows the path
(s1 , t2 ) −→ (A, t2 ) −→ (A, B) −→ (C, B) −→ (C, t1) −→ (D, t1 ) −→ (D, E).
This path alternates between the moves in S and the moves in T shown in Figure
3.2 below:
s2
t2
b
b
t3
2
Db
s3
B
5
t1
C
4
3
t2
s1
b
1
s2
A
b
s3
b
b
t1
s1
6
E
b
t3
Figure 3.2
Let’s look at this path in detail. The best response to s1 is t2 , so (s1 , t2 ) ∈ M.
The best response to t2 is s3 , so there is an edge in M leading away from (s1 , t2 ) that
increases the probability of s3 until (A, t2 ) is reached. We can’t continue further in
this direction because t2 would cease to be a best response. However, t1 becomes a
41
3.1. THE LEMKE-HOWSON ALGORITHM
best response at A, so there is the possibility of holding A fixed and moving away
from t2 along the edge of T between t1 and t2 . We can’t continue in this way past
B because s3 would no longer be a best response. However, at B both s2 and
s3 are best responses, so the conditions defining M place no constraints on agent
1’s mixed strategy. Therefore we can move away from (A, B) by holding B fixed
and moving into the interior of S in a way that obeys the constraints on agent 2’s
mixed strategy, which are that t1 and t2 are best responses. This edge bumps into
the boundary of S at C. Since the probability of s3 is now zero, we are no longer
required to have it be a best response, so we can continue from B along the edge of
T until we arrive at t1 . Since the probability of t2 is now zero, we can move away
from C along the edge between s1 and s2 until we arrive at D. Since t3 is now a
best response, we can move away from t1 along the edge between t1 and t3 until we
arrive at E. As we saw above, (D, E) = (σ ∗ , τ ∗ ) is a Nash equilibrium.
We now explain how this works in general. If Y is a proper subset of {1, . . . , m}
and D is a nonempty subset of {1, . . . , n}, let
SY (D) = { σ ∈ S : σi = 0 for all i ∈ Y and D ⊂ argmax
j=1,...,n
X
i
bij σi }
be the set of mixed strategies for agent 1 that assign zero probability to every pure
strategy in Y and make every pure strategy in D a best response. Evidently SY (D)
is a polytope.
It is now time to say what “typically” means. The matrix B is said to be in
Lemke-Howson general position if, for all Y and D, SY (D) is either empty or
(m − |D| − |Y |)-dimensional. That is, SY (D) has the dimensions one would expect
by counting equations and unknowns. In particular, if m < |D| + |Y |, then SY (D)
is certainly empty.
Similarly, if Z is a proper subset of {1, . . . , n} and C is a nonempty subset of
{1, . . . , m}, let
TZ (C) = { τ ∈ T : τj = 0 for all j ∈ Z and C ⊂ argmax
i=1,...,m
X
j
aij τj }.
The matrix A is said to be in Lemke-Howson general position if, for all Z and C,
TZ (C) is either empty or (n − |C| − |Z|)-dimensional. Through the remainder of
this section we assume that A and B are in Lemke-Howson general position.
The set of Nash equilibria is the union of the cartesian products SY (D) × TZ (C)
over all quadruples (Y, D, Z, C) with Y ∪ C = {1, . . . , m} and Z ∪ D = {1, . . . , n}.
The general position assumption implies that if such a product is nonempty, then
|Y | + |C| = m and |Z| + |D| = n, so that Y and C are disjoint, as are Z and D,
and SY (D) × TZ (C) is zero dimensional, i.e., a singleton. Thus the general position
assumption implies that there are finitely many equilibria.
In addition, we now have
[
M=
SY (D) × TZ (C)
(∗)
where the union is over all quadruples (Y, D, Z, C) such that:
42
CHAPTER 3. COMPUTING FIXED POINTS
(a) Y and Z are proper subsets of {1, . . . , m} and {1, . . . , n};
(b) C and D are nonempty subsets of {1, . . . , m} and {1, . . . , n};
(c) {2, . . . , m} ⊂ Y ∪ C;
(d) {1, . . . , n} = Z ∪ D;
(e) SY (D) and TZ (C) are nonempty.
A quadruple (Y, D, Z, C) satisfying these conditions is said to be qualified. A
vertex quadruple is a qualified quadruple (Y, D, Z, C) such that SY (D) × TZ (C)
is 0-dimensional. It is the starting point of the algorithm if Y = {2, . . . , m},
and it is a Nash equilibrium if 1 ∈ Y ∪ C.
An edge quadruple is a qualified quadruple (Y, D, Z, C) such that SY (D) ×
TZ (C) is 1-dimensional. A vertex quadruple (Y ′ , D ′ , Z ′ , C ′) is an endpoint of this
edge quadruple if Y ⊂ Y ′ , D ⊂ D ′ , Z ⊂ Z ′ , and C ⊂ C ′ . It is easy to see that
the edge quadruple has two endpoints: if SY (D) is 1-dimensional, then it has two
endpoints SY ′ (D ′ ) and SY ′′ (D ′′ ), in which case (Y ′ , D ′ , Z, C) and (Y ′′ , D ′′, Z, C) are
the two endpoints of (Y, C, Z, D), and similarly if TZ (C) is q-dimensional.
Evidently M is a graph. The picture we would like to establish is that it is
a union of loops, paths whose endpoints are the Nash equilibria and the starting
point of the algorithm, and possibly an isolated point if the starting point of the
algorithm happens to be a Nash equilibrium. If this is the case we can find a Nash
equilibrium by following the path leading away from the starting point until we
reach its other endpoint, which is necessarily a Nash equilibrium.
Put another way, we would like to show that a vertex quadruple is an endpoint
of zero, one, or two edge quadruples, and:
(i) if it is an endpoint of no edge quadruples, then it is both the starting point
of the algorithm and a Nash equilibrium;
(ii) if it is an endpoint of one edge quadruple, then it is either the starting point
of the algorithm, but not a Nash equilibrium, or a Nash equilibrium, but not
the starting point of the algorithm;
(iii) if it is an endpoint of two edge quadruples, then it is neither the starting point
of the algorithm nor a Nash equilibrium.
So, suppose that (Y, D, Z, C) is a vertex quadruple. There are two main cases
to consider, the first of which is that it is a Nash equilibrium, so that 1 ∈ Y ∪ C.
If 1 ∈ Y , then (Y \ {1}, D, Z, C) is the only quadruple that could be an edge
quadruple that has (Y, D, Z, C) as an endpoint, and it is in fact such a quadruple:
(a)-(d) hold obviously, and SY \{1} (D) is nonempty because SY (D) is a nonempty
subset. If 1 ∈ C, then (Y \ {1}, D, Z, C) is the only quadruple that could be an
edge quadruple that has (Y, D, Z, C) as an endpoint, and the same logic shows that
it is except when C = {1}, in which case Y = {2, . . . , m}, i.e., (Y, D, Z, C) is the
starting point of the algorithm. Summarizing, if (Y, D, Z, C) is a Nash equilibrium
vertex quadruple, it is an endpoint of precisely one edge quadruple except when it
3.1. THE LEMKE-HOWSON ALGORITHM
43
is the starting point of the algorithm, in which case it is not an endpoint of any
edge quadruple.
Now suppose that (Y, D, Z, C) is not a Nash equilibrium. Since SY (D) and
TZ (C) are 0-dimensional, |D| + |Y | = m and |C| + |Z| = n, so, in view of (e), one
of the two intersections Y ∩ C and Z ∩ D is a singleton while the other is empty.
First suppose that Z ∩ D = {j}. Then (Y, D, Z \ {j}, C) and (Y, D \ {j}, Z, C)
are the only quadruples that might be edge quadruples that have (Y, D, Z, C) as an
endpoint, and in fact both are: again (a)-(d) hold obviously (except that one must
note that |D| ≥ 2 because |Z ∪ D| = n, |Z| < n, and |Z ∩ D| = 1) and SY (D \ {j})
and TZ\{j} (C) are both nonempty because SY (D) and TZ (C) are nonempty subsets.
On the other hand, if Y ∩ C = {i}, then (Y \ {i}, D, Z, C) and (Y, D, Z, C \ {i})
are the only quadruples that might be edge quadruples that have (Y, D, Z, C) as an
endpoint. By the logic above, (Y \ {i}, D, Z, C) certainly is, and (Y, D, Z, C \ {i})
is if C 6= {i}, and not otherwise. When C = {i} we have Y ∪ C = {2, . . . , m}
and Y ∩ C = {i} = C, so Y = {2, . . . , m}, which is to say that (Y, D, Z, C) is the
starting point of the algorithm. In sum, if (Y, D, Z, C) is not a Nash equilibrium, it
is an endpoint of precisely two edge quadruples except when it is the starting point
of the algorithm, in which case is an endpoint of precisely one edge quadruple.
Taken together, these observations verify (i)-(iii), and complete the formal verification of the main properties of the Lemke-Howson algorithm. Two aspects of
the procedure are worth noting. First, when SY (D) × TZ (C) is a vertex that
is an endpoint of two edges, the two edges are either SY \{i} (D) × TZ (C) and
SY (D) × TZ (C \ {i}) for some i or SY (D) × TZ\{j} (C) and SY (D \ {j}) × TZ (C) for
some j. In both cases one of the edges is the cartesian product of a line segment
in S and a point in T while the other is the cartesian product of a point in S and
a line segment in T . Geometrically, the algorithm alternates between motion in S
and motion in T .
Second, although our discussion has singled out the first pure strategy of agent
1, this was arbitrary, and any pure strategy of either player could be designated for
this role. It is quite possible that different choices will lead to different equilibria.
In addition, although the algorithm was described in terms of starting at this pure
strategy and its best response, the path following procedure can be started at any
endpoint of a path in M. In particular, having computed a Nash equilibrium using
one designated pure strategy, we can then switch to a different designated pure
strategy and follow the path, for the new designated pure strategy, going away
from the equilibrium. This path may go to the starting point of the algorithm
for the new designated pure strategy, but it is also quite possible that it leads
to a Nash equilibrium that cannot be reached directly by the algorithm using any
designated pure strategy. Equilibria that can be reached by repeated applications of
this maneuver are said to be accessible. A famous example due to Robert Wilson
(reported in Shapley (1974))) shows that there can be inaccessible equilibria even
in games with a surprisingly small number of pure strategies.
44
3.2
CHAPTER 3. COMPUTING FIXED POINTS
Implementation and Degeneracy Resolution
We have described the Lemke-Howson algorithm geometrically, in terms that a
human can picture, but that it not quite the same thing as providing a description
in terms of concrete, fully elaborated, algebraic operations. This section provides
such a description. In addition, our discussion to this point has assumed a game
in Lemke-Howson general position. In order to prove that any game has a Nash
equilibrium it suffices to show that games in general position are dense in the set
of pairs (A, B) of m × n matrices, because it is easy to see that if (Ar , B r ) is a
sequence converging to (A, B), and for each r we have a Nash equilibrium (σ r , τ r )
of (the game with payoff matrices) (Ar , B r ), then along some subsequence we have
(σ r τ r ) → (σ, τ ), and (σ, τ ) is a Nash equilibrium of (A, B). However, we will do
something quite a bit more elegant and useful, providing a refinement of the LemkeHowson algorithm that works even for games that are not in Lemke-Howson general
position.
The formulation of the Nash equilibrium problem we have been working with
so far is a matter of finding u∗ , v ∗ ∈ R, s′ , σ ′ ∈ Rm , and t′ , τ ′ ∈ Rn such that:
Aτ ′ + s′ = u∗ em , B T σ ′ + t′ = v ∗ en , hs′ , σ ′ i = 0 = ht′ , τ ′ i, hσ ′ , em i = 1 = hτ ′ , en i,
s′ , σ ′ ≥ 0 ∈ Rm , t′ , τ ′ ≥ 0 ∈ Rn .
The set of Nash equilibria is unaffected if we add a constant to every entry in a
column of A, or to every entry of a row of B. Therefore we may assume that all
the entries of A and B are positive, and will do so henceforth. Now the equilibrium
utilities u∗ and v ∗ are necessarily positive, so we can divide in the system above,
obtaining the system
Aτ + s = em , B T σ + t = en , hs, σi = 0 = ht, τ i, s, σ ≥ 0 ∈ Rm , t, τ ≥ 0 ∈ Rn
together with the formulas hσ, em i = 1/v ∗ and hτ, en i = 1/u∗ for computing equilibrium expected payoffs. The components of s and t are called slack variables.
This new system is not quite equivalent to the one above because the one above
in effect requires that σ and τ each have some positive components. The new system
has another solution that does not come from a Nash equilibrium, namely σ = 0,
τ = 0, s = em , and t = en . It is called the extraneous solution. To see that
this is the only new solution consider that if σ = 0, then t = en , so that ht, τ i = 0
implies τ = 0, and similarly τ = 0 implies that σ = 0.
We now wish to see the geometry of the Lemke-Howson algorithm in the new
coordinate system. Let
S ∗ = { σ ∈ Rm : σ ≥ 0 and B T σ ≤ en } and T ∗ = { τ ∈ Rn : τ ≥ 0 and Aτ ≤ em }.
P
There is a bijection σ 7→ σ/ i σi between the points on the upper surface of S ∗ ,
namely those for which some component of en − B T σ is zero, and the points of S,
and similarly for T ∗ and T .
For the game studied in the last section the polytopes S ∗ and T ∗ are shown in
Figure 3.3 below. Note that the best response regions in Figure 3.1 have become
facets.
45
3.2. IMPLEMENTATION AND DEGENERACY RESOLUTION
τ2
σ2
S∗
T∗
t3
s3
t1
τ3
s1
σ3
s2
t2
σ1
τ1
Figure 3.3
We now transport the Lemke-Howson algorithm to this framework. Let M ∗ be
the set of (σ, τ ) ∈ S ∗ × T ∗ such that, when we set s = em − Aτ and t = en − B T σ,
we have
(a) for each i = 2, . . . , m, either σi = 0 or si = 0;
(b) for each j = 1, . . . , n, either τj = 0 or tj = 0.
For our running example we can follow a path in M ∗ from (0, 0) to the image of
the Nash equilibrium, as shown in Figure 3.4. This path has a couple more edges
than the one in Figure 3.2, but there is the advantage of starting at (0, 0), which is
a bit more canonical.
If we set
0 A
,
q = eℓ , y = (σ, τ ), and x = (s, t),
ℓ = m + n,
C=
BT 0
the system above is equivalent to
Cy + x = q
hx, yi = 0
x, y ≥ 0 ∈ Rℓ .
(∗)
This is called the linear complementarity problem. It arises in a variety of other
settings, and is very extensively studied. The framework of the linear complementarity problem is simpler conceptually and notationally, and it allows somewhat
greater generality, so we will work with it for the remainder of this section.
46
CHAPTER 3. COMPUTING FIXED POINTS
Let
P = { (x, y) ∈ Rℓ × Rℓ : x ≥ 0, y ≥ 0, and Cy + x = q }.
We will assume that all the components of q are positive, that all the entries of C
are nonnegative, and that each row of C has at least one positive entry, so that P
is bounded and thus a polytope. In general a d-dimensional polytope is said to be
simple if each of its vertices is in exactly d facets. The condition that generalizes
the general position assumption on A and B is that P is simple.
Let the projection of P onto the second copy of Rℓ be
Q = { y ∈ Rℓ : y ≥ 0 and Cy ≤ q }.
0 A
and q = eℓ , then Q = S ∗ × T ∗ , and each edge of Q is either the
If C =
BT 0
cartesian product of a vertex of S ∗ and an edge of T ∗ or the cartesian product of
an edge of S ∗ and a vertex of T ∗ .
τ2
σ2
t3
s3
b
t1
s1
σ3
b
b
s2
b
t2
σ1
τ1
Figure 3.4
Our problem is to find a (x, y) ∈ P such that x 6= 0 satisfying the “complementary slackness condition” hx, yi = 0. The algorithm follows the path starting at
(x, y) = (q, 0) in
M ∗∗ = { (x, y) ∈ P : x2 y2 + · · · + xℓ yℓ = 0 }.
The equation x2 y2 +· · ·+xℓ yℓ = 0 encodes the condition that for each j = 2, . . . , ℓ, either xj = 0 or yj = 0. Suppose we are at a vertex (x, y) of P satisfying this condition,
τ3
47
3.2. IMPLEMENTATION AND DEGENERACY RESOLUTION
but not x1 y1 = 0. Since P is simple, exactly ℓ of the variables x2 , . . . , xℓ , y2 , . . . , yℓ
vanish, so there is some i such that xi = 0 = yi . The portion of P where xi ≥ 0
and the other ℓ − 1 variables vanish is an edge of P whose other endpoint is the
first point where one of the ℓ variables that are positive at (x, y) vanishes. Again,
since P is simple, precisely one of those variables vanishes there.
How should we describe moving from one vertex to the next algebraically? Consider specifically the mave away from (0, q). Observe that P is the graph of the
function y 7→ q − Cy from Q to Rℓ . We explicitly write out the system of equations
describing this function:
x1 = q1 − c11 y1 − · · · − c1ℓ yℓ ,
..
..
..
..
.
.
.
.
xi = qi − ci1 y1 − · · · − ciℓ yℓ ,
..
..
..
..
.
.
.
.
xℓ = qℓ − cℓ1 y1 − · · · − cℓℓ yℓ .
As we increase y1 , holding 0 = y2 = · · · = yℓ , the constraint we bump into first is
the one requiring xi ≥ 0 for the i for which qi /ci1 is minimal. If i = 1, then the
point we arrived at is a solution and the algorithm halts, so we may suppose that
i ≥ 2.
We now want to describe P as the graph of a function with domain in the
xi , y2, . . . , yℓ coordinate subspace, and x1 , . . . , xi−1 , y1 , xi+1 , . . . , xℓ as the variables
parameterizing the range. To this end we rewrite the ith equation as
y1 =
1
1
ci2
ciℓ
qi − xi − y2 − · · · − yℓ .
ci1
ci1
ci1
ci1
Replacing the first equation above with this, and substituting it into the other
equations, gives
c11 ci2 c11 ciℓ c11 c11 xi − c12 −
y2 − · · · − c1ℓ −
yℓ ,
x1 = q1 −
qi − −
ci1
ci1
ci1
ci1
..
..
..
..
..
.
.
.
.
.
1
ci2
1
qi
− xi
− y2 −
···
−
ci1
ci1
ci1
..
..
..
.
.
.
cℓ1 cℓ1 cℓ1 ci2 = qℓ −
xi − cℓ2 −
y2 − · · · − cℓℓ −
qi − −
ci1
ci1
ci1
y1 =
..
.
xℓ
ciℓ
yℓ ,
ci1
..
.
cℓ1 ciℓ yℓ .
ci1
This is not exactly a thing of beauty, but it evidently has the same form as what
we started with. The data of the algorithm consists of a tableau [q ′ , C ′ ], a list
describing how the rows and the last ℓ columns of the tableau correspond to the
original variables of the problem, and the variable that vanished when we arrived
at the corresponding vertex. If this variable is either x1 or y1 we are done. Otherwise the data is updated by letting the variable that is complementary to this one
48
CHAPTER 3. COMPUTING FIXED POINTS
increase, finding the next variable that will vanish when we do so, then updating
the list and the tableau appropriately. This process is called pivoting.
We can now describe how the algorithm works in the degenerate case when P
is not necessarily simple. From a conceptual point of view, our method of handling
degenerate problems is to deform them slightly, so that they become nondegenerate,
but in the end we will have only a combinatoric rule for choosing the next pivot
variable. Let L = { (x, y) ∈ Rℓ × Rℓ : Cy + x = q }, let α1 , . . . , αℓ , β1 , . . . , βℓ be
distinct positive integers, and for ε > 0 let
Pε = { (x, y) ∈ L : xi ≥ −εαi and yi ≥ −εβi for all i = 1, . . . , ℓ }.
If (x, y) is a vertex of Pε , then there are ℓ variables, which we will describe as
“free variables,” whose corresponding equations xi = εαi and yi = εβi determine
(x, y) as the unique member of L satisfying them. At the point in L where these
equations are satisfied, the other variables can be written as linear combinations of
the free variables, and thus as polynomial functions of ε. Because the αi and βi
are all different, there are only finitely many values of ε such that any of the other
variables vanish at this vertex. Because there are finitely many ℓ-element subsets
of the 2ℓ variables, it follows that Pε is simple for all but finitely many values of ε.
In particular, for all ε in some interval (0, ε) the combinatoric structure of Pε will
be independent of ε. In addition, we do not actually need to work in Pε because the
pivoting procedure, applied to the polytope Pε for such ε, will follow a well defined
path that can be described in terms of a combinatoric procedure for choosing the
next pivot variable.
To see what we mean be this consider the problem of finding which xi first goes
below −εαi as we go out the line y1 ≥ −εβ1 , y2 = −εβ2 , . . . , yℓ = −εβℓ . This is
basically a process of elimination. If ci1 ≤ 0, then increasing y1 never leads to a
violation of the ith constraint, so we can begin by eliminating all those i for which
ci1 is not positive. Among the remaining i, the problem is to find the i for which
1
ci2
ciℓ
1
qi + εαi + εβ2 + · · · + εβℓ
ci1
ci1
ci1
ci1
is smallest for small ε > 0. The next step is to eliminate all i for which qi /c1i is not
minimal. For each i that remains the expression
1 αi ci2 β2
ciℓ
ε + ε + · · · + εβℓ
ci1
ci1
ci1
has a dominant term, namely the term, among those with nonzero coefficients,
whose exponent is smallest. The dominant terms are ordered according to their
values for small ε > 0:
(a) terms with positive coefficients are greater than terms with negative coefficients;
(b) among terms with positive coefficients, those with smaller exponents are
greater than terms with larger exponents, and if two terms have equal exponents they are ordered according to the coefficients;
3.3. USING GAMES TO FIND FIXED POINTS
49
(c) among terms with negative coefficients, those with larger exponents are greater
than terms with smaller exponents, and if two terms have equal exponents
they are ordered according to the coefficients.
We now eliminate all i for which the dominant term is not minimal. All remaining
i have the same dominant term, and we continue by subtracting off this term and
comparing the resulting expressions in a similar manner, repeating until only one i
remains. This process does necessarily continue until only one i remains, because
if other terms of the expressions above fail to distinguish between two possibilities,
eventually there will be a comparison involving the terms εαi /ci1 , and the exponents
α1 , . . . , αℓ , β1 , . . . , βℓ are distinct.
Let’s review the situation. We have given an algorithm that finds a solution
of the linear complementarity problem (∗) that is different from (q, 0). The assumptions that insure that the algorithm works are that q ≥ 0 and that P is
a polytope. In particular, these assumptions are satisfied when the linear complementarity problem is derived from a two person game with positive payoffs, in which
case any solution other than (q, 0) corresponds to a Nash equilibrium. Therefore
any two person game with positive payoffs has a Nash equilibrium, but since the
equilibrium conditions are unaffected by adding a constant to a player’s payoffs, in
fact we have now shown that any two person game has a Nash equilibrium.
There are additional issues that arise in connection with implementing the algorithm, since computers cannot do exact arithmetic on arbitrary real numbers.
One possibility is to require that the entries of q and C lie in a set of numbers
for which exact arithmetic is possible—usually the rationals, but there are other
possibilities, at least theoretically. Alternatively, one may work with floating point
numbers, which is more practical, but also more demanding because there are issues
associated with round-off error, and in particular its accumulation as the number of
pivots increases. The sort of pivoting we have studied here also underlies the simplex algorithm for linear programming, and the same sorts of ideas are applied to
resolve degeneracy. Numerical analysis for linear programming has a huge amount
of theory, much of which is applicable to the Lemke-Howson algorithm, but it is far
beyond our scope.
3.3
Using Games to Find Fixed Points
It is surprisingly easy to use the existence of equilibrium in two person games to
prove Kakutani’s fixed point theorem in full generality. The key idea has a simple
description. Fix a nonempty compact convex X ⊂ Rd , and let F : X → X be a (not
necessarily convex valued or upper semicontinuous) correspondence with compact
values. We can define a two person game with strategy sets S = T = X by setting
(
0, s 6= t,
u(s, t) = − min ks − xk2 and v(s, t) =
x∈F (t)
1, s = t.
If (s, t) is a Nash equilibrium, then s ∈ F (t) and t = s, so s = t is a fixed point.
Conversely, if x is a fixed point, then (x, x) is a Nash equilibrium.
50
CHAPTER 3. COMPUTING FIXED POINTS
Of course this observation does not prove anything, but it does point in a useful
direction. Let x1 , . . . , xn , y1 , . . . , yn ∈ X be given. We can define a finite two person
game with n × n payoff matrices A = (aij ) and B = (bij ) by setting
(
0, i 6= j,
aij = −kxi − yj k2 and bij =
1, i = j.
Let (σ, τ ) ∈ ∆n−1 ×∆n−1 be a mixed strategy profile. Clearly τ is a best response to
σ if and only if it assigns all probability to the strategies that are assigned maximum
probability by σ, which is to say that τj > 0 implies that σj ≥ σi for all i.
Understanding
when σ is a best response to τ requires a brief calculation. Let
P
z = nj=1 τj yj . For each i we have
X
j
aij τj = −
X
j
τj kxi − yj k2 = −
X τj xi − yj , xi − yj
j
X X X =−
τj xi , xi + 2
τj xi , yj −
τj yj , yj
j
j
j
= − xi , xi + 2 xi , z − hz, zi + C = −kxi − zk2 + C
P
where C = kzk2 − nj=1 τj kyj k2 is a quantity that does not depend on i. Therefore
σ is a best response to τ if and only if it assigns all probability to those i with xi as
close to z as possible. If y1 ∈ F (x1 ), . . . , yn ∈ F (xn ), then there is a sense in which
a Nash equilibrium may be regarded as a “point that is approximately fixed.”
We are going to make this precise, thereby proving Kakutani’s fixed point theorem. Assume now that F is upper semicontinuous with convex values. Define
sequences x1 , x2 , . . . and y1 , y2 , . . . inductively as follows. Choose x1 arbitrarily, and
let y1 be an element of F (x1 ). Supposing that x1 , . . . , xn and y1 , . . . , yn , have already been determined, let (σ n , τ n ) be a Nash equilibrium of the two person game
with payoff matrices An = (anij ) and B nP= (bnij ) where anij = −kxi − yj k2 and bnij is
1 if i = j and 0 otherwise. Let xn+1 = j τj yj , and choose yn+1 ∈ F (yn+1).
Let x∗ be an accumulation point of the sequence {xn }. To show that x∗ is a
fixed point of F it suffices to show that it is an element of the closure of any convex
neighborhood V of F (x∗ ). Choose
0 such that F (x) ⊂ V for all x ∈ Uδ (x∗ ).
Pδ >
Consider an n such that xn+1 = j τjn yj ∈ Uδ/3 (x∗ ) and at least one of x1 , . . . , xn
is also in this ball. Then the points in x1 , . . . , xn that are closest to xn+1 are in
U2δ/3 (xn+1 ) ⊂ Uδ (x∗ ), so xn+1 is a convex combination of points in V , and is
therefore in V . Therefore x∗ is in the closure of the set of xn that lie in V , and thus
in the closure of V .
In addition to proving the Kakutani fixed point theorem, we have accumulated
all the components of an algorithm for computing approximately fixed points of
a continuous function f : X → X. Specifically, for any error tolerance ε > 0 we
compute the sequences x1 , x2 , . . . and y1 , y2 , . . . with f in place of F , halting when
kxn+1 −f (xn+1 )k < ε. The argument above shows that this is, in fact, an algorithm,
in the sense that it is guaranteed to halt eventually. This algorithm is quite new.
Code implementing it exists, and the initial impression is that it performs quite
well. But it has not been extensively tested.
51
3.4. SPERNER’S LEMMA
There is one more idea that may have some algorithmic interest. As before, we
consider points x1 , . . . , xn , y1 , . . . , yn ∈ Rd . Define a correspondence Φ : Rd → Rd
by letting Φ(z) be the convex hull of { yj : j ∈ argmini kz − xi k } when z ∈ PJ .
(Evidently this construction is closely related to the Voronoi diagram determined by
x1 , . . . , xn . Recall that this is the polyhedral decomposition of Rd whose nonempty
polyhedra are the sets PJ = { z ∈ V : J ⊂ argmini kz − xi k } where ∅ =
6 J ⊂
{1, . . . , n}.) Clearly Φ is upper semicontinuous and convex valued.
Suppose that
P z is a fixed point of this correspondence. Then z is a convex
combination
/ argmini kz − xi k. Let J = { j : yj >
j τj yj with yj = 0 if j ∈
0 }. If σi = 1/|J| when i ∈ J and σi = 0 when i ∈
/ J, then (σ, τ ) is a Nash
equilibrium of the game derived from xP
,
.
.
.
,
x
,
y
,
.
.
.
, yn . Conversely, if (σ, τ ) is
1
n 1
a Nash equilibrium of this game, then j∈J τj yj is a fixed point of Φ. In a sense,
the algorithm described above approximates the given correspondence F with a
correspondence of a particularly simple type.
We may project the path of the Lemke-Howson algorithm, in its application to
the game derived from x1 , . . . , xn , y1, . . . , yn , into this setting. Define Φ1 : Rd → Rd
by letting Φ1 (z) be the convex hull of { yi : i ∈ {1}∪argmini kz−xi k }. Suppose that
(σ, τ ) is an element of the set M defined in Section 3.1, so that all the conditions
of Nash equilibrium are satisfied except that it may be the case that σ1 > 0Peven if
the first pure strategy is not optimal. Let J = { j : τj > 0 }, and let z = j τj yj .
Then J ⊂ { i : σi > 0 } ⊂ {1} ∪ argminj kz − xj k, so z ∈ Φ1 (z). Conversely,
suppose
P
z is a fixed point of Φ1 , and let J = argminj kz − xj k. Then z = j τj yj for some
τ ∈ ∆n−1 with τj = 0 for all j ∈
/ {1} ∪ J. If we let σ be the element of ∆n−1 such
that σi = 1/|{1} ∪ J| if i ∈ J and σi = 0 if i ∈
/ {1} ∪ J, then (σ, τ ) ∈ M.
If n is large one might guess that there is a sense in which operating in Rd might
be less burdensome than working in ∆n−1 × ∆n−1 , but it seems to be difficult to
devise algorithms that take concrete advantage of this. Nonetheless this setup does
give a picture of what the Lemke-Howson algorithm is doing that has interesting
implications. For example, if there is no point in Rd that is equidistant from more
than d + 1 points, then there is no point (σ, τ ) ∈ M with σi > 0 for more than
d + 2 indices. This gives a useful upper bound on the number of pivots of the
Lemke-Howson algorithm.
3.4
Sperner’s Lemma
Sperner’s lemma is the traditional method of proving Brouwer’s fixed point
theorem without developing the machinery of algebraic topology. It dates from the
late 1920’s, which was a period during which the methods developed by Poincaré
and Brouwer were being recast in algebraic terms.
Most of our work will take place in ∆d−1 . Let P be a triangulation of ∆d−1 . For
k = 0, . . . , d − 1 let P k be the set of k-dimensional elements of P. Let V = P 0 be
the set of vertices of P, and fix a function
ℓ : V → {1, . . . , d}.
We say that ℓ is a labelling for P, and we call ℓ(v) the label of v. If ℓ(v) 6= i
for all v ∈ V with vi = 0, then ℓ is a Sperner labelling. Let e1 , . . . , ed be the
52
CHAPTER 3. COMPUTING FIXED POINTS
standard unit basis vectors of Rd . Then ℓ is a Sperner labelling if ℓ(v) ∈ { i1 , . . . , ik }
whenever v is contained in the convex hull of ei1 , . . . , eik . We say that σ ∈ P d−1
with vertex set {v1 , . . . , vd } is completely labelled if
{ℓ(v1 ), . . . , ℓ(vd )} = {1, . . . , d}.
3
b
3
b
3
b
3
1
b
b
b
1
3
b
1
1
1
b
2
b
b
b
2
b
b
b
1
2
b
1
b
b
b
b
b
1
2
2
1
2
b
2
Figure 3.5
Theorem 3.4.1 (Sperner’s Lemma). If ℓ is a Sperner labelling, then the number
of completely labelled simplices is odd.
Before proving this, let’s see why it’s important:
Proof of Brower’s Theorem. Let f : ∆d−1 → ∆d−1 be a continuous function. Proposition 2.5.2 implies that there is a sequence P1 , P2 , . . . of triangulations whose meshes
converge to zero. For each r = 1, 2, . . . let V r be the set of vertices of Pr . If any
of the elements of V r is a fixed point we are done, and otherwise we can define
ℓr : V r → {0, . . . , d} by letting ℓr (v) be the smallest index i such that vi > fi (v).
Evidently ℓr is a Sperner labelling, so there is a completely labelled simplex with
vertices v1r , . . . , vdr where ℓr (vir ) = i. Passing to a subsequence, we may assume that
the sequences vi1 , vi2 , . . . have a common limit x. For each i we have
fi (x) = lim fi (v r ) ≤ lim vir = xi ,
and
P
i
fi (x) = 1 =
P
i
xi , so f (x) = x.
We will give two proofs of Sperner’s lemma. The first of these uses facts about
volume, and in this sense is less elementary than the second (which is given in the
next section) but it quickly gives both an intuition for why the result is true and
an important refinement.
We fix an affine isometry1 A : H d−1 → Rd−1 such that
D = det A(e2 ) − A(e1 ), . . . , A(ed ) − A(e1 ) > 0.
1
If (X, dX ) and (Y, dY ) are metric spaces, a function ι : X → Y is an isometry if
dY (ι(x), ι(x′ )) = dX (x, x′ ) for all x, x′ ∈ X.
53
3.4. SPERNER’S LEMMA
(We regard the determinant as a function of (d − 1)-tuples of elements of Rd−1 be
identifying the tuple with the matrix with those columns.) A theorem of Euclid is
that the volume of a pyramid is one third of the product of the height and the area
of the base. The straightforward2 generalization of this to arbitrary dimensions
implies that d!1 D is the volume of ∆d−1 .
For each v ∈ V there is an associated function v : [0, 1] → ∆d−1 given by
v(t) = (1 − t)v + teℓ(v) . Consider a simplex σ ∈ P d−1 that is the convex hull of
v1 , . . . , vd ∈ V , where these vertices are indexed in such a way that
det A(v2 ) − A(v1 ), . . . , A(vd ) − A(v1 ) > 0.
We define a function pσ : [0, 1] → R by setting
pσ (t) =
1
det A(v2 (t)) − A(v1 (t)), . . . , A(vd (t)) − A(v1 (t)) .
d!
For 0 ≤ t ≤ 1 let σ(t) be the convex hull of v1 (t), . . . , vd (t). Then pσ (t) is the
volume of σ(t) when t is small.
We have
1
pσ (1) = det A(eℓ(v2 ) ) − A(eℓ(v1 ) ), . . . , A(eℓ(vd ) ) − A(eℓ(v1 ) ) .
d!
If σ is not completely labelled, then pσ (1) = 0 because some A(eℓ(vi ) ) − A(eℓ(v1 ) )
is zero or two of them are equal. If σ is completely labelled, then we say that the
labelling is orientation preserving on σ if pσ (1) > 0, in which case pσ (1) = d!1 D,
and orientation reversing on σ if pσ (1) < 0, in which case pσ (1) = − d!1 D.
Let p : [0, 1] → R be the sum
X
p(t) =
pσ (t).
σ∈P d−1
Elementary properties of the determinant imply that each pσ and p are polynomial
functions. For sufficiently small t the simplices σ(t) are the (d − 1)-dimensional
simplices of a triangulation of ∆d−1 .3 Therefore p(t) is d!1 D for small t. Since p is a
2
Actually, it is straightforward if you know integration, but Gauss regarded this as “too heavy”
a tool, expressing a wish for a more elementary theory of the volume of polytopes. The third of
Hilbert’s famous problems asks whether it is possible, for any two polytopes of equal volume, to
triangulate the first in such a way that the pieces can be reassembled to give the second. This
was resolved negatively by Hilbert’s student Max Dehn within a year of Hilbert’s lecture laying
out the problems, and it remains the case today that there is no truly elementary theory of the
volumes of polytopes. In line with this, our discussion presumes basic facts about d-dimensional
measure of polytopes in Rd that are very well understood by people with no formal mathematical
training, but which cannot be justified formally without appealing to relatively advanced theories
of measure and integration.
3
This is visually obvious, and a formal proof would be tedious, so we provide only a sketch.
Suppose that for each v ∈ V we have a path connected neighborhood Uv of v in the interior of the
smallest face of ∆d−1 containing v, and this system of neighborhoods satisfies the condition that
for any simplex in P, say with vertices v1 , . . . , vk , if v1′ ∈ Uv1 , . . . , vk′ ∈ Uvk , then v1′ , . . . , vk′ are
affinely independent. We claim that a simplicial complex obtained by replacing each v with some
element of Uv is a triangulation of ∆d−1 ; note that this can be proved by moving one vertex at a
time along a path. Finally observe that because ℓ is a Sperner labelling, for each v and 0 ≤ t < 1,
v(t) is contained in the interior of the smallest face of ∆d−1 containing v.
54
CHAPTER 3. COMPUTING FIXED POINTS
polynomial function of t, it follows that it is constant, and in particular p(1) =
We have established the following refinement of Sperner’s lemma:
1
D.
d!
Theorem 3.4.2. If ℓ is a Sperner labelling, then the number of σ ∈ P d−1 such that
ℓ is orientation preserving on σ is one greater than the number of σ ∈ P d−1 such
that ℓ is orientation reversing on σ.
One of our major themes is that fixed points where the function or correspondence reverses orientation are different from those where orientation is preserved.
Much of what follows is aimed at keeping track of this difference in increasingly
general settings.
3.5
The Scarf Algorithm
The traditional proof of Sperner’s lemma is an induction on dimension, using
path following in a graph with maximal degree two to show that if the result is
true in dimension d − 2, then it is also true in dimension d − 1. In the late 1960’s
and early 1970’s Herbert Scarf and his coworkers pointed out that the graphs in
the various dimensions can be combined into a single graph with maximal degree
two that has an obvious vertex whose degree is either zero or one. If the labelling
is derived from a function f : ∆d−1 → ∆d−1 in the manner described in the proof of
Brouwer’s fixed point theorem in Section 3.4, then following the path in this graph
from this starting point to the other endpoint amounts to an algorithm for finding
a point that is approximately fixed for f .
Our exposition will follow this history, first presenting the inductive argument,
then combining the graphs in the various dimensions into a single graph that supports the algorithm. As before, we are given a triangulation P of ∆d−1 and a
Sperner labelling ℓ : V → {1, . . . , d} where V = P 0 = {v1 , . . . , vm } is the set of
vertices. For each k = 0, . . . , d − 1 a k-dimensional simplex σ ∈ P d with vertices
vi1 , . . . , vik+1 is said to be k-almost completely labelled if
{1, . . . , k} ⊂ {ℓ(vi1 ), . . . , ℓ(vik+1 )},
and it is k-completely labelled if
{ℓ(vi1 ), . . . , ℓ(vik+1 )} = {1, . . . , k + 1}.
Note that a k-completely labelled simplex is k-almost completely labelled. What
we were calling completely labelled simplices in the last section are now (d − 1)completely labelled simplices.
Suppose that σ ∈ P d−1 is (d − 1)-almost completely labelled. If it is (d − 1)completely labelled, then it has precisely one facet that is (d−2)-completely labelled,
namely the facet that does not include the vertex with label d. If σ is not (d − 1)completely labelled, then it has two vertices with the same label, and the facets
opposite these vertices are its (d − 2)-completely labelled facets, so it has precisely
two such facets.
For k = 0, . . . , d − 2 let ∆k ⊂ ∆d−1 be the convex hull of e1 , . . . , ek+1 . If one
of the (d − 2)-completely labelled facets of σ is contained in the boundary of ∆d−1 ,
55
3.5. THE SCARF ALGORITHM
then it must be contained in ∆d−2 because the labelling is Sperner. (Every other
facet of ∆d−1 lacks one of the labels 1, . . . , d − 1.) When σ has two such facets, it
is not possible that ∆d−2 contains both of them, of course, because σ is the convex
hull of these facets.
Suppose now that τ ∈ P d−2 is (d − 2)-completely labelled. Any element of P d−1
that has it as a facet is necessarily (d −1)-almost completely labelled. If τ intersects
the interior of ∆d−1 , then it is a facet of two elements of P d−1 . On the other hand, if
it is contained in the boundary of ∆d−1 , then it must be contained in ∆d−2 because
ℓ is a Sperner labelling, and it is a facet of precisely one element of P d−1 .
We define a graph Γd−1 = (V d−1 , E d−1 ) in which V d−1 be the set of (d−1)-almost
completely labelled elements of P d−1 , by declaring that two elements of V d−1 are
the endpoints of an edge in E d−1 if their intersection is a (d − 2)-completely labelled
element of P d−2 . Let σ be an element of V d−1 . Our remarks above imply that if
σ is (d − 1)-completely labelled, then it is an endpoint of no edges if its (d − 2)completely labelled facet is contained in ∆d−2 , and otherwise it is an endpoint of
exactly one edge. On the other hand, if σ is not (d − 1)-completely labelled, then
it is an endpoint of precisely on edge if one of its (d − 2)-completely labelled facets
is contained in ∆d−2 , and otherwise it is an endpoint of exactly two edges.
Thus Γd−1 has maximum degree two, so it is a union of isolated points, paths, and
loops. The isolated points are the (d−1)-completely labelled simplices whose (d−2)completely labelled facets are contained in ∆d−2 . The endpoints of paths are the
(d−1)-completely labelled simplices whose (d−2)-completely labelled facets are not
contained in ∆d−2 and the (d − 1)-almost completely labelled simplices that are not
completely labelled and have a (d−2)-completely labelled facet in ∆d−2 . Combining
this information, we find that the sum of the number of (d − 1)-completely labelled
simplices and the number of (d − 2)-completely labelled simplices contained in ∆d−2
is even, because every isolated point is associated with one element of each set, and
every path has two endpoints. If there are an odd number of (d − 2)-completely
labelled simplices contained in ∆d−2 , then there are necessarily an odd number of
(d − 1)-completely labelled simplices.
3
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Figure 3.6
Of course for each k = 0, . . . , d − 2 the set of simplices in P that lie in ∆k
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CHAPTER 3. COMPUTING FIXED POINTS
constitute a simplicial subdivision of ∆k , and it is easy to see that the restriction of
the labelling to the vertices that lie in ∆k is a Sperner labelling for that subdivision.
Thus Sperner’s lemma follows from induction if we can establish it when d − 1 = 0.
In this case ∆d−1 = ∆0 is a 0-dimensional simplex (i.e., a point) and the elements
of the triangulation P are necessarily this simplex and the empty set. The simplex
is 0-completely labelled, because 0 is the only available label, so the number of
0-completely labelled simplices is odd, as desired. Figure 3.6 shows the simplices in
Γ2 for the labelling of Figure 3.5.
In order to describe the Scarf algorithm we combine the graphs developed at
each stage of the inductive process to create a single graph with a path from a
known starting point to a (d − 1)-completely labelled simplex. Let V k be the set
of k-almost completely labelled simplices contained in ∆k . Define a graph Γk =
(V k , E k ) by specifying that two elements of V k are the endpoints of an edge in
E k if their intersection is a (k − 1)-completely labelled element of P k−1 . For each
k = 1, . . . , d −1, let F k be the set of unordered pairs {τ, σ} where τ ∈ V k−1 , σ ∈ V k ,
and τ is a facet of σ. Define a graph Γ = (V, E) by setting
V = V 0 ∪ · · · ∪ V d−1
and
E = E 0 ∪ F 1 ∪ E 1 ∪ · · · ∪ E d−2 ∪ F d−1 ∪ E d−1 .
In our analysis above we saw that the number of neighbors of σ ∈ V k in Γk is
two except that this number is reduced by one if σ has a facet in V k−1 , and it is
also reduced by one if σ is k-completely labelled. If 1 ≤ k ≤ d − 2, then the first
of these conditions is precisely the circumstance in which σ is an endpoint of an
edge in F k , and the second is precisely the circumstance in which σ is an endpoint
of an edge in F k+1. Therefore every element of V 1 ∪ · · · ∪ V d−2 has precisely two
neighbors in Γ.
Provided that d ≥ 1, every completely labelled simplex in V d−1 has precisely
one neighbor in Γ, and every d-almost completely labelled simplex in V d−1 that
is not completely labelled has two neighbors in Γ that are associated with its two
(d−1)-completely labelled facets. Again provided that d ≥ 1, the unique element of
V 0 has exactly one neighbor in V 1 . Thus the completely labelled elements of V d−1
and the unique element of V 0 each have one neighbor in Γ, and every other element
of V has exactly two neighbors in Γ. Consequently the path in Γ that begins at
the unique element of V 0 ends at a completely labelled element of V d−1 . Figure 3.7
shows the simplices in Γ for the labelling of Figure 3.5, which include points and
line segments in addition to those shown in Figure 3.6.
Conceptually, the Scarf algorithm is the process of following this path. An
actual implementation requires a computational description of a triangulation of
∆d−1 . That is, there must be a triangulation and an algorithm such that if we are
given a k-simplex in of this simplex in ∆k , the algorithm will compute the (k + 1)simplex in ∆k+1 that has the given simplex as a facet (provided that k < d − 1) and
if we are given a vertex of the given simplex, the algorithm will return the other
k-simplex in ∆k that shares the facet of the given simplex opposite the given vertex
(provided that this facet is not contained in the boundary of ∆k ). In addition, we
need an algorithm that computes the label of a given vertex; typically this would
be derived from an algorithm for computing a given function f : ∆d−1 → ∆d−1 , as
in the proof of Brouwer’s theorem. Given these resources, if we are at an element of
57
3.5. THE SCARF ALGORITHM
V, we can compute the simplices of its neighbors in Γ and the labels of the vertices
of these simplices. If we remember which of these neighbors we were at prior to
arriving at the current element of V, then the next step in the algorithm is to go to
the other neighbor. Such a step along the path of the algorithm is called a pivot.
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Figure 3.7
At this point we remark on a few aspects of the Scarf algorithm, and later
we will compare it with various alternatives. The first point is that it necessarily
moves through ∆d−1 rather slowly. Consider a k-almost completely labelled simplex
σ. Each pivot of the algorithm drops one of the vertices of the current simplex,
possibly adding a new vertex, or possibly dropping down to a lower dimensional
face. Therefore a minimum of k pivots are required before one can possibly arrive
at a simplex that has no vertex in common with σ. If the grid is fine, the algorithm
will certainly require many pivots to arrive a fixed point far from the algorithm’s
starting point.
This suggests the following strategy. We first apply the Scarf algorithm to a
coarse given triangulation of ∆d−1 , thereby arriving at a completely labelled simplex
that is hopefully a rough approximation of a fixed point. We then subdivide the
given triangulation of ∆d−1 , using barycentric subdivision or some other method.
If we could somehow “restart” the algorithm in the fine triangulation, near the
completely labelled simplex in the coarse triangulation, it might typically be the
case that the algorithm did not have to go very far to find a completely labelled
simplex in the fine triangulation. Restart methods do exist (see, e.g., Merrill (1972),
Kuhn and MacKinnon (1975), and van der Laan and Talman (1979)) but it remains
the case that the Scarf algorithm has not proved to be very useful in practice,
perhaps due in part to its difficulties with high dimensional problems.
There is one more feature of the Scarf algorithm that is worth mentioning. In
our description of the algorithm the ordering of the vertices plays an explicit role,
and can easily make a difference to the outcome. If one wishes to find more than
one completely labelled simplex, or perhaps as many as possible, or perhaps even all
of them, there is the following strategy. Having followed the algorithm for the given
ordering of the indices to its terminus, now proceed from that completely labelled
simplex in the graph Γ′ associated with some different ordering. This might lead
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CHAPTER 3. COMPUTING FIXED POINTS
back to the starting point of the algorithm in Γ′ , but it is also quite possible that
it might lead to some completely labelled simplex that cannot be reached directly
by the algorithm under any ordering of the indices. A completely labelled simplex
σ is accessible if it is reachable by the algorithm in this more general sense: there
is path going to σ from the starting point of the algorithm for some ordering of the
indices, along a path that is a union of maximal paths of the various graphs Γ′ for
the various orderings of the indices.
3.6
Homotopy
Let f : X → X be a continuous function, and let x0 be an element of X. We
let h : X × [0, 1] → X be the homotopy
h(x, t) = (1 − t)x0 + tf (x).
Here we think of the variable t at time, and let ht = h(·, t) : X → X be the function
“at time t.” In this way we imagine deforming the constant function with value x0
at time zero into the function f at time one.
Let g : X × [0, 1] → X be the function g(x, t) = h(x, t) − x. The idea of the
homotopy method is to follow a path in Z = g −1 (0) starting at (x0 , 0) until we reach
a point of the form (x∗ , 1). As a practical matter it is necessary to assume that f is
C 1 , so that h and g are C 1 . It is also necessary to assume that the derivative of g has
full rank at every point of Z, and that the derivative of the map x 7→ f (x) − x has
full rank at each of the fixed points of f . As we will see later in the book, there is a
sense in which this is typically the case, so that these assumptions are mild. With
these assumptions Z will be a union of finitely many curves. Some of these curves
will be loops, while others will have two endpoints in X × {0, 1}. In particular,
the other endpoint of the curve beginning at (x0 , 0) cannot be in X × {0}, because
there is only one point in Z ∩ (X × {0}), so it must be (x∗ , 1) for some fixed point
x∗ of f .
We now have to tell the computer how to follow this path. The standard computational implementation of curve following is called the predictor-corrector
method. Suppose we are at a point z0 = (x, t) ∈ Z. We first need to compute a
vector v that is tangent to Z at z0 . Algebraicly this amounts to finding a nonzero
linear combination of the columns of the matrix of Dg(z0 ) that vanishes. For this
it suffices to express one of the columns as a linear combination of the others, and,
roughly speaking, the Gram-Schmidt process can be used to do this. We can divide
any vector we obtain this way by its norm, so that v becomes a unit vector. There is
a parameter of the procedure called the step size that is a number ∆ > 0, and the
“predictor” part of the process is completed by passing to the point z1 = z0 + ∆v.
The “corrector” part of the process uses the Newton method to pass from z1 to
a new point in Z, or at least very close to it. The first step is to find a vector w1
that is orthogonal to v such that g(z1 ) + Dg(z1)w1 = 0. To do this we can use the
Gram-Schmidt process to find a basis for the orthogonal complement of v, compute
the matrix M of the derivative of g with respect to this basis, compute the inverse
of M, and then set w1 = −M −1 g(z1 ). We then set z2 = z1 + w1 , find a vector w2
3.7. REMARKS ON COMPUTATION
59
orthogonal to v such that g(z2 ) + Dg(z2 )w2 = 0, set z3 = z2 + w2 , and continue in
this manner until g(zn ) is acceptably small. The net effect of the predictor followed
by the corrector is to move us from one point on Z to another a bit further down.
By repeating this one can go from one end of the curve to the other.
Probably the reader has sensed that the description above is a high level overview
that glides past many issues. In fact it is difficult to regard the homotopy method
as an actual algorithm, in the sense of having precisely defined inputs and being
guaranteed to eventually halt at an output of the promised sort. One issue is
that the procedure might accidentally hop from one component of Z to another,
particularly if ∆ is large. There are various things that might be done about this,
for instance trying to detect a likely failure and starting over with a smaller ∆, but
these issues, and the details of round off error that are common to all numerical
software, are really in the realm of engineering rather than computational theory.
As a practical matter, the homotopy method is highly successful, and is used to
solve systems of equations from a wide variety of application domains.
3.7
Remarks on Computation
We have now seen three algorithms for computing points that are approximately
fixed. How good are these, practically and theoretically? The first algorithm we saw,
in Section 3.3, is new. It is simple, and can be applied to a wide variety of settings.
Code now exists, but there has been little testing or practical experience. The Scarf
algorithm has not lived up to the hopes it raised when it was first developed, and is
not used in practical computation. Homotopy methods are restricted to problems
that are smooth. As we mentioned above, within this domain they have an extensive
track record with considerable success.
More generally, what can we reasonably hope for from an algorithm that computes points that are approximately fixed, and what sort of theoretical concepts can
we bring to bear on these issues? These question has been the focus of important
recent advances in theoretical computer science, and in this section we give a brief
description of these developments. The discussion presumes little in the way of
prior background in computer science, and is quite superficial—a full exposition of
this material is far beyond our scope. Interested readers can learn much more from
the cited references, and from textbooks such as Papadimitriou (1994a) and Arora
and Boaz (2007).
Theoretical analyses of algorithms must begin with a formal model of computation. The standard model is the Turing machine, which consists of a processor with
finitely many states connected by an input-output device to a unbounded one dimensional storage medium that records data in cells, on each of which one can write
an element of a finite alphabet that includes a distinguished character ‘blank.’ At
the beginning of the computation the processor is in a particular state, the storage
medium has a finitely many cells that are not blank, and the input-output device
is positioned at a particular cell in storage. In each step of the computation the
character at the input-output device’s location is read. The Turing machine is essentially defined by functions that take state-datum pairs as their arguments and
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CHAPTER 3. COMPUTING FIXED POINTS
compute:
• the next state of the processor,
• a bit that will be written at the current location of the input-output device
(overwriting the bit that was just read) and
• a motion (forward, back, stay put) of the input-output device.
The computation ends when it reaches a particular state of the machine called
“Halt.” Once that happens, the data in the storage device is regarded as the
output of the computation.
As you might imagine, an analysis based on a concrete and detailed description
of the operation of a Turing machince can be quite tedious. Fortunately, it is
rarely necessary. Historically, other models of computation were proposed, but were
subsequently found to be equivalent to the Turing model, and the Church-Turing
thesis is the hypothesis that all “reasonable” models of computation are equivalent,
in the sense that they all yield the same notion of what it means for something to be
“computable.” This is a metamathematical assertion: it can never be proved, and a
refutation would not be logical, but would instead be primarily a social phenomenon,
consisting of researchers shifting their focus to some inequivalent model.
Once we have the notion of a Turing machine, we can define an algorithm to
be a Turing machine that eventually halts, for any input state of the storage device.
A subtle distinction is possible here: a Turing machine that always halts is not
necessarily the same thing as a Turing machine that can be proved to halt, regardless
of the input. In fact one of the most important early theorems of computer science
is that there is no algorithm that has, as input, a description of a Turing machine
and a particular input, and decides whether the Turing machine with that input will
eventually halt. As a practical matter, one almost always works with algorithms
that can easily be proved to be such, in the sense that it is obvious that they
eventually halt.
A computational problem is a rule that associates a nonempty set of outputs
with each input, where the set of possible inputs and outputs is the set of pairs
consisting of a position of the input-output device and a state of the storage medium
in which there are finitely many nonblank cells. (Almost always the inputs of
interest are formatted in some way, and this definition implicitly makes checking the
validity of the input part of the problem.) A computational problem is computable
if there is an algorithm that passes from each input to one of the acceptable outputs.
The distinction between computational problems that are computable and those
that are not is fundamental, with many interesting and important aspects, but in
our discussion here we will focus exclusively on problems that are known to be
computable.
For us the most important distinctions is between those computable computational problems that are “easy” and those that are “hard,” where the definitions
of these terms remain to be specified. In order to be theoretically useful, the easiness/hardness distinction should not depend on the architecture of a particular
machine or the technology of a particular era. In addition, it should be robust, at
least in the sense that a composition of two easy computational problems, where
3.7. REMARKS ON COMPUTATION
61
the output of the first is the input of the second, should also be easy, and possibly in other senses as well. For these reasons, looking at the running time of an
algorithm on a particular input is not very useful. Instead, it is more informative
to think about how the resources (time and memory) consumed by a computation
increase as the size of the input grows. In theoretical computer science, the most
useful distinction is between algorithms whose worst case running time is bounded
by a polynomial function of the size of the output, and algorithms that do not
have this property. The class of computational problems that have polynomial time
algorithms is denoted by P. If the set of possible inputs of a computational problem is finite, then the problem is trivially in P, and in fact we will only consider
computational problems with infinite sets of inputs.
There are many kinds of computational problems, e.g., sorting, function evaluation, optimization, etc. For us the most important types are decision problems ,
which require a yes or no answer to a well posed question, and search problems,
which require an instance of some sort of object or a verification that no such object exists. An important example of a decision problem is Clique: given a simple
undirected graph G and an integer k, determine whether G has a clique with k
nodes, where a clique is a collection of vertices such that G has an edge between
any two of them. An example of a search problem is to actually find such a clique
or to certify that no such clique exists.
There is a particularly important class of decision problems called NP, which
stands for “nondeterministic polynomial time.” Originally NP was thought of as
the class of decision problems for which a Turing machine that chose its next state
randomly has a positive probability of showing that the answer is “Yes” when this
is the case. For example, if a graph has a k-clique, an algorithm that simply guesses
which elements constitute the clique has a positive probability of stumbling onto
some k-clique. The more modern way of thinking about NP is that it is the class of
decision problems for which a “Yes” answer has a certificate or witness that can
be verified in polynomial time. In the case of Clique an actual k-clique is such a
witness. Factorization of integers is another algorithmic issue which easily generates
decision problems—for example, does a given number have a prime factor whose
first digit is 3?—that are in NP because a prime factorization is a witness for them.
(One of the historic recent advances in mathematics is the discovery of a polynomial
time algorithm for testing whether a number is prime. Thus it is possible to verify
the primality of the elements of a factorization in polynomial time.)
An even larger computational class is EXP, which is the class of computational
problems that have algorithms with running times that are bounded above by a
function of the form exp(p(s)), where s is the size of the problem and p is a polynomial function. Instead of using time to define a computational class, we can
also use space, i.e., memory; PSPACE is the class of computational problems that
have algorithms that use an amount of memory that is bounded by a polynomial
function of the size of the input. The sizes of the certificates for a problem in
NP are necessarily bounded by some polynomial function of the size of the input,
and the problem can be solved by trying all possible certificates not exceeding this
bound, so any problem in NP is also in PSPACE. In turn, the number of processor
state-memory state pairs during the run of a program using polynomially bounded
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CHAPTER 3. COMPUTING FIXED POINTS
memory an exponential function of the polynomial, so any problem in PSPACE
is also in EXP. Thus
P ⊂ NP ⊂ PSPACE ⊂ EXP.
Computational classes can also be defined in relation to an oracle which is
assumed to perform some computation. The example of interest to us is an oracle
that evaluates a continuous function f : X → X. How hard is it to find a point
that is approximately fixed using such an oracle? Hirsch et al. (1989) showed that
any algorithm that does this has an exponential worst case running time, because
some functions require exponentially many calls to the oracle. Once you commit to
an algorithm, the Devil can devise a function for which your algorithm will make
exponentially many calls to the oracle before finding an approximate fixed point.
An important aspect of this result is that the oracle is assumed to be the only
source of information about the function. In practice the function is specified by
code, and in principle an algorithm could inspect the code and use what it learned to
speed things up. For linear functions, and certain other special classes of functions,
this is a useful approach, but it seems quite farfetched to imagine that a fully general
algorithm could do this fruitfully. At the same time it is hard to imagine how we
might prove that this is impossible, so we arrive at the conclusion that even though
we do not quite have a theorem, finding fixed points almost certainly has exponential
worst case complexity.
Even if finding fixed points is, in full generality, quite hard, it might still be the
case that certain types of fixed point problems are easier. Consider, in particular,
finding a Nash equilibrium of a two person game. Savani and von Stengel (2006)
(see also McLennan and Tourky (2010)) showed that the Lemke-Howson algorithm
has exponential worst case running time, but the algorithm is in many ways similar
to the simplex algorithm for linear programming, not least because both algorithms
tend to work rather well in practice. The simplex algorithm was shown by Klee
and Minty (1972) to have exponential case running time, but later polynomial time
algorithms were developed by Khachian (1979) and Karmarkar (1984). Whether or
not finding a Nash equilibrium of a two person game is in P was one of the outstanding open problems of computer science for over a decade. Additional concepts
are required in order to explain how this issue was resolved.
A technique called reduction can be used to show that some computational
problems are at least as hard as others, in a precise sense. Suppose that A and B
are two computational problems, and we have two algorithms, guaranteed to run
in polynomial time, the first of which converts the input encoding an instance of
problem A into the input encoding an instance of problem B, and the second of
which converts the desired output for the derived instance of problem B into the
desired output for the given instance of problem A. Then problem B is at least as
hard as problem A because one can easily turn an algorithm for problem B into
an algorithm for problem A that is “as good,” in any sense that is invariant under
these sorts of polynomial time transformations.
A problem is complete for a class of computational problems if it is at least as
hard, in this sense, as any other member of the class. One of the reasons that NP
is so important is there are numerous NP-complete problems, many of which arise
3.7. REMARKS ON COMPUTATION
63
naturally; Clique is one of them. One of the most famous problems in contemporary mathematics is to determine whether NP is contained in P. This question
boils down to deciding whether Clique (or any other NP-complete problem) has
a polynomial time algorithm. This is thought to be highly unlikely, both because a
lot of effort has gone into designing algorithms for these problems, and because the
existence of such an algorithm would have remarkable consequences. It should be
mentioned that this problem is, to some extent at least, an emblematic representative of numerous open questions in computer science that have a similar character.
In fact, one of the implicit conventions of the discipline is to regard a computational
problem as hard if, after some considerable effort, people haven’t been able to figure
out whether it is hard or easy.
For any decision problem in NP there is an associated search problem, namely
to find a witness for an affirmative answer or verify that the answer is negative.
For Clique this means not only showing that a clique of size k exists, but actually
producing one. The class of search problems associated with decision problems is
called FNP. (The ‘F’ stands for “function.”) For Clique the search problem is
not much harder than the decision problem, in the following sense: if we had a
polynomial time algorithm for the decision problem, we could apply it to the graph
with various vertices removed, repeatedly narrowing the focus until we found the
desired clique, thereby solving the search problem is polynomial time.
However, there is a particular class of problems for which the search problem
is potentially quite hard, even though the decision problem is trivial because the
answer is known to be yes. This class of search problems is called TFNP. (The
’T’ stands for “total.”) There are some “trivial” decision problems that give rise
to quite famous problems in this class:
• “Does a integer have a prime factorization?” Testing primality can now be
done in polynomial time, but there is still no polynomial time algorithm for
factoring.
• “Given a set of positive integers {a1 , . . . , an } with ai < 2n /n for all i, do there
exist two different subsets with the same sum?” There are 2n different subsets,
and the sum of any one of them is less than 2n − n + 1, so the pigeonhole
principle implies that the answer is certainly yes.
• “Does a two person game have sets of pure strategies for the agents that are
the supports4 of a Nash equilibrium?” Verifying that a pair of sets are the
support of a Nash equilibrium is a computation involving linear algebra and a
small number of inequality verifications that can be performed in polynomial
time.
Problems involving a function defined on some large space must be specified
with a bit more care, because if the function is given by listing its values, then the
problem is easy, relative to the size of the input, because the input is huge. Instead,
one takes the input to be a Turing machine that computes (in polynomial time) the
value of the function at any point in the space.
4
The support of a mixed strategy is the set of pure strategies that are assigned positive
probability.
64
CHAPTER 3. COMPUTING FIXED POINTS
• “Given a Turing machine that computes a real valued function at every vertex
of a graph, is there a vertex where the function’s value is at least as large as
the function’s value at any of the vertex’ neighbors in the graph?” Since the
graph is finite, the function has a global maximum and therefore at least one
local maximum.
• “Given a Turing machine that computes the value of a Sperner labelling at
any vertex in a triangulation of the simplex, does there exist a completely
labelled subsimplex?”
Mainly because the class of problems in NP that always have a positive answer is
defined in terms of a property of the outputs, rather than a property of the inputs
(but also in part because factoring seems so different from the other problems)
experts expect that TFNP does not contain any problems that are complete for
the class. In view of this, trying to study the class as a whole is unlikely to be
very fruitful. Instead, it makes sense to define and study coherent subclasses, and
Papadimitriou (1994b) advocates defining subclasses in terms of the proof that a
solution exists. Thus PPP (“polynomial pigeonhole principle”) is (roughly) the
class of problems for which existence is guaranteed by the pigeonhole principle, and
PLS (“polynomial local search”) is (again roughly) the set of problems requesting
a local maximum of a real valued function defined on a graph by a Turing machine.
For us the most important subclass of TFNP is PPAD (“polynomial parity
argument directed”) which is defined by abstracting certain features of the algorithms we have seen in this chapter. The computational problem EOTL (“end of
the line”) is defined by a Turing machine that defines a directed graph5 of maximal
degree two in a space that may, without loss of generality, be taken to be the set
{0, 1}k of bit strings of length k, where k is bounded by a polynomial function of
the size of the input. For each v ∈ {0, 1}k the Turing machine specifies whether v is
a vertex in the graph. If it is, the Turing machine computes its predecessor, if it has
one, and its successor, if it has one. When it exists, the predecessor of v must be a
vertex, and its successor must be v. Similarly, when v has a successor, it must be a
vertex, and its predecessor must be v. Finally, we require that (0, . . . , 0) is a vertex
that has a successor but no predecessor. The problem is to find another “leaf” of
the graph, by which we mean either a vertex with a predecessor but no successor,
or a vertex with a successor but no predecessor. Of course the existence of such a
leaf follows from Lemma 2.6.1, generalized in the obvious way to handle directed
graphs. The class of computational problems that have reductions to EOTL is
PPAD (“polynomial parity problem directed”).
The Lemke-Howson algorithm passes from a two person game to an instance
of EOTL, then solves it by following the path in the graph to its other endpoint.
Similarly, the Scarf algorithm has as input the algorithms for navigating in a triangulation of ∆d−1 and generating the labels of the vertices, and if follows a path in
a graph from one endpoint to another. (It would be difficult to describe homotopy
in exactly these terms, but there is an obvious sense in which it has this character.)
5
A directed graph is a pair G = (V, E) where V is a finite set of vertices and E is a finite
set of ordered pairs of elements of V . That is, in a directed graph each edge has a source and a
target.
3.7. REMARKS ON COMPUTATION
65
There is a rather subtle point that is worth mentioning here. In our descriptions
of Lemke-Howson, Scarf, and homotopy, we implicitly assumed that the algorithm
used its memory of where it had been to decide which direction to go in the graph,
but the definition of EOTL requires that the graph be directed, which means in
effect that if we begin at any point on the path, we can use local information to decide which of the two directions in the graph constitutes forward motion. It turns
out that each of our three algorithms has this property; a proper explanation of
this would require more information about orientation than we have developed at
this point. The class of problems that can be reduced to the computational problem that has the same features as EOTL, except that the graph is undirected, is
PPA. Despite the close resemblance to PPAD, the theoretical properties of the
two classes differ in important ways.
In a series of rapid developments in 2005 and 2006 (Daskalakis et al. (2006);
Chen and Deng (2006b,a)) it was shown that computing a Nash equilibrium of a
two player game is PPAD-complete, and also that the two dimensional Sperner
problem is PPAD-complete. This means that computing a Nash equilibrium of a
two player game is almost certainly hard, in the sense that there is no polynomial
time algorithm for the problem, because computing general fixed points is almost
certainly hard. Since this breakthrough many other computational problems have
been shown to be PPAD-complete, including finding Walrasian equilibria in seemingly quite simple exchange economies. In various senses the problem does not go
away if we relax the problem, asking for a point that is ε-approximately fixed for
an ε that is significantly greater than zero.
The current state of theory presents a contrast between theoretical concepts
that classify even quite simple fixed point problems as intractable, and algorithms
that often produce useful results in a reasonable amount of time. A recent result
presents an even more intense contrast. The computational problem OEOTL has
the same given data as EOTL, but now the goal is to find the other end of the path
beginning at (0, . . . , 0), and not just any second leaf of the graph. Goldberg et al.
(2011) show that OETL is PSPACE-complete, even though the Lemke-Howson
algorithm, the Scarf algorithm, and many specific instances of homotopy procedures
can be recrafted as algorithms for OEOTL.
Recent developments have led to a rich and highly interesting theory explaining
why the problem of finding an approximate fixed point is intractable, in the sense
that there is almost certainly no algorithm that always finds an approximate fixed
point in a small amount of time. What is missing at this point are more tolerant
theoretical concepts that give an account of why the algorithms that exist are as
useful as they are in fact, and how they might be compared with each other, and
with theoretical ideals that have not yet been shown to be far out of reach.
Chapter 4
Topologies on Spaces of Sets
The theories of the degree and the index involve a certain kind of continuity
with respect to the function or correspondence in question, so we need to develop
topologies on spaces of functions and correspondences. The main idea is that one
correspondence is close to another if its graph is close to the graph of the second
correspondence, so we need to have topologies on spaces of subsets of a given space.
In this chapter we study such spaces of sets, and in the next chapter we apply these
results to spaces of functions and correspondences. There are three basic set theoretic operations that are used to construct new functions or correspondences from
given ones, namely restriction to a subdomain, cartesian products, and composition, and our agenda here is to develop continuity results for elementary operations
on sets that will eventually support continuity results for those operations.
To begin with Section 4.1 reviews some basic properties of topological spaces
that hold automatically in the case of metric spaces. In Section 4.2 we define
topologies on spaces of compact and closed subsets of a general topological space.
Section 4.3 presents a nice result due to Vietoris which asserts that for one of these
tolopogies the space of nonempty compact subsets of a compact space is compact.
Economists commonly encounter this in the context of a metric space, in which
case the topology is induced by the Hausdorff distance; Section 4.4 clarifies the
connection. In Section 4.5 we study the continuity properties of basic operations
for these spaces. Our treatment is largely drawn from Michael (1951) which contains
a great deal of additional information about these topologies.
4.1
Topological Terminology
Up to this point the only topological spaces we have encountered have been
subsets of Euclidean spaces. Now it will be possible that X lacks some of the
properties of metric spaces, in part because we may ultimately be interested in some
spaces that are not metrizable, but also in order to clarify the logic underlying our
result.
Throughout this chapter we work with a fixed topological space X. We say that
X is:
(a) a T1 -space if, for each x ∈ X, {x} is closed;
66
4.2. SPACES OF CLOSED AND COMPACT SETS
67
(b) Hausdorff if any two distinct points have disjoint neighborhoods;
(c) regular if every neighborhood of a point contains a closed neighborhood of
that point;
(d) normal if, for any two disjoint closed sets C and D, there are disjoint open
sets U and V with C ⊂ U and D ⊂ V .
In a Hausdorff space the complement of a point is a neighborhood of every other
point, so a Hausdorff space is T1 . It is an easy exercise to show that a metric space
is normal and T1 . Evidently a normal T1 space is Hausdorff and regular.
A collection B of subsets of X is a base of a topology if the open sets are all
unions of elements of B. Note that B is a base of a topology if and only if all the
elements of B are open and the open sets are those U ⊂ X such that for every
x ∈ U there there is a V ∈ B with x ∈ V ⊂ U. We say that B is a subbase of
the topology if the open sets are the unions of finite intersections of elements of B.
Equivalently, each element of B is open and for each open U and x ∈ U there are
V1 , . . . , Vk ∈ B such that x ∈ V1 ∩ · · · ∩ Vk ⊂ U.
It is often easy to define or describe a topology by specifying a subbase—in which
case we way that the topology of X is generated by B—so we should understand
what properties a collection B of subsets of X has to have in order for this to
work. Evidently the collection of all unions of finite intersections of elements of B
is closed under finite intersection and arbitrary union. We may agree, as a matter
of convention if you like, that the empty set is a finite intersection of elements of
B. Then the only real requirement is that the union of all elements of B is X, so
that X itself is closed.
4.2
Spaces of Closed and Compact Sets
There will be a number of topologies, and in order to define them we need the
corresponding subbases. For each open U ⊂ X let:
• ŨU = { K ⊂ U : K is compact };
• UU = ŨU \ {∅};
• VU = { K ⊂ X : K is compact and K ∩ U 6= ∅ };
• ŨU0 = { C ⊂ U : C is closed };
• UU0 = ŨU0 \ {∅};
• VU0 = { C ⊂ X : C is closed and C ∩ U 6= ∅ }.
We now have the following spaces:
• K̃(X) is the space of compact subsets of X endowed with the topology generated by the subbase { ŨU : U ⊂ X is open }.
68
CHAPTER 4. TOPOLOGIES ON SPACES OF SETS
• K(X) is the space of nonempty compact subsets of X endowed with the
subspace topology inherited from K̃(X).
• H(X) is the space of nonempty compact subsets of X endowed with the
topology generated by the subbase
{ UU : U ⊂ X is open } ∪ { VU : U ⊂ X is open }.
• K̃0 (X) is the space of closed subsets of X endowed with the topology generated
by the base { ŨU0 : U ⊂ X is open }.
• K0 (X) is the space of nonempty closed subsets of X endowed with the subspace topology inherited from K̃0 (X).
• H0 (X) is the space of nonempty closed subsets of X endowed with the topology generated by the subbase
{ UU0 : U ⊂ X is open } ∪ { VU0 : U ⊂ X is open }.
The topologies of H(X) and H0 (X) are both called the Vietoris topology.
Roughly, a neighborhood of K in K̃(X) or K(X) consists of those K ′ that
are close to K in the sense that every point in K ′ is close to some point of K.
A neighborhood of K ∈ H(X) consists of those K ′ that are close in this sense,
and also in the sense that every point in K is close to some point of K ′ . Similar
remarks pertain to K̃0 (X), K0 (X), and H0 (X). Section 4.4 develops these intuitions
precisely when X is a metric space.
Compact subsets of Hausdorff spaces are closed, so “for practical purposes” (i.e.,
when X is Hausdorff) every compact set is closed. In this case K̃(X), K(X), and
H(X) have the subspace topologies induced by the topologies of K̃0 (X), K0 (X), and
H0 (X). Of course it is always the case that K(X) and K0 (X) have the subspace
topologies induced by K̃(X) and K̃0 (X) respectively.
It is easy to see that { UU : U ⊂ X is open } is a base for K(X) and { UU0 :
U ⊂ X is open } is a base for K0 (X). Also, for any open U1 , . . . , Uk we have
ŨU1 ∩ . . . ∩ ŨUk = ŨU1 ∩...∩Uk ,
and similarly for UU , ŨU0 , and UU0 , so the subbases of K̃(X), K(X), K̃0 (X), and
K0 (X) are actually bases.
4.3
Vietoris’ Theorem
An interesting fact, which was proved already in Vietoris (1923), and which is
applied from time to time in mathematical economics, is that H(X) is compact
whenever X is compact. We begin the argument with a technical lemma.
Lemma 4.3.1. If X has a subbase such that any cover of X by elements of the
subbase has a finite subcover, then X is compact.
69
4.4. HAUSDORFF DISTANCE
Proof. Say that a set is basic if it is a finite intersection of elements of the subbasis.
Any open cover is refined by the collection of basic sets that are subsets of its
elements. If a refinement of an open cover has a finite subcover, then so does the
cover, so it suffices to show that any open cover of X by basic sets has a finite
subcover.
A collection of open covers is a chain if it is completely ordered by inclusion:
for any two covers in the chain, the first is a subset of the second or vice versa. If
each open cover in a chain consists of basic sets, and has no finite subcover, then
the union of the elements of the chain also has these properties (any finite subset
of the union is contained in some member of the chain) so Zorn’s lemma implies
that if there is one open cover with these properties, then there is a maximal such
cover, say {Uα : α ∈ A}.
Suppose, for some β ∈ A, that Uβ = V1 ∩ . . . ∩ Vn where V1 , . . . , Vn are in the
subbasis. If, for each i = 1, . . . , n, {Uα : α ∈ A} ∪ {Vi } has a finite subcover Ci ,
then each Ci \ {Vi } covers X \ Vi , so
(C1 \ {V1 }) ∪ . . . ∪ (Cn \ {Vn }) ∪ {Uβ }
is a finite subcover from {Uα : α ∈ A}. Therefore there is at least one i such that
{Uα : α ∈ A}∪{Vi } has no finite subcover, and maximality implies that Vi is already
in the cover. This argument shows that each element Uβ of the cover is contained
in a subbasic set that is also in the cover, so the subbasic sets in {Uα : α ∈ A} cover
X, and by hypothesis there must be a finite subcover after all.
Theorem 4.3.2. If X is compact, then H(X) is compact.
Proof. Suppose that { UUα : α ∈SA} ∪ { VVβ : β ∈ B} is an open cover of H(X)
by subbasic sets. Let D := X \ β Vβ ; since D is closed and X is compact, D is
compact. We may assume that D is nonempty because otherwise X = Vβ1 ∪. . .∪Vβn
for some β1 , . . . , βn , in which case H(X) = VVβ1 ∪ . . . ∪ VVβn . In addition, D must
be contained in some Uα because otherwise D would not be an element of any UUα
or any VVβ . But then {Uα } ∪ {Vβ : β ∈ B} has a finite subcover, so, for some
β1 , . . . , βn , we have
H(X) = UUα ∪ VVβ1 ∪ . . . ∪ VVβn .
4.4
Hausdorff Distance
Economists sometimes encounter spaces of compacts subsets of a metric space,
which are frequently topologized with the Hausdorff metric. In this section we clarify
the relationship between that approach and the spaces introduced above. Suppose
that X is a metric space with metric d. For nonempty compact sets K, L ⊂ X let
δK (K, L) := max min d(x, y).
x∈K y∈L
Then for any K and ε > 0 we have
{ L : δK (L, K) < ε } = { L : L ⊂ Uε (K) } = UUε (K) .
(∗)
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CHAPTER 4. TOPOLOGIES ON SPACES OF SETS
On the other hand, whenever K ⊂ U with K compact and U open there is some
ε > 0 such that Uε (K) ⊂ U (otherwise we could take sequences x1 , x2 , . . . in L
and y1 , y2 , . . . in X \ U with d(xi , yi ) → 0, then take convergent subsequences) so
{ L : δK (L, K) < ε } ⊂ UU . Thus:
Lemma 4.4.1. When X is a metric space, the sets of the form { L : δK (L, K) < ε }
constitute a base of the topology of K(X).
The Hausdorff distance between nonempty compact sets K, L ⊂ X is
δH (K, L) := max{δK (K, L), δK (L, K)}.
This is a metric. Specifically, it is evident that δH (K, L) = δH (L, K), and that
δH (K, L) = 0 if and only if K = L. If M is a third compact set, then
δK (K, M) ≤ δK (K, L) + δK (L, M),
from which it follows easily that the Hausdorff distance satisfies the triangle inequality.
There is now an ambiguity in our notation, insofar as Uε (L) might refer either to
the the union of the ε-balls around the various points of L or to the set of compact
sets whose Hausdorff distance from L is less than ε. Unless stated otherwise, we
will always interpret it in the first way, as a set of points and not as a set of sets.
Proposition 4.4.2. The Hausdorff distance induces the Vietoris topology on H(X).
Proof. Fix a nonempty compact K. We will show that any neighborhood of K in
one topology contains a neighborhood in the other topology.
S
First consider some ε > 0. Choose x1 , . . . , xn ∈ K such that K ⊂ i Uε/2 (xi ).
If L ∩ Uε/2 (xi ) 6= ∅ for all i, then δK (L, K) < ε, so, in view of (∗),
K ∈ UUε (K) ∩ VUε/2 (x1 ) ∩ . . . ∩ VUε/2 (xn ) ⊂ { L : δH (K, L) < ε }.
We now show that any element of our subbasis for the Vietoris topology contains
{ L : δH (K, L) < ε } for some ε > 0. If U is an open set containing K, then (as we
argued above) Uε (K) ⊂ U for some ε > 0, so that
K ∈ { L : δH (L, K) < ε } ⊂ { L : δK (L, K) < ε } ⊂ UU .
If V is open with K ∩ V 6= ∅, then we can choose x ∈ K ∩ V and ε > 0 small enough
that Uε (x) ⊂ V . Then
K ∈ { L : δH (K, L) < ε } ⊂ { L : δK (K, L) < ε } ⊂ VV .
4.5. BASIC OPERATIONS ON SUBSETS
4.5
71
Basic Operations on Subsets
In this section we develop certain basic properties of the topologies defined in
Section 4.2. To achieve a more unified presentation, it will be useful to let T denote
a generic element of {K̃, K, H, K̃0 , K0 , H0 }. This is, T (X) will denote one of the
spaces K̃(X), K(X), H(X), K̃0 (X), H0 (X), and H0 (X), with the range of allowed
interpretations indicated in each context. Similarly, W will denote a generic element
of {Ũ, U, V, Ũ 0 , U 0 , V 0 }.
We will frequently apply the following simple fact.
Lemma 4.5.1. If Y is a second topological space, f : Y → X is a function, and B
is a subbase for X such that f −1 (V ) is open for every V ∈ B, then f is continuous.
T
T
Proof. For any sets S1 , . . . , Sk ⊂ X we have f −1 (S i Si ) =S i f −1 (Si ), and for
any collection {Ti }i∈I of subsets of X we have f −1 ( i Ti ) = i f −1 (Ti ). Thus the
preimage of a union of finite intersections of elements of B is open, because it is a
union of finite intersections of open subsets of Y .
4.5.1
Continuity of Union
The function taking a pair of sets to their union is as well behaved as one might
hope.
Lemma 4.5.2. For any T ∈ {K̃, K, H, K̃0 , K0 , H0 } the function υ : (K1 , K2 ) 7→
K1 ∪ K2 is a continuous function from T (X) × T (X) to T (X).
Proof. Applying Lemma 4.5.1, it suffices to show that preimages of subbasic open
sets are open. For T ∈ {K̃, K, K̃0 , K0 } it suffices to note that
υ −1(WU ) = WU × WU
for all four W ∈ {Ũ, U, Ũ 0 , U 0 }. For T ∈ {H, H0 } we also need to observe that
υ −1(WU ) = (WU × H(X)) ∪ (H(X) × WU )
for both W ∈ {V, V 0 }.
4.5.2
Continuity of Intersection
Simple examples show that intersection is not a continuous operation for the
topologies H and H0 , so the only issues here concern K̃, K, K̃0 , and K0 .
Lemma 4.5.3. If A ⊂ X is closed, the function K 7→ K ∩ A from K̃A (X) to K̃(A)
0
and the function C 7→ C ∩ A from K̃A
(X) to K̃0 (A) are continuous.
Proof. If V ⊂ A is open, then the set of compact K such that K ∩ A ⊂ V is
UV ∪(X\A) . This establishes the first asserted continuity, and a similar argument
establishes the second.
72
CHAPTER 4. TOPOLOGIES ON SPACES OF SETS
0
For a nonempty closed set A ⊂ X let KA (X) and KA
(X) be the sets of compact
and closed subsets of X that have nonempty intersection with A. Since the topologies of K(X) and K0 are the subspace topologies inherited from K̃(X) and K̃0 (X),
last result has the following immediate consequence.
Lemma 4.5.4. The function K 7→ K ∩ A from KA (X) to K(A) and the function
0
C 7→ C ∩ A from KA
(X) to K0 (A) are continuous.
Joint continuity of the map (C, D) 7→ C ∩ D requires an additional hypothesis.
Lemma 4.5.5. If X is a normal space, then ι : (C, D) 7→ C ∩ D is a continuous
function from K̃0 (X) × K̃0 (X) to K̃0 (X). If, in addition, X is a T1 space, then
ι : K̃(X) × K̃(X) → K̃(X) is continuous.
Proof. By Lemma 4.5.1 it suffices to show that, for any open U ⊂ X, ι−1 (UU0 ) is
open. For any (C, D) in this set normality implies that there are disjoint open sets
V and W containing C \ U and D \ U respectively. Then (U ∪ V ) ∩ (U ∪ W ) = U,
so
(C, D) ∈ (UU0 ∪V × UU0 ∪W ) ∩ I 0 (X) ⊂ ι−1 (UU0 ).
If X is also T1 , it is a Hausdorff space, so compact sets are closed. Therefore
ι : K̃(X) × K̃(X) → K̃(X) is continuous because its domain and range have the
subspace topologies inherited from K̃0 (X) × K̃0 (X) and K̃0 (X).
Let I(X) (resp. I 0 (X)) be the set of pairs (K, L) of compact (resp. closed)
subsets of X such that K ∩ L 6= ∅, endowed with the topology it inherits from the
product topology of K(X) × K(X) (resp. K0 (X) × K0 (X)). The relevant topologies
are relative topologies obtained from the spaces in the last result, so:
Lemma 4.5.6. If X is a normal space, then ι : (C, D) 7→ C ∩ D is a continuous
function from I 0 (X) to K0 (X). If, in addition, X is a T1 space, then ι : I(X) →
K(X) is continuous.
4.5.3
Singletons
Lemma 4.5.7. The function η : x 7→ {x} is a continuous function from X to
T (X) when T ∈ {K, H}. If, in addition, X is a T1 -space, then it is continuous
when T ∈ {K0 , H0 }.
Proof. Singletons are always compact, so for any open U we have η −1 (UU ) =
η −1 (VU ) = U. If X is T1 , then singletons are closed, so η −1 (UU0 ) = η −1 (VU0 ) = U.
4.5.4
Continuity of the Cartesian Product
In addition to X, we now let Y be another given topological space. A simple
example shows that the cartesian product π 0 : (C, D) 7→ C × D is not a continuous
function from H0 (X) × H0 (Y ) to H0 (X × Y ). Suppose X = Y = R, (C, D) =
(X, {0}), and
W = { (x, y) : |y| < (1 + x2 )−1 }.
4.5. BASIC OPERATIONS ON SUBSETS
73
It is easy to see that there is no neighborhood V ⊂ H0 (Y ) of D such that π 0 (C, D ′ ) ∈
UW (that is, R × D ′ ⊂ W ) for all D ′ ∈ V .
For compact sets there are positive results. In preparation for them we recall a
basic fact about the product topology.
Lemma 4.5.8. If K ⊂ X and L ⊂ Y are compact, and W ⊂ X × Y is a neighborhood of K × L, then there are neighborhoods U of K and V of L such that
U × V ⊂ W.
Proof. By the definition of the product topology, for each (x, y) ∈ K × L there
are neighborhoods U(x,y) and V(x,y) of x and y such that
S U(x,y) × V(x,y) ⊂ W . For
each x ∈ KT we can find y1 , . . . , yn such that L ⊂ Vx := j V(x,yj ) , and
S we can then
let UxT:= j U(x,yj ) . Now choose x1 , . . . , xm such that K ⊂ U := i Uxi , and let
V := i Vxi .
Proposition 4.5.9. For T ∈ {K̃, K, H} the function π : (K, L) 7→ K × L is a
continuous function from T (X) × T (Y ) to T (X × Y ).
Proof. Let K ⊂ X and L ⊂ Y be compact. If W is a neighborhood of K × L and
U and V are open neighborhoods of K and L with U × V ⊂ W , then
(K, L) ∈ UU × UV ⊂ π −1 (UW ).
By Lemma 4.5.1, this establishes the asserted continuity when T ∈ {K̃, K}.
To demonstrate continuity when T = H we must also show that π −1 (VW ) is open
in H(X) × H(Y ) whenever W ⊂ X × Y is open. Suppose that (K × L) ∩ W 6= ∅.
Choose (x, y) ∈ (K × L) ∩ W , and choose open neighborhoods U and V of x and y
with U × V ⊂ W . Then
K × L ∈ VU × VV ⊂ π −1 (VW ).
4.5.5
The Action of a Function
Now fix a continuous function f : X → Y . Then f maps compact sets to
compact sets while f −1 (D) is closed whenever D ⊂ Y is closed. The first of these
operations is as well behaved as one might hope.
Lemma 4.5.10. If T ∈ {K̃, K, H}, then φf : K 7→ f (K) is a continuous function
from T (X) to T (Y ).
Proof. Preimages of subbasic open sets are open: for any open V ⊂ Y we have
φ−1
f (WV ) = Wf −1 (V ) for all W ∈ {Ũ, U, V}.
There is the following consequence for closed sets.
Lemma 4.5.11. If X is compact, Y is Hausdorff, and T ∈ {K̃, K, H}, then φf :
K 7→ f (K) is a continuous function from T 0 (X) to T 0 (Y ).
74
CHAPTER 4. TOPOLOGIES ON SPACES OF SETS
Proof. Recall that a closed subset of a compact space X is compact1 so that
T 0 (X) ⊂ T (X). As we mentioned earlier, T 0 (X) has the relative topologies induced by the topology of T (X), so the last result implies that φf is a continuous
function from T 0 (X) to T (Y ). The proof is completed by recalling that a compact
subset of a Hausdorff space Y is closed2 , so that T (Y ) ⊂ T 0 (Y ).
Since preimages of closed sets are closed, there is a well defined function ψf :
D 7→ f −1 (D) from K̃0 (Y ) to K̃0 (X). We need an additional hypothesis to guarantee
that it is continuous. Recall that a function is closed if it is continuous and maps
closed sets to closed sets.
Lemma 4.5.12. If f is a closed map, then ψf : D 7→ f −1 (D) is a continuous
function from K̃0 (Y ) to K̃0 (X).
Proof. For an open U ⊂ X, we claim that ψf−1 (UU0 ) = UY0 \f (X\U ) . First of all,
Y \ f (X \ U) is open because f is a closed map. If D ⊂ Y \ f (X \ U) is closed, then
f −1 (D) is a closed subset of U. Thus UY0 \f (X\U ) ⊂ ψf−1 (UU0 ). On the other hand,
if D ⊂ Y is closed and f −1 (D) ⊂ U, then D ∩ f (X \ U) = ∅. Thus ψf−1 (UU0 ) ⊂
UY0 \f (X\U ) .
Of course if f is closed and surjective, then ψf restricts to a continuous map
from K0 (Y ) to K0 (X). When X is compact and Y is Hausdorff, any continuous f :
X → Y is closed, because any closed subset of X is compact, so its image is compact
and consequently closed. Here is an example illustrating how the assumption that
f is closed is indispensable.
Example 4.5.13. Suppose 0 < ε < π, let X = (−ε, 2π + ε) and Y = { z ∈
C : |z| = 1 }, and let f : X → Y be the function f (t) := eit . The function
ψf : D 7→ f −1 (D) is discontinuous at D0 = { eit : ε ≤ t ≤ 2π − ε } because for any
open V containing D0 there are closed D ⊂ V such that f −1 (D) includes points far
from f −1 (D0 ) = [ε, 2π − ε].
4.5.6
The Union of the Elements
Whenever we have a set of subsets ofSsome space, we can take the union of its
elements. For any open U ⊂ X we have K∈UU K = U because for each x ∈ U, {x}
is compact. Since the sets UU are a base for the topology of K(X), it follows that
the union of all elements of an open subset of K(X) is open. If U and V1 , . . . , Vk
are open, then UU ∩ VV1 ∩ · · · ∩ VVk = ∅ if there is some j with U ∩ Vj = ∅, and
otherwise
{x, y1 , . . . , yk } ∈ UU ∩ VV1 ∩ · · · ∩ VVk
whenever x ∈ U and y1 ∈ V1 ∩ U, . . . , yk ∈ Vk ∩ U, so the union of all K ∈
UU ∩ VV1 ∩ · · · ∩ VVk is again U. Therefore the union of all the elements of an open
1
Proof: an open cover of the subset, together with its complement, is an open cover of the
space, any finite subcover of which yields a finite subcover of the subset.
2
Proof: fixing a point y in the complement of the compact set K, for each x ∈ K there are
disjoint neighborhoods of Ux of x and Vx of y, {Ux } is an open cover of K, and if Ux1 , . . . , Uxn is
a finite subcover, then Vx1 ∩ . . . ∩ Vxn is a neighborhood of y that does not intersect K.
4.5. BASIC OPERATIONS ON SUBSETS
75
subset of H(X) is open. If X is either T1 or regular, then similar logic shows that
for either T ∈ {K0 , H0 } the union of the elements of an open subset of T (X) is
open.
If a subset C of H(X) or H0 (X) is compact, then it is automatically compact in
the coarser topology of K(X) or K0 (X). Therefore the following two results imply
the analogous claims for the H(X) and H0 (X), which are already interesting.
S
Lemma 4.5.14. If S ⊂ K(X) is compact, then L := K∈S K is compact.
Proof. Let {Uα : α ∈ A} be an open cover of L. For each K ∈ S let VK be the
union of the elements of some finite subcover. Then K ∈ UVK , so { UVK :SK ∈ S }
is an open cover of S; let UVK1 , . . . , UVKr be a finite subcover. Then L ⊂ ri=1 VKi ,
and the various sets from {Uα } that were united to form the VKi are the desired
finite subcover of L.
S
Lemma 4.5.15. If X is regular and S ⊂ K0 (X) is compact, then D := C∈S C is
closed.
Proof. We will show that X \ D is open; let x be a point in this set. Each element
of S is a closed set that does not contain x, so (since X is regular) it is an element
0
of UX\N
for some closed neighborhood N of x. Since S is compact we have S ⊂
0
0
UX\N1 ∪ . . . ∪ UX\N
for some N1 , . . . , Nk . Then N1 ∩ . . . ∩ Nk is a neighborhood
k
of x that does not intersect any element of S, so x is in the interior of X \ D as
desired.
Chapter 5
Topologies on Functions and
Correspondences
In order to study of robustness of fixed points, or sets of fixed points, with respect
to perturbations of the function or correspondence, one must specify topologies on
the relevant spaces of functions and correspondences. We do this by identifying
a function or correspondence with its graph, so that the topologies from the last
chapter can be invoked. The definitions of upper and lower semicontinuity, and their
basic properties, are given in Section 5.1. There are two topologies on the space of
upper semicontinuous correspondences from X to Y . The strong upper topology,
which is defined and discussed in Section 5.2, turns out to be rather poorly behaved,
and the weak upper topology, which is usually at least as coarse, is presented in
Section 5.3. When X is compact the strong upper topology coincides with the weak
upper topology.
We will frequently appeal to a perspective in which a homotopy h : X × [0, 1] →
Y is understood as a continuous function t 7→ ht from [0, 1] to the space of continuous functions from X to Y . Section 5.4 presents the underlying principle in full
generality for correspondences. The specializations to functions of the strong and
weak upper topologies are known as the strong topology and the weak topology
respectively. If X is regular, then the weak topology coincides with the compactopen topology, and when X is compact the strong and weak topologies coincide.
Section 5.5 discusses these matters, and presents some results for functions that are
not consequences of more general results pertaining to correspondences.
The strong upper topology plays an important role in the development of the
topic, and its definition provides an important characterization of the weak upper
topology when the domain is compact, but it does not have any independent significance. Throughout the rest of the book, barring an explicit counterindication, the
space of upper semicontinuous correspondences from X to Y will be endowed with
the weak upper topology, and the space of continuous functions from X to Y will be
endowed with the weak topology.
76
5.1. UPPER AND LOWER SEMICONTINUITY
5.1
77
Upper and Lower Semicontinuity
Let X and Y be topological spaces. Recall that a correspondence F : X → Y
maps each x ∈ X to a nonempty F (x) ⊂ Y . The graph of F is
Gr(F ) = { (x, y) ∈ X × Y : y ∈ F (x) }.
If each F (x) is compact (closed, convex, etc.) then F is compact valued
(closed valued, convex valued, etc.). We say that F is upper semicontinuous if it
is compact valued and, for any x ∈ X and open set V ⊂ Y containing F (x), there
is a neighborhood U of x such that F (x′ ) ⊂ V for all x′ ∈ U. When F is compact
valued, it is upper semi-continuous if and only if F −1 (UV ) is a open whenever V ⊂ Y
is open. Thus:
Lemma 5.1.1. A compact valued correspondence F : X → Y is upper semicontinuous if and only if it is continuous when regarded as a function from X to
K(Y ).
In economics literature the graph being closed in X × Y is sometimes presented
as the definition of upper semicontinuity. Useful intuitions and simple arguments
flow from this point of view, so we should understand precisely when it is justified.
Proposition 5.1.2. If F is upper semicontinuous and Y is a Hausdorff space, then
Gr(F ) is closed.
Proof. We show that the complement of the graph is open. Suppose (x, y) ∈
/ Gr(F ).
Since Y is Hausdorff, y and each point z ∈ F (x) have disjoint neighborhoods Vz
and Wz . Since F (x) is compact, F (x) ⊂ Wz1 ∪ · · · ∪ Wzk for some z1 , . . . , zk . Then
V := Vz1 ∩ · · · ∩ Vzk and W := Wz1 ∪ · · · ∪ Wzk are disjoint neighborhoods of y and
F (x) respectively. If U is a neighborhood of x with F (x′ ) ⊂ W for all x′ ∈ U, then
U × V is a neighborhood (x, y) that does not intersect Gr(F ).
If Y is not compact, then a compact valued correspondence F : X → Y with a
closed graph need not be upper semicontinuous. For example, suppose X = Y = R,
F (0) = {0}, and F (t) = {1/t} when t 6= 0.
Proposition 5.1.3. If Y is compact and Gr(F ) is closed, then F is upper semicontinuous.
Proof. Fix x ∈ X. Since (X × Y ) \ Gr(F ) is open, for each y ∈ Y \ V we can choose
neighborhoods
S Uy of x and Vy of y such that (Uy × Vy ) ∩ Gr(F ) = ∅. In particular,
Y \ F (x) = y∈Y \F (x) Vy is open, so F (x) is closed and therefore compact. Thus F
is compact valued.
Now fix an open neighborhood V of F (x). Since Y \ V is a closed subset of a
compact space, hence compact, there are y1 , . . . , yk such that Y \ V ⊂ Vy1 ∪. . . ∪Vyk .
Then F (x′ ) ⊂ V for all x′ ∈ Uy1 ∩ . . . ∩ Uyk .
Proposition 5.1.4. If F is upper semicontinuous and X is compact, then Gr(F )
is compact.
78CHAPTER 5. TOPOLOGIES ON FUNCTIONS AND CORRESPONDENCES
Proof. We have the following implications of earlier results:
• Lemma 4.5.7 implies that the function x 7→ {x} ∈ K(X) is continuous;
• Lemma 5.1.1 implies that F is continuous, as a function from X to K(Y );
• Proposition 4.5.9 states that (K, L) 7→ K × L is a continuous function from
K(X) × K(Y ) to K(X × Y ).
Together these imply that F̃ : x 7→ {x} × F (x) is continuous, as a function from
X to K(X × Y ). Since X is compact, it follows that
S F̃ (X) is a compact subset of
K(X × Y ), so Lemma 4.5.14 implies that Gr(F ) = x∈X F̃ (x) is compact.
We say that F is lower semicontinuous if, for each x ∈ X, y ∈ F (x), and
neighborhood V of y, there is a neighborhood U of x such that F (x′ ) ∩ V 6= ∅ for
all x′ ∈ U. If F is both upper and lower semi-continuous, then it is said to be
continuous. When F is compact valued, it is lower semicontinuous if and only if
F −1 (VV ) is open whenever V ⊂ Y is open. Combining this with Lemma 5.1.1 gives:
Lemma 5.1.5. A compact valued correspondence F : X → Y is continuous if and
only if it is continuous when regarded as a function from X to H(Y ).
5.2
The Strong Upper Topology
Let X and Y be topological spaces with Y Hausdorff, and let U(X, Y ) be the set
of upper semicontinuous correspondences from X to Y . Proposition 5.1.2 insures
that the graph of each F ∈ U(X, Y ) is closed, so there is an embedding F 7→ Gr(F )
of U(X, Y ) in K0 (X × Y ). The strong upper topology is the topology induced
by this embedding when the image has the subspace topology. Let US (X, Y ) be
U(X, Y ) endowed with this topology. Since {UV0 : V ⊂ X × Y is open } is a subbase
for K0 (X × Y ), there is a subbase of US (X, Y ) consisting of the sets of the form
{ F : Gr(F ) ⊂ V }.
Naturally the following result is quite important.
Theorem 5.2.1. If Y is a Hausdorff space and X is a compact subset of Y , then
F P : US (X, Y ) → K̃(X)
is continuous.
Proof. Since Y is Hausdorff, X and ∆ = { (x, x) : x ∈ X } are closed subsets of
Y and X × Y respectively. For each F ∈ US (X, Y ), F P(F ) is the projection of
Gr(F ) ∩ ∆ onto the first coordinate. Since Gr(F ) is compact (Proposition 5.1.4) so
is Gr(F )∩∆, and the projection is continuous, so F P(F ) is compact. The definition
of the strong topology implies that Gr(F ) is a continuous function of F . Since ∆ is
closed in X × Y , Lemma 4.5.3 implies that Gr(F ) ∩ ∆ is a continuous function of F ,
after which Lemma 4.5.10 implies that F P(F ) is a continuous function of F .
5.2. THE STRONG UPPER TOPOLOGY
79
The basic operations for combining given correspondences to create new correspondences are restriction to a subset of the domain, cartesian products, and
composition. We now study the continuity of these constructions.
Lemma 5.2.2. If A is a closed subset of X, then the map F 7→ F |A is continuous
as a function from US (X, Y ) to US (A, Y ).
Proof. Since A × Y is a closed subset of X × Y , continuity as a function from
US (X, Y ) to US (A, Y )—that is, continuity of Gr(F ) 7→ Gr(F ) ∩ (A × Y )—follows
immediately from Lemma 4.5.4.
An additional hypothesis is required to obtain continuity of restriction to a
compact subset of the domain, but in this case we obtain a kind of joint continuity.
Lemma 5.2.3. If X is regular, then the map (F, K) 7→ Gr(F |K ) is a continuous
function from US (X, Y ) × K(X) to K(X × Y ). In particular, for any fixed K the
map F 7→ F |K is a continuous function from US (X, Y ) to US (K, Y ).
Proof. Fix F ∈ US (X, Y ), K ∈ K(X), and an open neighborhood W of Gr(F |K ).
For each x ∈ K Lemma 4.5.8 gives neighborhoods Ux of x and Vx of F (x) with
Ux × Vx ⊂ W . Choose x1 , . . . , xk such that U := Ux1 ∪ . . . ∪ Uxk contains K.
Since X is regular, each point in K has a closed neighborhood contained in U, and
the interiors of finitely many of these cover K, so K has a closed neighborhood C
contained in U. Let
W ′ := (Ux1 × Vx1 ) ∪ . . . ∪ (Uxk × Vxk ) ∪ ((X \ C) × Y ).
Then (K, Gr(F )) ∈ Uint C × UW ′ , and whenever (K ′ , Gr(F ′ )) ∈ Uint C × UW ′ we have
Gr(F ′|K ′ ) ⊂ W ′ ∩ (C × Y ) ⊂ (Ux1 × Vx1 ) ∪ . . . ∪ (Uxk × Vxk ) ⊂ W.
Let X ′ and Y ′ be two other topological spaces with Y ′ Hausdorff. Since the map
(C, D) 7→ C × D is not a continuous operation on closed sets, we should not expect
the function (F, F ′ ) 7→ F ×F ′ from US (X, Y )×US (X ′ , Y ′ ) to US (X×X ′ , Y ×Y ′ ) to be
continuous, and indeed, after giving the matter a bit of thought, the reader should
be able to construct a neighborhood of the graph of the function (x, x′ ) 7→ (0, 0)
that shows that the map (F, F ′) 7→ F × F ′ from US (R, R) × US (R, R) to US (R2 , R2 )
is not continuous.
We now turn our attention to composition. Suppose that, in addition to X and
Y , we have a third topological space Z that is Hausdorff. (We continue to assume
that Y is Hausdorff.) We can define a composition operation from (F, G) 7→ G ◦ F
from U(X, Y ) × U(Y, Z) to U(X, Z) by letting
[
G(F (x)) :=
F (y).
y∈F (x)
That is, G(F (x)) is the projection onto Z of Gr(G|F (x)), which is compact by
Proposition 5.1.4, so G(F (x)) is compact. Thus G ◦ F is compact valued. To show
80CHAPTER 5. TOPOLOGIES ON FUNCTIONS AND CORRESPONDENCES
that G◦F is upper semicontinuous, consider an x ∈ X, and let W be a neighborhood
′
of G(F (x)). For each y ∈ F (x)
S there is open neighborhood Vy such that G(y ) ⊂ W
′
for all y ∈ Vy . Setting V := y∈F (x) Vy , we have G(y) ⊂ W for all y ∈ V . If U is a
neighborhood of x such that F (x′ ) ⊂ V for all x′ ∈ U, then G(F (x′ )) ⊂ W for all
x′ ∈ U.
We can also define G ◦ F to be the correspondence whose graph is
πX×Z ((Gr(F ) × Z) ∩ (X × Gr(G)))
where πX×Z : X × Y × Z → X × Z is the projection. This definition involves set
operations that are not continuous, so we should suspect that (F, G) 7→ G ◦ F is
not a continuous function from US (X, Y ) × US (Y, Z) to US (X, Z). For a concrete
example let X = Y = Z = R, and let f and g be the constant function with value
zero. If U and V are neighborhoods of the graph of f and g, there are δ, ε > 0 such
that (−δ, δ) × (−ε, ε) ⊂ V , and consequently the set of g ′ ◦ f ′ with Gr(f ′ ) ⊂ U and
Gr(g ′ ) ⊂ V contains the set of all constant functions with values in (−ε, ε), but of
course there are neighborhoods of the graph of g ◦ f that do not contain this set of
functions for any ε.
5.3
The Weak Upper Topology
As in the last section, X and Y are topological spaces with Y Hausdorff. There
is another topology on U(X, Y ) that is in certain ways more natural and better
behaved than the strong upper topology. Recall that if {Bi }i∈I is a collection of
topological spaces and { fi : A → Bi }i∈I is a collection of functions, the quotient
topology on A induced by this data is the coarsest topology such that each fi
is continuous. The weak upper topology on U(X, Y ) is the quotient topology
induced by the functions F 7→ F |K ∈ US (K, Y ) for compact K ⊂ X. Since a
function is continuous if and only if the preimage of every subbasic subset of the
range is open, a subbase for the weak upper topology is given by the sets of the
form { F : Gr(F |K ) ⊂ V } where K ⊂ X is compact and V is a (relatively) open
subset of K × Y .
Let UW (X, Y ) be U(X, Y ) endowed with the weak upper topology. As in the
last section, we study the continuity of basic constructions.
Lemma 5.3.1. If A is a closed subset of X, then the map F 7→ F |A is continuous
as a function from UW (X, Y ) to UW (A, Y ).
Proof. If A has the quotient topology induced by { fi : A → Bi }i∈I , then a function
g : Z → A is continuous if each composition fi ◦ g is continuous. (The sets of the
form fi−1 (Vi ), where Vi ⊂ Bi is open, constitute a subbase of the quotient topology,
so this follows from Lemma 4.5.1.) To show that the composition F 7→ F |A 7→ F |K
is continuous whenever K is a compact subset of A we simply observe that K is
compact as a subset of X, so this follows directly from the definition of the topology
of UW (X, Y ).
Lemma 5.3.2. If every compact set in X is closed (e.g., because X is Hausdorff )
then the topology of UW (X, Y ) is at least as coarse as the topology of US (X, Y ). If,
in addition, X is itself compact, then the two topologies coincide.
5.3. THE WEAK UPPER TOPOLOGY
81
Proof. We need to show that the identity map from US (X, Y ) to UW (X, Y ) is continuous, which is to say that for any given compact K ⊂ X, the map Gr(F ) →
Gr(F |K ) = Gr(F ) ∩ (K × Y ) is continuous. This follows from Lemma 5.3.1 because
K × Y is closed in X × Y whenever K is compact.
If X is compact, the continuity of the identity map from UW (X, Y ) to US (X, Y )
follows directly from the definition of the weak upper topology.
There is a useful variant of Lemma 5.2.3.
Lemma 5.3.3. If X is normal, Hausdorff, and locally compact, then the function
(K, F ) 7→ Gr(F |K ) is a continuous function from K(X) × UW (X, Y ) to K(X × Y ).
Proof. We will demonstrate continuity at a given point (K, F ) in the domain. Local
compactness implies that there is a compact neighborhood C of K. The map
F ′ 7→ F ′ |C from U(X, Y ) to US (C, Y ) is a continuous function by virtue of the
definition of the topology of U(X, Y ). Therefore Lemma 5.2.3 implies that the
composition (K ′ , F ′ ) → (K ′ , F ′ |C ) → Gr(F |K ′ ) is continuous, and of course it
agrees with the function in question on a neighborhood of (K, F ).
In contrast with the strong upper topology, for the weak upper topology cartesian products and composition are well behaved. Let X ′ and Y ′ be two other spaces
with Y ′ Hausdorff.
Lemma 5.3.4. If X and X ′ are Hausdorff, then the function (F, F ′ ) 7→ F × F ′
from UW (X, Y ) × UW (X ′ , Y ′ ) to UW (X × X ′ , Y × Y ′ ) is continuous.
Proof. First suppose that X and X ′ are compact. Then, by Proposition 5.1.4,
the graphs of upper semicontinuous functions with these domains are compact,
and continuity of the function (F, F ′) 7→ F × F ′ from US (X, Y ) × US (X ′ , Y ′ ) to
US (X × X ′ , Y × Y ′ ) follows from Proposition 4.5.9.
Because UW (X × X ′ , Y × Y ′ ) has the quotient topology, to establish the general case we need to show that (F, F ′ ) 7→ F × F ′ |C is a continuous function from
UW (X, Y ) × UW (X ′ , Y ′ ) to US (C, Y × Y ′ ) whenever C ⊂ X × X ′ is compact. Let
K and K ′ be the projections of C onto X and X ′ respectively; of course these sets
are compact. The map in question is the composition
(F, F ′ ) → (F |K , F ′ |K ′ ) → F |K × F ′ |K ′ → (F |K × F ′ |K ′ )|C .
The continuity of the second map has already been established, and the continuity
of the first and third maps follows from Lemma 5.3.1, because compact subsets of
Hausdorff spaces are closed and products of Hausdorff spaces are Hausdorff1 .
Suppose that, in addition to X and Y , we have a third topological space Z that
is Hausdorff.
Lemma 5.3.5. If K ⊂ X is compact, Y is normal and locally compact, and X ×
Y × Z is normal, then
(F, G) 7→ Gr(G ◦ F |K )
is a continuous function from UW (X, Y ) × UW (Y, Z) to K(X × Z).
1
I do not know if the compact subsets of X × X ′ are closed when X and X ′ are compact spaces
whose compact subsets are closed.
82CHAPTER 5. TOPOLOGIES ON FUNCTIONS AND CORRESPONDENCES
Proof. The map F 7→ Gr(F |K ) is a continuous function from UW (X, Y ) to K(X ×Y )
by virtue of the definition of the weak upper topology, and the natural projection of
X × Y onto Y is continuous, so Lemma 4.5.10 implies that im(F |K ) is a continuous
function of (K, F ). Since Y is normal and locally compact, Lemma 5.3.3 implies
that (F, G) 7→ Gr(G|im(F |K ) ) is a continuous function from UW (X, Y ) × UW (Y, Z) to
K(X × Z), and again (F, G) 7→ im(G|im(F |K ) ) is also continuous. The continuity of
cartesian products of compact sets (Proposition 4.5.9) now implies that
Gr(F |K ) × im(G|im(F |K ) ) and K × Gr(G|im(F |K ) )
are continuous functions of (K, F, G). Since X is T1 while Y and Z are Hausdorff,
X × Y × Z is T1 , so Lemma 4.5.6 implies that the intersection
{ (x, y, z) : x ∈ K, y ∈ F (x), and z ∈ G(y) }
of these two sets is a continuous function of (K, F, G), and Gr(G ◦ F |K ) is the
projection of this set onto X × Z, so the claim follows from another application of
Lemma 4.5.10.
As we explained in the proof of Lemma 5.3.1, the continuity of (F, G) 7→ G◦F |K
for each compact K ⊂ X implies that (F, G) 7→ G ◦ F is continuous when the range
has the weak upper topology, so:
Proposition 5.3.6. If X is T1 , Y is normal and locally compact, and X × Y × Z is
normal, then (F, G) 7→ G ◦ F is a continuous function from UW (X, Y ) × UW (Y, Z)
to UW (X, Z).
5.4
The Homotopy Principle
Let X, Y , and Z be topological spaces with Z Hausdorff, and fix a compact
valued correspondence F : X × Y → Z. For each x ∈ X let Fx : Y → Z be
the derived correspondence y 7→ F (x, y). Motivated by homotopies, we study the
relationship between the following two conditions:
(a) x 7→ Fx is a continuous function from X to US (Y, Z);
(b) F is upper semi-continuous.
If F : X × Y → Z is upper semicontinuous, then x 7→ Fx will not necessarily be
continuous without some additional hypothesis. For example, let X = Y = Z = R,
and suppose that F (0, y) = {0} for all y ∈ Y . Without F being in any sense poorly
behaved, it can easily happen that for x arbitrarily close to 0 the graph of Fx is not
contained in { (y, z) : |z| < (1 + y 2 )−1 }.
Lemma 5.4.1. If Y is compact and F is upper semicontinuous, then x 7→ Fx is a
continuous function from X to US (Y, Z).
5.5. CONTINUOUS FUNCTIONS
83
Proof. For x ∈ X let F̃x : Y → Y × Z be the correspondence F̃x (y) := {y} × Fx (y).
Clearly F̃x is compact valued and continuous as a function from Y to K(Y × Z).
Since Y isScompact, the image of F̃x is compact, so Lemma 4.5.14 implies that
Gr(Fx ) = y∈Y F̃x (y) is compact, and Lemma 4.5.15 implies that it is closed.
Since Z is a Hausdorff space, Proposition 5.1.2 implies that Gr(F ) is closed.
Now Proposition 5.1.3 implies that x 7→ Gr(Fx ) is upper semicontinuous, which is
the same (by Lemma 5.1.1) as it being a continuous function from X to K(Y × Z).
But since Gr(Fx ) is closed for all x, this is the same as it being a continuous function
from X to K0 (Y × Z), and in view of the definition of the topology of US (Y, Z),
this is the same as x 7→ Fx being continuous.
Lemma 5.4.2. If Y is regular and x 7→ Fx is a continuous function from X to
US (Y, Z), then F is upper semicontinuous.
Proof. Fix (x, y) ∈ X × Y and a neighborhood W ⊂ Z of F (x, y). Since Fx is upper
semicontinuous, there is neighborhood V of y such that F (x, y ′) ⊂ W for all y ′ ∈ V .
Applying the regularity of Y , let Ṽ be a closed neighborhood of y contained in V .
Since x 7→ Fx is continuous, there is a neighborhood U ⊂ X of x such that
Gr(Fx′ ) ⊂ (V × W ) ∪ ((Y \ Ṽ ) × Z)
for all x′ ∈ U. Then F (x′ , y ′) ⊂ W for all (x′ , y ′ ) ∈ U × Ṽ .
For the sake of easier reference we combine the last two results.
Theorem 5.4.3. If Y is regular and compact, then F is upper semicontinuous if
and only if x 7→ Fx is a continuous function from X to US (Y, Z).
5.5
Continuous Functions
If X and Y are topological spaces with Y Hausdorff, CS (X, Y ) and CW (X, Y )
will denote the space of continuous functions with the topologies induced by the
inclusions of C(X, Y ) in US (X, Y ) and UW (X, Y ). In connection with continuous
functions, these topologies are know as the strong topology and weak topology
respectively. Most of the properties of interest are automatic corollaries of our earlier
work; this section contains a few odds and ends that are specific to functions.
If K ⊂ X is compact and V ⊂ Y is open, let CK,V be the set of continuous
functions f such that f (K) ⊂ V . The compact-open topology is the topology
generated by the subbasis
{ CK,V : K ⊂ X is compact, V ⊂ Y is open },
and CCO (X, Y ) will denote the space of continuous functions from X to Y endowed
with this topology. The set of correspondences F : X → Y with Gr(F |K ) ⊂ K × V
is open in UW (X, Y ), so the compact-open topology is always at least as coarse as
the topology inherited from UW (X, Y ).
Proposition 5.5.1. Suppose X is regular. Then the compact-open topology coincides with the weak topology.
84CHAPTER 5. TOPOLOGIES ON FUNCTIONS AND CORRESPONDENCES
Proof. What this means concretely is that whenever we are given a compact K ⊂ X,
an open set W ⊂ K × Y , and a continuous f : X → Y with Gr(f |K ) ⊂ W , we can
find a compact-open neighborhood of f whose elements f ′ satisfy Gr(f ′ |K ) ⊂ W .
For each x ∈ K the definition of the product topology gives open sets Ux ⊂ K and
Vx ⊂ Y such that (x, f (x)) ∈ Ux × Vx ⊂ W . Since f is continuous, by replacing Ux
with a smaller open neighborhood if necessary, we may assume that f (Ux ) ⊂ Vx .
Since X is regular, x has a closed neighborhood Cx ⊂ Ux , and Cx is compact because
it is a closed subset of a compact set. Then f ∈ CCx ,Vx for each x. We can find
x1 , . . . , xn such that K = Cx1 ∪ . . . ∪ Cxn , and clearly Gr(f ′ |K ) ⊂ W whenever
f ′ ∈ CCx1 ,Vx1 ∩ . . . ∩ CCxn ,Vxn .
For functions there is a special result concerning continuity of composition.
Proposition 5.5.2. If X is compact and f : X → Y is continuous, then g 7→ g ◦ f
is a continuous function from CCO (Y, Z) → CCO (X, Z).
Proof. In view of the subbasis for the strong topology, it suffices to show, for a given
continuous g : Y → Z and an open V ⊂ X × Z containing the graph of g ◦ f , that
N = { (y, z) ∈ Y × Z : f −1 (y) × {z} ⊂ V }
is a neighborhood of the graph of g. If not, then some point (y, g(y)) is an accumulation point of points of the form (f (x′ ), z) where (x′ , z) ∈
/ V . Since X is compact,
it cannot be the case that for each x ∈ X there are neighborhoods A of x and B of
(y, g(y)) such that
{ (x′ , z) ∈ (A × Z) \ V : (f (x′ ), z) ∈ B } = ∅.
Therefore there is some x ∈ X such that for any neighborhoods A of x and B
of (y, g(y)) there is some x′ ∈ A and z such that (x′ , z) ∈
/ V and (f (x′ ), z) ∈ B.
Evidently f (x) = y. To obtain a contradiction choose neighborhoods A of x and
W of g(y) such that A × W ⊂ V , and set B = Y × W .
The following simple result, which does not depend on any additional assumptions on the spaces, is sometimes just what we need.
Proposition 5.5.3. If g : Y → Z is continuous, then f 7→ g ◦ f is a continuous
function from CS (X, Y ) to CS (X, Z).
Proof. If U ⊂ X × Z is open, then so is (IdX × g)−1(U).
Chapter 6
Metric Space Theory
In this chapter we develop some advanced results concerning metric spaces.
An important tool, partitions of unity, exist for locally finite open covers of
a normal space: this is shown in Section 6.2. But sometimes we will be given
a local cover that is not necessarily locally finite, so we need to know that any
open cover has a locally finite refinement. A space is paracompact if this is the
case. Paracompactess is studied in Section 6.1; the fact that metric spaces are
paracompact will be quite important.
Section 6.3 describes most of the rather small amount we will need to know
about topological vector spaces. Of these, the most important for us are the locally
convex spaces, which have many desirable properties. One of the larger themes of
this study is that the concepts and results of fixed point theory extend naturally to
this level of generality, but not further.
Two important types of topological vector spaces, Banach spaces and Hilbert
spaces, are introduced in Section 6.4. Results showing that metric spaces can be
embedded in such linear spaces are given in Section 6.5. Section 6.6 presents an
infinite dimensional generalization of the Tietze extension theorem due to Dugundji.
6.1
Paracompactness
Fix a topological space X. A family {Sα }α∈A of subsets of X is locally finite if
every x ∈ X has a neighborhood W such that there are only finitely many α with
W ∩ Sα 6= ∅. If {Uα }α∈A is a cover of X, a second cover {Vβ }β∈B is a refinement of
{Uα }α∈A if each Vβ is a subset of some Uα . The space X is paracompact if every
open cover is refined by an open cover that is locally finite. This section is devoted
to the proof of:
Theorem 6.1.1. A metric space is paracompact.
This result is due to Stone (1948). At first the proofs were rather complex, but
eventually Rudin (1969) found a brief and simple argument. A well ordering of a
set Z is a complete ordering ≤ such that any A ⊂ Z has a least element. That any
set Z has a well ordering is the assertion of the well ordering theorem, which
is a simple consequence of Zorn’s lemma. Let O be the set of all pairs (Z ′ , ≤′ )
where Z ′ ⊂ Z and ≤′ is a well ordering of Z. We order O by specifying that
85
86
CHAPTER 6. METRIC SPACE THEORY
(Z ′ , ≤′ ) (Z ′′ , ≤′′ ) if Z ′ ⊂ Z ′′ , ≤′ is the restriction of ≤′′ to Z ′ and z ′ ≤′′ z ′′ for
all z ′ ∈ Z ′ and z ′′ ∈ Z ′′ \ Z ′ . Any chain in O has an upper bound in O (just take
the union of all the sets and all the orderings) so Zorn’s lemma implies that O has
a maximal element (Z ∗ , ≤∗ ). If there was a z ∈ Z \ Z ∗ we could extend ≤∗ to
a well ordering of Z ∗ ∪ {z} by specifying that every element of Z ∗ is less than z.
This would contradict maximality, so we must have Z ∗ = Z. (The axiom of choice,
Zorn’s lemma, and the well ordering theorem are actually equivalent; cf. Kelley
(1955).)
Proof of Theorem 6.1.1. Let {Uα }α∈A be an open cover of X where A is a well
ordered set. We define sets Vαn for α ∈ A and n = 1, 2, . . ., inductively (over n) as
follows: let Vαn be the union of the balls U2−n (x) for those x such that:
(a) α is the least element of A such that x ∈ Uα ;
S
(b) x ∈
/ j<n,β∈A Vβj ;
(c) U3·2−n (x) ⊂ Uα .
For each x there is a least α such that x ∈ Uα and an n large enough that (c) holds,
so x ∈ Vαn unless x ∈ Vβj for some β and j < n. Thus {Vαn } is a cover of X, and
of course each Vαn is open and contained in Uα , so it is a refinement of {Uα }.
To prove that the cover is locally finite we fix x, let α be the least element of
A such that x ∈ Vαn for some n, and choose j such that U2−j (x) ⊂ Vαn . We claim
that U2−n−j (x) intersects only finitely many Vβi .
If i > j and y satisfies (a)-(c) with β and i in place of α and n, then U2−n−j (x) ∩
/ Vαn , and n + j, i ≥ j + 1. Therefore
U2−i (y) = ∅ because U2−j (x) ⊂ Vαn , y ∈
U2−n−j (x) ∩ Vβi = ∅.
For i ≤ j we will show that there is at most one β such that U2−n−j (x) intersects
Vβi . Suppose that y and z are points satisfying (a)-(c) for β and γ, with i in place
of j. Without loss of generality β preceeds γ. Then U3·2−i (y) ⊂ Uβ , z ∈
/ Uβ , and
n + j > i, so U2−n−j (x) cannot intersect both U2−i (y) and U2−i (z). Since this is
the case for all y and z, U2−n−j (x) cannot intersect both Vβi and Vγi .
6.2
Partitions of Unity
We continue to work with a fixed topological space X. This section’s central
concept is:
Definition 6.2.1. Let {Uα }α∈A be a locally finite open cover of X. A partition
of unity subordinate to {Uα } is a collection ofP
continuous functions {ψα : X →
[0, 1]} such that ψα (x) = 0 whenever x ∈
/ Uα and α∈A ψα (x) = 1 for each x.
The most common use of a partition of unity is to construct a global function
or correspondence with particular properties. Typically locally defined functions
or correspondences are given or can be shown to exist, and the global object is
constructed by taking a “convex combination” of the local objects, with weights
that vary continuously. Of course to apply this method one must have results
guaranteeing that suitable partitions of unity exist. Our goal in this section is:
87
6.2. PARTITIONS OF UNITY
Theorem 6.2.2. For any locally finite open cover {Uα }α∈A of a normal space X
there is a partition of unity subordinate to {Uα }.
A basic tool used in the constructive proof of this result, and many others, is:
Lemma 6.2.3 (Urysohn’s Lemma). If X is a normal space and C ⊂ U ⊂ X with C
closed and U open, then there is a continuous function ϕ : X → [0, 1] with ϕ(x) = 0
for all x ∈ C and ϕ(x) = 1 for all x ∈ X \ U.
Proof. Since X is normal, whenever C ′ ⊂ U ′ , with C ′ closed and U ′ open, there
exist a closed C ′′ and an open U ′′ such that C ′ ⊂ U ′′ , X \ U ′ ⊂ X \ C ′′ , and
U ′′ ∩ (X \ C ′′ ) = ∅, which is to say that C ′ ⊂ U ′′ ⊂ C ′′ ⊂ U ′ . Let C0 := C and
U1 := U. Choose an open U1/2 and a closed C1/2 with C0 ⊂ U1/2 ⊂ C1/2 ⊂ U1 .
Choose an open U1/4 and a closed C1/4 with C0 ⊂ U1/4 ⊂ C1/4 ⊂ U1/2 , and choose
an open U3/4 and a closed C3/4 with C1/2 ⊂ U3/4 ⊂ C3/4 ⊂ U1 . Continuing in
this fashion, we obtain a system of open sets Ur and a system of closed sets Cr for
rationals r ∈ [0, 1] of the form k/2m (except that C1 and U0 are undefined) with
Ur ⊂ Cr ⊂ Us ⊂ Cs whenever r < s.
For x ∈ X let
(
S
inf{ r : x ∈ Cr }, x ∈ r Cr
ϕ(x) :=
1,
otherwise.
Clearly ϕ(x) = 0 for all x ∈ C and ϕ(x) = 1 for all x ∈ X \ U. Any open subset
of [0, 1] is a union of finite intersections of sets of the form [0, a) and (b, 1], where
0 < a, b < 1, and
[
[
ϕ−1 [0, a) =
Ur and ϕ−1 (b, 1] = (X \ Cr )
r<a
r>b
are open, so ϕ is continuous.
Below we will apply Urysohn’s lemma to a closed subset of each element of a
locally finite open cover. We will need X to be covered by these closed sets, as per
the next result.
Proposition 6.2.4. If X is a normal space and {Uα }α∈A is a locally finite cover
of X, then there is an open cover {Vα }α∈A such that for each α, the closure of Vα
is contained in Uα .
Proof. A partial thinning of {Uα }α∈A is a function F from a subset B of A to the
open sets of X such that:
(a) for each β ∈ B, the closure of F (β) is contained in Uβ ;
S
S
(b) β∈B F (β) ∪ α∈A\B Uα = X.
Our goal is to find such an F with B = A. The partial thinnings can be partially
ordered as follows: F < G if the domain of F is a proper subset of the domain of
G and F and G agree on this set. We will show that this ordering has maximal
elements, and that the domain of a maximal element is all of A.
88
CHAPTER 6. METRIC SPACE THEORY
Let {Fι }ι∈I be a chain of partial thinnings. That is, for all distinctS ι, ι′ ∈ I,
either Fι < Fι′ or Fι′ < Fι . Let the domain of each Fι be Bι , let B := ι Bι , and
for β ∈ B let F (β) be the common value of Fι (β) for those ι with β ∈ Bι . For each
x ∈ X there is some ι with Fι (β) = F (β) for all β ∈ B such that x ∈ Uβ because
there are only finitely many α with x ∈ Uα . Therefore F satisfies (b). We have
shown that any chain of partial thinnings has an upper bound, so Zorn’s lemma
implies that the set of all partial thinnings has a maximal element.
If F is a partial thinning with domain B and α′ ∈ A \ B, then
[
[
Uα
F (β) ∪
X\
β∈B
α∈A\B,α6=α′
is a closed subset of Uα , so it has an open superset Vα′ whose closure is contained in
Uα . We can define a partial thinning G with domain B∪{α′ } by setting G(α′ ) := Vα′
and G(β) := F (β) for β ∈ B. Therefore F cannot be maximal unless its domain is
all of A.
Proof of Theorem 6.2.2. The result above gives a closed cover {Cα }α∈A of X with
Cα ⊂ Uα for each α. For each α let ϕα : X → [0, 1] be continuous
with ϕα (x) = 0
P
for all x ∈ X \ Uα and ϕα (x) = 1 for all x ∈ Cα . Then α ϕα is well defined
and continuous everywhere since {Uα } is locally finite, and it is positive everywhere
since {Cα } covers X. For each α ∈ A set
ϕα
.
α′ ϕα′
ψα := P
6.3
Topological Vector Spaces
Since we wish to develop fixed point theory in as much generality as is reasonably
possible, infinite dimensional vector spaces will inevitably appear at some point. In
addition, these spaces will frequently be employed as tools of analysis. The result
in the next section refers to such spaces, so this is a good point at which to cover
the basic definitions and elementary results.
A topological vector space V is a vector space over the real numbers1 that is
endowed with a topology that makes addition and scalar multiplication continuous,
and makes {0} a closed set. Topological vector spaces, and maps between them,
are the objects studied in functional analysis. Over the last few decades functional
analysis has grown into a huge body of mathematics; it is fortunate that our work
here does not require much more than the most basic definitions and facts.
We now lay out elementary properties of V . For any given w ∈ V the maps
v 7→ v + w and v 7→ v − w are continuous, hence inverse homeomorphisms. That
is, the topology of V is translation invariant. In particular, the topology of V is
1
Other fields of scalars, in particular the complex numbers, play an important role in functional
analysis, but have no applications in this book.
6.3. TOPOLOGICAL VECTOR SPACES
89
completely determined by a neighborhood base of the origin, which simplifies many
proofs.
The following facts are basic.
Lemma 6.3.1. If C ⊂ V is convex, then so is its closure C.
Proof. Aiming at a contradiction, suppose that v = (1 − t)v0 + tv1 is not in C even
though v0 , v1 ∈ C and 0 < t < 1. Let U be a neighborhood of v that does not
intersect C. The continuity of addition and scalar multiplication implies that there
are neighborhoods U0 and U1 of v0 and v1 such that (1 − t)v0′ + tv1′ ∈ U for all
v0′ ∈ U0 and v1′ ∈ U1 . Since U0 and U1 contain points in C, this contradicts the
convexity of C.
Lemma 6.3.2. If A is a neighborhood of the origin, then there is closed neighborhood of the origin U such that U + U ⊂ A.
Proof. Continuity of addition implies that there are neighborhoods of the origin
B1 , B2 with B1 + B2 ⊂ A, and replacing these with their intersection gives a neighborhood B such that B +B ⊂ A. If w ∈ B, then w −B intersects any neighborhood
of the origin, and in particular (w − B) ∩ B 6= ∅. Thus B ⊂ B + B ⊂ A. Applying
this argument again gives a closed neighborhood U of the origin with U ⊂ B.
We can now establish the separation properties of V .
Lemma 6.3.3. V is a regular T1 space, and consequently a Hausdorff space.
Proof. Since {0} is closed, translation invariance implies that V is T1 . Translation
invariance also implies that to prove regularity, it suffices to show that any neighborhood of the origin, say A, contains a closed neighborhood, and this is part of
what the last result asserts. As has been pointed out earlier, a simple and obvious
argument shows that a regular T1 space is Hausdorff.
We can say slightly more in this direction:
Lemma 6.3.4. If K ⊂ V is compact and U is a neighborhood of K, then there is
a closed neighborhood W of the origin such that K + W ⊂ U.
Proof. For each v ∈ K Lemma 6.3.2 gives a closed neighborhood Wv of the origin,
which is convex if V is locally convex, such that v + Wv + Wv ⊂ U. Then there are
v1 , . . . , vn such that v1 +Wv1 , . . . , vn +Wvn is a cover of K. Let W := Wv1 ∩. . .∩Wvn .
For any v ∈ K there is an i such that v ∈ vi + Wi , so that
v + W ⊂ vi + Wvi + Wvi ⊂ U.
A topological vector space is locally convex if every neighborhood of the origin
contains a convex neighborhood. In several ways the theory of fixed points developed in this book depends on local convexity, so for the most part locally convex
topological vector spaces represent the outer limits of generality considered here.
90
CHAPTER 6. METRIC SPACE THEORY
Lemma 6.3.5. If V is locally convex and A is a neighborhood of the origin, then
there is closed convex neighborhood of the origin W such that W + W ⊂ A.
Proof. Lemma 6.3.2 gives a closed neighborhood U of the origin such that U + U ⊂
A, the definition of local convexity gives a convex neighborhood of the origin W
that is contained in U. If we replace W with its closure, it will still be convex due
to Lemma 6.3.1.
6.4
Banach and Hilbert Spaces
We now describe two important types of locally convex spaces. A norm on V
is a function k · k : V → R≥ such that:
(a) kvk = 0 if and only if v = 0;
(b) kαvk = |α| · kvk for all α ∈ R and v ∈ V ;
(c) kv + wk ≤ kvk + kwk for all v, w ∈ V .
Condition (c) implies that the function (v, w) 7→ kv − wk is a metric on V , and we
endow V with the associated topology. Condition (a) implies that {0} is closed because every other point has a neighborhood that does contain the origin. Conditions
(b) and (c) give the calculations
kα′ v ′ − αvk ≤ kα′ v ′ − α′ vk + kα′ v − αvk = |α′| · kv ′ − vk + |α′ − α| · kvk
and
k(v ′ + w ′ ) − (v + w)k ≤ kv ′ − vk + kw ′ − wk,
which are easily seen to imply that scalar multiplication and addition are continuous.
A vector space endowed with a norm and the associated metric and topology is
called a normed space.
For a normed space the calculation
k(1 − α)v + αwk ≤ k(1 − α)vk + kαwk = (1 − α)kvk + αkwk ≤ max{kvk, kwk}
shows that for any ε > 0, the open ball of radius ε centered at the origin is convex.
The open ball of radius ε centered at any other point is the translation of this ball,
so a normed space is locally convex.
A sequence {vm } in a topological vector space V is a Cauchy sequence if, for
each neighborhood A of the origin, there is an integer N such that vm − vn ∈ A for
all m, n ≥ N. The space V is complete if its Cauchy sequences are all convergent.
A Banach space is a complete normed space.
For the most part there is little reason to consider topological vector spaces
that are not complete except insofar as they occur as subspaces of complete spaces.
The reason for this is that any topological vector space V can be embedded in
a complete space V whose elements are equivalence classes of Cauchy sequences,
where two Cauchy sequence {vm } and {wn } are equivalent if, for each neighborhood
A of the origin, there is an integer N such that vm − wn ∈ A for all m, n ≥ N. (This
6.4. BANACH AND HILBERT SPACES
91
relation is clearly reflexive and symmetric. To see that it is transitive, suppose {uℓ }
is equivalent to {vm } which is in turn equivalent to {wn }. For any neighborhood A
of the origin the continuity of addition implies that there are neighborhoods B, C of
the origin such that B + C ⊂ A. There is N such that uℓ −vm ∈ B and vm −wn ∈ C
for all ℓ, m, n ≥ N, whence uℓ − wn ∈ A.) Denote the equivalence class of {vm } by
[vm ]. The vector operations have the obvious definitions: [vm ] + [wn ] := [vm + wm ]
and α[vm ] := [αvm ]. The open sets of V are the sets of the form
{ [vm ] : vm ∈ A for all large m }
where A ⊂ V is open. (It is easy to see that the condition “vm ∈ A for all large
m” does not depend on the choice of representative {vm } of [vm ].) A complete
justification of this definition would require verifications of the vector space axioms,
the axioms for a topological space, the continuity of addition and scalar multiplication, and that {0} is a closed set. Instead of elaborating, we simply assert that the
reader who treats this as an exercise will find it entirely straightforward. A similar
construction can be used to embed any metric space in a “completion” in which all
Cauchy sequences (in the metric sense) are convergent.
As in the finite dimensional case, the best behaved normed spaces have inner
products. An inner product on a vector space V is a function h·, ·i : V × V → R
that is symmetric, bilinear, and positive definite:
(i) hv, wi = hw, vi for all v, w ∈ V ;
(ii) hαv + v ′ , wi = αhv, wi + hv ′ , wi for all v, v ′ , w ∈ V and α ∈ R;
(iii) hv, vi ≥ 0 for all v ∈ V , with equality if and only if v = 0.
We would like to define a norm by setting kvk := hv, vi1/2. This evidently satisfies
(a) and (b) of the definition of a norm. The verification of (c) begins with the
computation
0 ≤ hv, viw − hv, wiv, hv, viw − hv, wiv = hv, vi hv, vihw, wi − hv, wi2 ,
which implies the Cauchy-Schwartz inequality: hv, wi ≤ kvk · kwk for all v, w ∈
V . This holds with equality if v = 0 or hv, viw − hv, wiv, which is the case if
and only if w is a scalar multiple of v, and otherwise the inequality is strict. The
Cauchy-Schwartz inequality implies the inequality in the calculation
kv + wk2 = hv + w, v + wi = kvk2 + 2hv, wi + kwk2 ≤ (kvk + kwk)2 ,
which implies (c) and completes the verification and k · k is a norm. A vector space
endowed with an inner product and the associated norm and topology is called an
inner product space. A Hilbert space is a complete inner product space.
Up to linear isometry there is only one separable2 Hilbert space. Let
H := { s = (s1 , s2 , . . .) ∈ R∞ : s21 + s22 + · · · < ∞ }
2
Recall that a metric space is separable if it contains a countable set of points whose closure
is the entire space.
92
CHAPTER 6. METRIC SPACE THEORY
P
be the Hilbert space of square summable sequences. Let hs, ti := i si ti be the usual
inner product; the Cauchy-Schwartz inequality implies that this sum is convergent.
For any Cauchy sequence in H and for each i, the sequence of ith components is
Cauchy, and the element of R∞ whose ith component is the limit of this sequence is
easily shown to be the limit in H of the given sequence. Thus H is complete. The
set of points with only finitely many nonzero components, all of which are rational,
is a countable dense subset, so H is separable.
We wish to show that any separable Hilbert space is linearly isomorphic to H, so
let V be a separable Hilbert space, and let {v1 , v2 , . . . } be a countable dense subset.
The span of this set is also dense, of course. Using the Gram-Schmidt process, we
may pass from this set to a countable sequence w1 , w2, . . . of orthnormal vectors
that has the same span. It is now easy to show that s 7→ s1 w1 + s2 w2 + · · · is a
linear isometry between H and V .
6.5
EmbeddingTheorems
An important technique is to endow metric spaces with geometric structures by
embedding them in normed spaces. Let (X, d) be a metric space, and let C(X) be
the space of bounded continuous real valued functions on X. This is, of course, a
vector space under pointwise addition and scalar multiplication. We endow C(X)
with the norm
kf k∞ = sup |f (x)|.
x∈X
Lemma 6.5.1. C(X) is a Banach space.
Proof. The verification that k · k∞ is actually a norm is elementary and left to
the reader. To prove completeness suppose that {fn } is a Cauchy sequence. This
sequence has a pointwise limit f because each {fn (x)} is Cauchy, and we need
to prove that f is continuous. Fix x ∈ X and ε > 0. There is an m such that
kfm −fn k < ε/3 for all n ≥ m, and there is a δ > 0 such that |fm (x′ ) −fm (x)| < ε/3
for all x′ ∈ Uδ (x). For such x′ we have
|f (x′ ) − f (x)| ≤ |f (x′ ) − fm (x′ )| + |fm (x′ ) + fm (x)| + |fm (x) − f (x)| < ε.
Theorem 6.5.2 (Kuratowski (1935), Wojdyslawski (1939)). X is homeomorphic
to a relatively closed subset of a convex subset of C(X). If X is complete, then it
is homeomorphic to a closed subset of C(X).
Proof. For each x ∈ X let fx ∈ C(X) be the function fx (y) := min{1, d(x, y)}; the
map h : x 7→ fx is evidently an injection from X to C(X). For any x, y ∈ X we
have
kfx −fy k∞ = sup | min{1, d(x, z)}−min{1, d(y, z)}| ≤ sup |d(x, z)−d(y, z)| ≤ d(x, y),
z
z
so h is continuous. On the other hand, if {xn } is a sequence such that fxn → fx ,
then min{1, d(xn , x)} = |fxn (x) − fx (x)| ≤ kfxn − fx k∞ → 0, so xn → x. Thus the
inverse of h is continuous, so h is a homeomorphism.
93
6.6. DUGUNDJI’S THEOREM
Now suppose that fxn converges to an element f =
of h(X). We have kfxn − f k∞ → 0 and
Pk
i=1 λi fyi
of the convex hull
kfxn − f k∞ ≥ |fxn (xn ) − f (xn )| = |f (xn )|,
so f (xn ) → 0. For each i we have 0 ≤ fyi (xn ) ≤ f (xn )/λi → 0, which implies that
xn → yi , whence f = fy1 = · · · = fyk ∈ h(X). Thus h(X) is closed in the relative
topology of its convex hull.
Now suppose that X is complete, and that {xn } is a sequence such that fxn → f .
Then as above, min{1, d(xm , xn )} ≤ kfxm − fxn k∞ , and {fxn } is a Cauchy sequence,
so {xn } is also Cauchy and has a limit x. Above we saw that fxn → fx , so fx = f .
Thus h(X) is closed in C(X).
The so-called Hilbert cube is
I ∞ := { s ∈ H : |si | ≤ 1/i for all i = 1, 2, . . . }.
For separable metric spaces we have the following refinement of Theorem 6.5.2.
Theorem 6.5.3. (Urysohn) If (X, d) is a separable metric space, there is an embedding ι : X → I ∞ .
Proof. Let { x1 , x2 , . . . } be a countable dense subset of X. Define ι : X → I ∞ by
setting
ιi (x) := min{d(x, xi ), 1/i}.
Clearly ι is a continuous injection. To show that the inverse is continuous, suppose
that {xj } is a sequence with ι(xj ) → ι(x). If it is not the case that xj → x, then
there is a neighborhood U that (perhaps after passing to a subsequence) does not
have any elements of the sequence. Choose xi in that neighborhood. The sequence
of numbers min{d(xj , xi ), 1/i} is bounded below by a positive number, contrary to
the assumption that ι(xj ) → ι(x).
6.6
Dugundji’s Theorem
The well known Tietze extension theorem asserts that if a topological space X
is normal and f : A → [0, 1] is continuous, where A ⊂ X is closed, then f has a
continuous extension to all of X. A map into a finite dimensional Euclidean space
is continuous if its component functions are each continuous, so Tietze’s theorem
is adequate for finite dimensional applications. Mostly, however, we will work with
spaces that are potentially infinite dimensional, for which we will need the following
variant due to Dugundji (1951).
Theorem 6.6.1. If A is a closed subset of a metric space (X, d), Y is a locally
convex topological vector space, and f : A → Y is continuous, then there is a
continuous extension f : X → Y whose image is contained in the convex hull of
f (A).
94
CHAPTER 6. METRIC SPACE THEORY
Proof. The sets Ud(x,A)/2 (x) are open and cover X \ A. Theorem 6.1.1 implies the
existence of an open locally finite refinement {Wα }α∈I . Theorem 6.2.2 implies the
existence of a partition of unity {ϕα }α∈I subordinate to {Wα }α∈I . For each α choose
aα ∈ A with d(aα , Wα ) < 2d(A, Wα ), and define the extension by setting
X
f (x) :=
ϕα (x)f (aα ) (x ∈ X \ A).
α∈I
Clearly f is continuous at every point of X \ A and at every interior point of A.
Let a be a point in the boundary of A, let U be a neighborhood of f (a), which we
may assume to be convex, and choose δ > 0 small enough that f (a′ ) ∈ U whenever
a′ ∈ Uδ (a) ∩ A. Consider x ∈ Uδ/7 (a) ∩ (X \ A). For any α such that x ∈ Wα and
x′ such that Wα ⊂ Ud(x′ ,A)/2 (x′ ) we have
d(aα , Wα ) ≥ d(aα , x′ ) − d(x′ , A)/2 ≥ d(aα , x′ ) − d(x′ , aα )/2 = d(aα , x′ )/2
and
d(x′ , x) ≤ d(x′ , A)/2 ≤ d(Wα , A) ≤ d(Wα , aα ),
so
d(aα , x) ≤ d(aα , x′ ) + d(x′ , x) ≤ 3d(aα , Wα ) ≤ 6d(A, Wα ) ≤ 6d(a, x).
Thus d(aα , a) ≤ d(aα , x)+d(x, a) ≤ 7d(x, a) < δ whenever x ∈ Wα , so f (x) ∈ U.
Chapter 7
Retracts
This chapter begins with Kinoshita’s example of a compact contractible space
that does not have the fixed point property. The example is elegant, but also rather
complex, and nothing later depends on it, so it can be postponed until the reader is
in the mood for a mathematical treat. The point is that fixed point theory depends
on some additional condition over and above compactness and contractibility.
After that we develop the required material from the theory of retracts. We
first describe retracts in general, and then briefly discuss Euclidean neighborhood
retracts, which are retracts of open subsets of Euclidean spaces. This concept is
quite general, encompassing simplicial complexes and (as we will see later) smooth
manifolds.
The central concept of the chapter is the notion of an absolute neighborhood
retract (ANR) which is a metrizable space whose image, under any embedding as a
closed subset of a metric space, is a retract of some neighborhood of itself. The two
key characterization results are that an open subset of a convex subset of a locally
convex linear space is an absolute neighborhood retract, and that an ANR can be
embedded in a normed linear space as a retract of an open subset of a convex set.
An absolute retract (AR) is a space that is a retract of any metric space it
is embedded in as a closed subset. It turns out that the ARs are precisely the
contractible ANRs.
The extension of fixed point theory to infinite dimensional settings ultimately
depends on “approximating” the setting with finite dimensional objects. Section
7.6 provides one of the key results in this direction.
7.1
Kinoshita’s Example
This example came to be known as the “tin can with a roll of toilet paper.” As
you will see, this description is apt, but does not do justice to the example’s beauty
and ingenuity.
Polar coordinates facilitate the description. Let P = [0, ∞) × R, with (r, θ) ∈ P
identified with the point (r cos θ, r sin θ). The unit circle and the open unit disk are
C = { (r, θ) : r = 1 }
and
95
D = { (r, t) : r < 1 }.
96
CHAPTER 7. RETRACTS
Let ρ : [0, ∞) → [0, 1) be a homeomorphism, let s : [0, ∞) → P be the function
s(t) := (ρ(t), t), and let
S = { s(t) : t ≥ 0 }.
Then S is a curve that spirals out from the origin, approaching C asymptotically.
The space of the example is
X = (C × [0, 1]) ∪ (D × {0}) ∪ (S × [0, 1]) ⊂ R3 .
Here C × [0, 1] is the cylindrical side of the tin can, D × {0} is its base, and S × [0, 1]
is the roll of toilet paper. Evidently X is closed, hence compact, and there is an
obvious contraction of X that first pushes the cylinder of the tin can and the toilet
paper down onto the closed unit disk and then contracts the disk to the origin.
We are now going to define functions
f1 : C × [0, 1] → X,
f2 : D × {0} → X,
f3 : S × [0, 1] → X
which combine to form a continuous function f : X → X with no fixed points.
Fix a number ε > 0 that is not an integral multiple of 2π; imagining that ε is
small may help to visualize f as a motion of X. Also, fix a continuous function
κ : [0, 1] → [0, 1] with κ(0) = 0, κ(1) = 1, and κ(z) > z for all 0 < z < 1.
The first function is given by the formula
f1 (1, θ, z) := (1, θ − (1 − 2z)ε, κ(z)).
This is evidently well defined and continuous. The point (1, θ, z) cannot be fixed
because κ(z) = z implies that z = 0 or z = 1 and ε is not a multiple of 2π.
Observe that D = { (ρ(t), θ) : t ≥ 0, θ ∈ R }. The second function is
(
(0, 0, 1 − t/ε),
0 ≤ t ≤ ε,
f2 (ρ(t), θ, 0) :=
(ρ(t − ε), θ − ε, 0), ε ≤ t.
This is well defined because ρ is invertible and the two formulas give the origin
as the image when t = ε. It is continuous because it is continuous on the two
subdomains, which are closed and cover D. It does not have any fixed points
because the coordinate of f2 (ρ(t), θ, 0) is less than ρ(t) except when t = 0, and
f2 (ρ(0), θ, 0) = (0, 0, 1).
The third function is
(
(s((t + ε)z), 1 − (1 − κ(z))t/ε), 0 ≤ t ≤ ε,
f3 (s(t), z) :=
(s(t − (1 − 2z)ε), κ(z)),
ε ≤ t.
This is well defined because s is invertible and the two formulas give (s(2εz), κ(z))
as the image when t = ε. It is continuous because it is continuous on the two
subdomains, which are closed and cover S × [0, 1]. Since f2 (s(t), 0) = f3 (s(t), 0)
for all t, f2 and f3 combine to define a continuous function on the union of their
domains.
Can (s(t), z) be a fixed point of f3 ? If t < ε, then the equation
z = 1 − (1 − κ(z))t/ε
7.2. RETRACTS
97
is equivalent to (1 − κ(z))t = (1 − z)ε, which is impossible if z < 1 due to the
conditions on κ. When t < ε and z = 1, we have s(t) 6= s(t + ε) because s is
injective. On the other hand, when t ≥ ε the equation κ(z) = z implies that either
z = 0, in which case s(t) 6= s(t − ε), or z = 1, in which case s(t) 6= s(t + ε).
We have now shown that f is well defined and has no fixed points, and that it is
continuous on (S ×[0, 1])∪(D×{0}) and on C ×[0, 1]. To complete the verification of
continuity, first consider a sequence {(ρ(ti ), θi , 0)} in D × {0} converging to (1, θ, 0).
Clearly
f2 (ρ(ti ), θi , 0) = (ρ(ti − ε), θi − ε, 0) → (1, θ − ε, 0) = f1 (1, θ, 0).
Now consider a sequence {(s(ti ), zi )} converging to a point (1, θ, z). In order for f
to be continuous it must be the case that
f3 (s(ti ), zi ) = (s(ti − (1 − 2zi )ε), κ(zi )) → (1, θ − (1 − 2z)ε, κ(z)) = f1 (1, θ, z).
Since s(ti ) → (1, θ) means precisely that ti → ∞ and ti mod 2π → θ mod 2π, again
this is clear.
7.2
Retracts
This section prepares for later material by presenting general facts about retractions and retracts. Let X be a metric space, and let A be a subset of X such that
there is a continuous function r : X → A with r(a) = a for all a ∈ A. We say that
A is a retract of X and that r is a retraction. Many desirable properties that X
might have are inherited by A.
Lemma 7.2.1. If X has the fixed point property, then A has the fixed point property.
Proof. If f : A → A is continuous, then f ◦ r necessarily has a fixed point, say a∗ ,
which must be in A, so that a∗ = f (r(a∗ )) = f (a∗ ) is also a fixed point of f .
Lemma 7.2.2. If X is contractible, then A is contractible.
Proof. If c : X × [0, 1] → X is a contraction X, then so is (a, t) 7→ r(c(a, t)).
Lemma 7.2.3. If X is connected, then A is connected.
Proof. We show that if A is not connected, then X is not connected. If U1 and
U2 are nonempty open subsets of A with U1 ∩ U2 = ∅ and U1 ∪ U2 = A, then
r −1 (U1 ) and r −1 (U2 ) are nonempty open subsets of X with r −1 (U1 ) ∩ r −1 (U2 ) = ∅
and r −1 (U1 ) ∪ r −1 (U2 ) = X.
Here are two basic observations that are too obvious to prove.
Lemma 7.2.4. If s : A → B is a second retraction, then s ◦ r : X → B is a
retraction, so B is a retract of X.
Lemma 7.2.5. If A ⊂ Y ⊂ X, then the restriction of r to Y is a retraction, so A
is a retract of Y .
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CHAPTER 7. RETRACTS
We say that A is a neighborhood retract in X if A is a retract of an open
U ⊂ X. We note two other simple facts, the first of which is an obvious consequence
of the last result:
Lemma 7.2.6. Suppose that A is not connected: there are disjoint open sets
U1 , U2 ⊂ X such that A ⊂ U1 ∪ U2 with A1 := A ∩ U1 and A2 := A ∩ U2 both
nonempty. Then A is a neighborhood retract in X if and only if both A1 and A2
are neighborhood retracts in X.
Lemma 7.2.7. If A is a neighborhood retract in X and B is a neighborhood retract
in A, then B is a neighborhood retract in X.
Proof. Let r : U → A and s : V → B be retractions, where U is a neighborhood of
A and V ⊂ A is a neighborhood of B in the relative topology of A. The definition
of the relative topology implies that there is a neighborhood W ⊂ X of B such that
V = A ∩ W . Then U ∩ W is a neighborhood of B in X, and the composition of s
with the restriction of r to U ∩ W is a retraction onto B.
A set A ⊂ X is locally closed if it is the intersection of an open set and a
closed set. Equivalently, it is an open subset of a closed set, or a closed subset of
an open set.
Lemma 7.2.8. A neighborhood retract is locally closed.
Proof. If U ⊂ X is open and r : U → A is a retraction, A is a closed subset of U
because it is the set of fixed points of r.
This terminology ‘locally closed’ is further explained by:
Lemma 7.2.9. If X is a topological space and A ⊂ X, then A is locally closed if
and only if each point x ∈ A has a neighborhood U such that U ∩ A is closed in U.
Proof. If A = U ∩ C where U is open and C is closed, then U is a neighborhood
of each x ∈ A, and A is closed in U. On the other hand suppose that each x ∈ A
has a neighborhood Ux suchSthat Ux ∩ A S
is closed in Ux ,Swhich is to say that
Ux ∩ A = Ux ∩ A. Then A = x (Ux ∩ A) = x (Ux ∩ A) =
x Ux ∩ A.
Corollary 7.2.10. If X is locally compact, a set A ⊂ X is locally closed if and
only if each x ∈ A has a compact neighborhood.
Proof. If A = U ∩ C, x ∈ A, and K is a compact neighborhood of x contained in
U, then K ∩ C is a compact neighborhood in A. On the other hand, if x ∈ A and
K is a compact neighborhood of x in A, then K = A ∩ V for some neighborhood
V of x in X. Let U be the interior of V . Then U ∩ A = U ∩ K is closed in U. This
shows that if every point in K has a compact neighborhood, then the condition in
the last result holds.
7.3. EUCLIDEAN NEIGHBORHOOD RETRACTS
7.3
99
Euclidean Neighborhood Retracts
A Euclidean neighborhood retract (ENR) is a topological space that is
homeomorphic to a neighborhood retract of a Euclidean space. If a subset of a
Euclidean space is homeomorphic to an ENR, then it is a neighborhood retract:
Proposition 7.3.1. Suppose that U ⊂ Rm is open, r : U → A is a retraction,
B ⊂ Rn , and h : A → B is a homeomorphism. Then B is a neighborhood retract.
Proof. Since A is locally closed and Rm is locally compact, each point in A has
a closed neighborhood that contains a compact neighborhood. Having a compact
neighborhood is an intrinsic property, so every point in B has such a neighborhood,
and Corollary 7.2.10 implies that B is locally closed. Let V ⊂ Rn be an open set
that has B as a closed subset. The Tietze extension theorem gives an extension of
h−1 to a map j : V → Rm . After replacing V with j −1 (U), V is still an open set
that contains B, and h ◦ r ◦ j : V → B is a retraction.
Note that every locally closed set A = U ∩ C ⊂ Rm is homeomorphic to a
closed subset of Rm+1 , by virtue of the embedding x 7→ (x, d(x, Rm \ U)−1 ), where
d(x, Rm \ U) is the distance from x to the nearest point not in U. Thus a sufficient
condition for X to be an ENR is that it is homeomorphic to a neighborhood retract
of a Euclidean space, but a necessary condition is that it homeomorphic to a closed
neighborhood retract of a Euclidean space.
In order to expand the scope of fixed point theory, it is desirable to show that
many types of spaces are ENR’s. Eventually we will see that a smooth submanifold
of a Euclidean space is an ENR. At this point we can show that simplicial complexes
have this property.
Lemma 7.3.2. If K ′ = (V ′ , C ′ ) is a subcomplex of a simplicial complex K = (V, C),
then |K ′ | is a neighborhood retract in |K|.
Proof. To begin with suppose that there are simplices of positive dimension in K
that are not in K ′ . Let σ be such a simplex of maximal dimension, and let β be
the barycenter of |σ|. Then |K| \ {β} is a neighborhood of |K| \ int |σ|, and there
is a retraction r of the former set onto the latter that is the identity on the latter,
of course, and which maps (1 − t)x + tβ to x whenever x ∈ |∂σ| and 0 < t < 1.
Iterating this construction and applying Lemma 7.2.7 above, we find that there
is a neighborhood retract of |K| consisting of |K ′ | and finitely many isolated points.
Now Lemma 7.2.6 implies that |K ′ | is a neighborhood retract in |K|.
Proposition 7.3.3. If K = (V, C) is a simplicial complex, then |K| is an ENR.
Proof. Let ∆ be the convex hull of the set of unit basis vectors in R|V | . After
repeated barycentric subdivision of ∆ there is a (|V | − 1)-dimensional simplex σ
in the interior of ∆. (This is a consequence of Proposition 2.5.2.) Identifying the
vertices of σ with the elements of V leads to an embedding of |K| as a subcomplex
of this subdivision, after which we can apply the result above.
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CHAPTER 7. RETRACTS
Giving an example of a closed subset of a Euclidean space that is not an ENR
is a bit more difficult. Eventually we will see that a contractible ENR has the
fixed point property, from which it follows that Kinoshita’s example is not an ENR.
A simpler example is the Hawaiian earring H, which is the union over all n =
1, 2, . . . of the circle of radius 1/n centered at (1/n, 0). If there was a retraction
r : U → H of a neighborhood U of H, then for small n the entire disk of radius
1/n centered at (1/n, 0) would be contained in U, and we would have a violation
of the following result, which is actually a quite common method of applying the
fixed point principle.
Theorem 7.3.4 (No Retraction Theorem). If D n is the closed unit disk centered
at the origin in Rn , and S n−1 is its boundary, then there does not exist a continuous
r : D n → Rn \ D n with r(x) = x for all s ∈ S n−1 .
Proof. Suppose that such an r exists, and let g : D n → S n−1 be the function that
takes each x ∈ S n−1 to itself and takes each x ∈ D n \ S n−1 to the point where the
line segment between r(x) and x intersects S n−1 . An easy argument shows that g
is continuous at each x ∈ D n \ S n−1 , and another easy argument shows that g is
continuous at each x ∈ S n−1 , so g is continuous. If a : S n−1 → S n−1 is the antipodal
map a(x) = −x, then a ◦ g gives a map from D n to itself that does not have a fixed
point, contradicting Brouwer’s fixed point theorem.
7.4
Absolute Neighborhood Retracts
A metric space A is an absolute neighborhood retract (ANR) if h(A) is a
neighborhood retract whenever X is a metric space, h : A → X is an embedding,
and h(A) is closed. This definition is evidently modelled on the description of ENR’s
we arrived at in the last section, with ‘metric space’ in place of ‘Euclidean space.’
We saw above that if A ⊂ Rm is a neighborhood retract, then any homeomorphic
image of A in another Euclidean space is also a neighborhood retract, and some
such homeomorphic image is a closed subset of the Euclidean space. Thus a natural,
and at least potentially more restrictive, extension of the concept is obtained by
defining an ANR to be a space A such that h(A) is a neighborhood retract whenever
h : A → X is an embedding of A is a metric space X, even if h(A) is not closed.
There is a second sense in which the definition is weaker than it might be. A
topological space is completely metrizable if its topology can be induced by a
complete metric. Since an ENR is homeomorphic to a closed subset of a Euclidean
space, an ENR is completely metrizable. Problem 6K of Kelley (1955) shows that a
topological space A is completely metrizable if and only if, whenever h : A → X is
an embedding of A in a metric space X, h(A) is a Gδ . The set of rational numbers
is an example of a space that is metrizable, but not completely metrizable, because
it is
T not a Gδ as a subset of R. To see this observe that the set of irrational numbers
is r∈Q R \ {r}, so if Q was a countable intersection of open sets, then ∅ would be a
countable intersection of open sets, contrary to the Baire category theorem (p. 200
of Kelley (1955)). The next result shows that the union of { eπir : r ∈ Q } with the
open unit disk in C is an ANR, but this space is not completely metrizable, so it is
not an ENR. Thus there are finite dimensional ANR’s that are not ENR’s.
7.4. ABSOLUTE NEIGHBORHOOD RETRACTS
101
By choosing the least restrictive definition we strengthen the various results
below. However, these complexities are irrelevant to compact ANR’s, which are,
for the most part, the only ANR’s that will figure in our work going forward. Of
course the homeomorphic image h(A) of a compact metric space A in any metric
space is compact and consequently closed, and of course h(A) is also complete.
At first blush being an ANR might sound like a remarkable property that can
only be possessed by quite special spaces, but this is not the case at all. Although
ANR’s cannot exhibit the “infinitely detailed features” of the tin can with a roll of
toilet paper, the concept is not very restrictive, at least in comparison with other
concepts that might serve as an hypothesis of a fixed point theorem.
Proposition 7.4.1. A metric space A is an ANR if it (or its homeomorphic image)
is a retract of an open subset of a convex subset of a locally convex linear space.
Proof. Let r : U → A be a retraction, where U is an open subset of a convex
set C. Suppose h : A → X maps A homeomorphically onto a closed subset h(A)
of a metric space X. Dugundji’s theorem implies that h−1 : h(A) → U has a
continuous extension j : X → C. Then V = j −1 (U) is a neighborhood of h(A), and
h ◦ r ◦ j|V : V → h(A) is a retraction.
Corollary 7.4.2. An ENR is an ANR.
The proposition above gives a sufficient condition for a space to be an ANR.
There is a somewhat stronger necessary condition.
Proposition 7.4.3. If A is an ANR, then there is a homeomorphic image of A
that is a retract of an open subset of a convex subset of Banach space.
Proof. Theorem 6.5.2 gives a map h : A → Z, where Z is a Banach space, such that
h maps A homeomorphically onto h(A) and h(A) is closed in the relative topology
of its convex hull C. Since A is an ANR, there is a relatively open U ⊂ C and a
retraction r : U → h(A).
Since compact metric spaces are separable, compact ANR’s satisfy a more demanding embedding condition than the one given by Proposition 7.4.3.
Proposition 7.4.4. If A is a compact ANR, then there exists an embedding ι :
A → I ∞ such that ι(A) is a neighborhood retract in I ∞ .
Proof. Urysohn’s Theorem guarantees the existence of an embedding of A in I ∞ .
Since A is compact, h(A) is closed in I ∞ , and since A is an ANR, h(A) is a neighborhood retract in I ∞ .
The simplicity of an open subset of I ∞ is the ultimate source of the utility of
ANR’s in the theory of fixed points. To exploit this simplicity we need analytic
tools that bring it to the surface. Fix a compact metric space (X, d), and let
∆ = { (x, x) : x ∈ X }
be the diagonal in X × X. We say that (X, d) is uniformly locally contractible
if, for any neighborhood V ⊂ X × X of ∆ there is a neighborhood W of ∆ and a
map γ : W × [0, 1] → X such that:
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CHAPTER 7. RETRACTS
(a) γ(x, x′ , 0) = x′ and γ(x, x′ , 1) = x for all (x, x′ ) ∈ W ;
(b) γ(x, x, t) = x for all x ∈ X and t ∈ [0, 1];
(c) (x, γ(x, x′ , t)) ∈ V for all (x, x′ ) ∈ W and t ∈ [0, 1].
Proposition 7.4.5. A compact ANR A is uniformly locally contractible.
Proof. By Proposition 7.4.4 we may assume that A ⊂ I ∞ , and that there is a
retraction r : U → A where U ⊂ I ∞ is open. Fix a neighborhood V ⊂ A × A of the
diagonal, and let Ṽ = (IdA × r)−1 (V ) ⊂ A × U. The distance from x to the nearest
point in I ∞ \ U is a positive continuous function on A, which attains its minimum
since A is compact, so there is some δ > 0 such that
∆δ = { (x, x′ ) ∈ A × U : kx′ − xk < δ } ⊂ Ṽ .
Let W = ∆δ ∩ (A × A), and let γ : W × [0, 1] → A be the function
γ(x, x′ , t) = r(tx + (1 − t)x′ ).
Evidently γ has all the required properties.
A topological space X is locally path connected if, for each x ∈ X, each
neighborhood Y of x contains a neighborhood U such that for any x0 , x1 ∈ U there
is a continuous path γ : [0, 1] → Y with γ(0) = x0 and γ(1) = x1 . At first sight
this seems less straightforward than requiring that any neighborhood of x contain a
pathwise connected neighborhood, but the weaker condition given by the definition
is sometimes much easier to verify, and it usually has whatever implications are
desired.
Corollary 7.4.6. A compact ANR A is locally path connected.
Proof. The last result (with V = A × A) gives a neighborhood W ⊂ A × A of the
diagonal and a function γ : W × [0, 1] → A satisfying (a) and (b). Fix x ∈ A,
and let Y be a neighborhood of x. There is a neighborhood U of x such that
U × U ⊂ W and γ(U × U × [0, 1]) ⊂ Y . (Combining (b) and the continuity of γ,
for each t ∈ [0, 1] there is a neighborhood Ut and εt > 0 such that Ut × Ut ⊂ W
and γ(Ut × Ut × (tS− εt , t + εt )) ⊂ Y . Since [0,T
1] is compact there are t1 , . . . , tk
such that [0, 1] ⊂ i (ti − εti , ti + εti ). Let U = i Uti .) Then for any x0 , x1 ∈ U,
t 7→ γ(x1 , x0 , t) is a path in Y going from x0 to x1 .
7.5
Absolute Retracts
A metric space A is an absolute retract (AR) if h(A) is a retract of X whenever
X is a metric space, h : A → X is an embedding, and h(A) is closed. Of course
an AR is an ANR. Below we will see that an ANR is an AR if and only if it is
contractible, so compact convex sets are AR’s. Eventually (Theorem 14.1.5) we
will show that nonempty compact AR’s have the fixed point property. In this sense
AR’s fulfill our goal of replacing the assumption of a convex domain in Kakutani’s
theorem with a topological condition.
The embedding conditions characterizing AR’s parallel those for ANR’s, with
some simplifications.
7.5. ABSOLUTE RETRACTS
103
Proposition 7.5.1. If a metric space A is a retract of a convex subset C of a locally
convex linear space, then it is an ANR.
Proof. Suppose h : A → X maps A homeomorphically onto a closed subset h(A)
of a metric space X. Dugundji’s theorem implies that h−1 : h(A) → C has a
continuous extension j : X → C. Let r : C → A be a retraction. Then q := h ◦ r ◦ j
is a retraction of X onto h(A).
Proposition 7.5.2. If A is an AR, then there is a homeomorphic image of A that
is a retract of a convex subset of a Banach space.
Proof. Theorem 6.5.2 gives a map h : A → Z, where Z is a Banach space, such that
h maps A homeomorphically onto h(A) and h(A) is closed in the relative topology
of its convex hull C. Since A is an AR, there is a retraction r : C → h(A).
The remainder of the section proves:
Proposition 7.5.3. An ANR is an AR if and only if it is contractible.
In preparation for the proof we introduce an important concept of general topology. A pair of topological spaces X, A with A ⊂ X are said to have the homotopy
extension property with respect to the class ANR if, whenever:
(a) Y is an ANR,
(b) f : X → Y is continuous,
(c) η : A × [0, 1] → Y is a homotopy, and
(d) η(·, 0) = f |A ,
there is a continuous η : X × [0, 1] → Y with η(·, 0) = f and η|A×[0,1] = η.
Proposition 7.5.4. If X is a metric space and A is a closed subset of X, then X
and A have the homotopy extension property with respect to ANR’s.
We separate out one of the larger steps in the argument.
Lemma 7.5.5. Let X be a metric space, let A be a closed subset of X, and let
Z := (X × {0}) ∪ (A × [0, 1]).
Then for every neighborhood V ⊂ X × [0, 1] of Z there is a map ϕ : X × [0, 1] → V
that agrees with the identity on Z.
Proof. For each (a, t) ∈ A × [0, 1] choose a product neighborhood
U(a,t) × (t − ε(a,t) , t + ε(a,t) ) ⊂ V
where U(a,t) ⊂ X is open and ε > 0. For any particular a the cover of {a}×[0, 1] has
a finite subcover, and the intersection of S
its first cartesian factors is a neighborhood
Ua of a with Ua × [0, 1] ⊂ V . Let U := a Ua . Thus there is a neighborhood U of
A such that U × [0, 1] ⊂ V .
Urysohn’s lemma gives a function α : X → [0, 1] with α(x) = 0 for all x ∈ X \ U
and α(a) = 1 for all a ∈ A, and the function ϕ(x, t) := (x, α(x)t) satisfies the
required conditions.
104
CHAPTER 7. RETRACTS
Proof of Proposition 7.5.4. Let Y , f : X → Y , and h : A × [0, 1] → Y satisfy (a)(d) above. By Theorem 6.5.2 we may assume without loss of generality that Y is
contained in a Banach space S, and is a relatively closed subset of its convex hull C.
Let Z := (X ×{0})∪(A×[0, 1]), and define g : Z → Y by setting g(x, 0) = f (x) and
g(a, t) = h(a, t). Dugundji’s theorem implies that there is a continuous extension
g : X × [0, 1] → C of g. Let W ⊂ C be a neighborhood of Y for which there is a
retraction r : W → Y , let V := g −1 (W ), and let ϕ : X × [0, 1] → V be a continuous
map that is the identity on Z, as per the result above. Clearly η := r ◦ g ◦ ϕ has
the indicated properties.
We now return to the characterization of AR’s.
Proof of Proposition 7.5.3. Let A be an ANR. By Theorem 6.5.2 we may embed A
as a relatively closed subset of a convex subset C of a Banach space.
If A is an AR, then it is a retract of C. A convex set is contractible, and a
retract of a contractible set is contractible (Lemma 7.2.2) so A is contractible.
Suppose that A is conractible. By Proposition 7.5.1 it suffices to show that A is
a retract of C. Let c : A×[0, 1] → A be a contraction, and let a1 be the “final value”
a1 , by which we mean that c(a, 1) = a1 for all a ∈ A. Set Z := (C ×{0})∪(A×[0, 1]),
and define f : Z → A by setting f (x, 0) := a1 for x ∈ C and f (a, t) := c(a, 1 − t) for
(a, t) ∈ A × [0, 1]. Proposition 7.5.4 implies the existence of a continuous extension
f : C × [0, 1] → A. Now r := f (·, 1) : C → A is the desired retraction.
7.6
Domination
In our development of the fixed point index an important idea will be to pass
from a theory for certain simple or elementary spaces to a theory for more general
spaces by showing that every space of the latter type can be “approximated” by a
simpler space, in the sense of the following definitions. Fix a metric space (X, d).
Definition 7.6.1. If Y is a topological space and ε > 0, a homotopy η : Y ×[0, 1] →
X is an ε-homotopy if
d η(y, s), η(y, t) < ε
for all y ∈ Y and all 0 ≤ s, t ≤ 1. We say that η0 and η1 are ε-homotopic.
Definition 7.6.2. For ε > 0, a topological space D ε-dominates C ⊂ X if there
are continuous functions ϕ : C → D and ψ : D → X such that ψ ◦ ϕ : C → X is
ε-homotopic to IdC .
This section’s main result is:
Theorem 7.6.3 (Domination Theorem). If X is a separable ANR and C ⊂ X is
compact, then for any ε > 0 there is a simplicial complex that ε-dominates C.
Proof. If C = ∅, then for any ε > 0 it is ε-dominated by ∅, which we consider
to be a simplicial complex. Similarly, if C is a singleton, then for any ε > 0 it is
ε-dominated by the simplicial complex consisting of a single point. Therefore we
may assume that C has more than one point.
105
7.6. DOMINATION
In view of Proposition 7.4.3 we may assume that X is a retract of an open set
U of a convex subset S of a Banach space. Let r : U → X be the retraction, and
let d be the metric on S derived from the norm of the Banach space. Fix ε > 0
small enough that C is not contained in the ε/2-ball around any of its points. Let
r : U → X be a retraction of a neighborhood onto X. For x ∈ C let
ρ(x) := 12 d x, S \ r −1 (Uε/2 (x) ∩ X) .
Choose x1 , . . . , xn ∈ C such that
U1 := Uρ(x1 ) (x1 ), . . . , Un := Uρ(xn ) (xn )
is an open cover of C.
Let e1 , . . . , en be the standard unit basis vectors of Rn . The nerve of the open
cover is
[
[
N(U1 ,...,Un) =
conv({ ej : x ∈ Uj }) =
conv(ej1 , . . . , ejk ).
x∈X
Vj1 ∩...∩Vjk 6=∅
Of course it is a (geometric) simplicial complex. There are functions α1 , . . . , αn :
C → [0, 1] given by
d(x, X \ Ui )
αi (x) := Pn
.
j=1 d(x, X \ Uj )
Of course the denominator is always positive, so these functions are well defined
and continuous. There is a continuous function ϕ : C → N(U1 ,...,Un ) given by
ϕ(x) :=
n
X
αj (x)ej .
j=1
We would like to define a function ψ : N(U1 ,...,Un ) → X be setting
ψ
n
X
j=1
αj ej = r
n
X
j=1
αj xj .
Pn
Consider a point y =
j=1 αj ej ∈ N(U1 ,...,Un ) . Let j1 , . . . , jk be the indices j
such that αj > 0, ordered so that ρ(xj1 ) ≥ max{ρ(xj2 ), . . . , ρ(xjk )}. Let
Tp B :=
U2ρ(xj1 ) (xj1 ). The definition of N(U1 ,...,Un ) implies that there is a point z ∈ h=0 Ujh .
For all h = 1, . . . , k we have xjh ∈ B because
d(z, xjh ) < ρ(xjh ) ≤ ρ(xj1 ).
Now note that
B ⊂ r −1 (Uε/2 (xj1 ) ∩ X) ⊂ U.
P
Since B is convex, it contains kh=1 αjh xjh , so ψ is well defined.
Now we would like to define a homotopy η : C × [0, 1] → X by setting
X
η(x, t) = r (1 − t)
αj (x)xj + tx ,
j
106
CHAPTER 7. RETRACTS
so suppose that y = ϕ(x) for some x ∈ C. Then x ∈ Uj1 ∩ . . . ∩ Ujk . In particular
B := U2ρ(xj1 ) (xj1 ) ⊃ Uρ(xj1 ) (xj1 ) = Uj1 , so B contains x. Again, since B is convex
P
P
it contains the line segment between x and nj=1 αj (x)xj = kh=1 αjh xjh , so η is
well defined. Evidently η is continuous with η0 = ψ ◦ ϕ and η1 = IdC . In addition,
since B ⊂ U we have
η(x, t) ∈ r(B) ⊂ Uε/2 (xj1 ) ⊂ Uε (x)
for all 0 ≤ t ≤ 1.
Sometimes we will need the following variant.
Theorem 7.6.4. If X is a separable ANR and C ⊂ X is compact, then for any
ε > 0 there is an open U ⊂ Rm , for some m, such that U is compact and ε-dominates
C.
Proof. Fixing ε > 0, let P ⊂ Rm be a simplicial complex that ε-dominates C by
virtue of the maps ϕ : C → P and ψ : P → X. Since P is an ENR (Proposition
7.3.3) it is a neighborhood retract. Let r : U ′ → P be a retraction of a neighborhood.
For sufficiently small ε > 0 the closed ε-ball around P is contained in U ′ . Let U be
the open ε-ball around P . Of course U is compact. Let ϕ′ : C → U be ϕ interpreted
as a function with range U , and let ψ ′ = ψ ◦ r : U → X. Since ψ ′ ◦ ϕ′ = ψ ◦ ϕ, C is
ε-dominated by U .
Chapter 8
Essential Sets of Fixed Points
Figure 2.1 shows a function f : [0, 1] → [0, 1] with two fixed points, s and t.
Intuitively, they are qualitatively different, in that a small perturbation of f can
result in a function that has no fixed points near s, but this is not the case for t.
This distinction was recognized by Fort (1950) who described s as inessential, while
t is said to be essential.
1
b
b
0
0
s
t
1
Figure 1.1
In game theory one often deals with correspondences with sets of fixed points
that are infinite, and include continua such as submanifolds. As we will see, the
definition proposed by Fort can be extended to sets of fixed points rather easily:
roughly, a set of fixed points is essential if every neighborhood of it contains fixed
points of every “sufficiently close” perturbation of the given correspondence. (Here
one needs to be careful, because in the standard terminology of game theory, following Jiang (1963), essential Nash equilibria, and essential sets of Nash equilibria,
are defined in terms of perturbations of the payoffs. This is a form of Q-robustness,
which is studied in Section 8.3.) But it is easy to show that the set of all fixed
107
108
CHAPTER 8. ESSENTIAL SETS OF FIXED POINTS
points is essential, so some additional condition must be imposed before essential
sets can be used to distinguish some fixed points from others.
The condition that works well, at least from a mathematical viewpoint, is connectedness. This chapter’s main result, Theorem 8.3.2, which is due to Kinoshita
(1953), asserts that minimal (in the sense of set inclusion) essential sets are connected. The proof has the following outline. Let K be a minimal essential set of
an upper semicontinuous convex valued correspondence F : X → X, where X is a
compact, convex subset of a locally convex toplogical vector space. Suppose that
K is disconnected, so there are disjoint open sets U1 , U2 such that K1 := K ∩ U1
and K2 := K ∩ U2 are nonempty and K1 ∪ K2 = K. Since K is minimal, K1 and
K2 are not essential, so there are perturbations F1 and F2 of F such that each Fi
has no fixed points near Ki . Let α1 , α2 : X → [0, 1] be continuous functions such
that each αi vanishes outside Ui and is identically 1 near Ki , and let α : X → [0, 1]
be the function α(x) := 1 − α1 (x) − α2 (x). Then α, α1 , α2 is a partition of unity
subordinate to the open cover X \ K, U1 , U2 . The correspondence
x 7→ α(x)F (x) + α1 (x)F1 (x) + α2 (x)F2 (x)
is then a perturbation of F that has no fixed points near K, which contradicts the
assumption that K is essential. Much of this chapter is concerned with filling in
the technical details of this argument.
Turning to our particular concerns, Section 8.1 gives the Fan-Glicksberg theorem, which is the extension of the Kakutani fixed point theorem to infinite dimensional sets. Section 8.2 shows that convex valued correspondences can be approximated by functions, and defines convex combinations of convex valued correspondences, with continuously varying weights. Section 8.3 then states and proves
Kinoshita’s theorem, which implies that minimal connected sets exist. There remains the matter of proving that minimal essential sets actually exist, which is also
handled in Section 8.3.
8.1
The Fan-Glicksberg Theorem
We now extend the Kakutani fixed point theorem to correspondences with infinite dimensional domains. The result below was proved independently by Fan
(1952) and Glicksberg (1952) using quite similar methods; our proof is perhaps a
bit closer to Fan’s. In a sense the result was already known, since it can be derived
from the Eilenberg-Montgomery theorem, but the proof below is much simpler.
Theorem 8.1.1 (Fan, Glicksberg). If V is a locally convex topological vector space,
X ⊂ V is nonempty, convex, and compact, and F : X → X is an upper semicontinuous convex valued correspondence, then F has a fixed point.
We treat two technical points separately:
Lemma 8.1.2. If V is a (not necessarily locally convex) topological vector space
and K, C ⊂ V with K compact and C closed, then K + C is closed.
109
8.1. THE FAN-GLICKSBERG THEOREM
Proof. We will show that the compliment is open. Let y be a point of V that is not
in K + C. For each x ∈ K, translation invariance of the topology of V implies that
x + C is closed, so Lemma 6.3.2 gives a neighborhood Wx of the origin such that
(y + Wx + Wx ) ∩ (x + C) = ∅. Since we can replace Wx with −Wx ∩ Wx , we may
assume that −Wx = Wx , so that (y + Wx ) ∩ (x + C + Wx ) = ∅. Choose x1 , . . . , xk
such that the sets xi + Wxi cover K, and let W = Wx1 ∩ . . . ∩ Wxk . Now
[
(y + W ) ∩ (K + C) ⊂ (y + W ) ∩ (xi + C + Wxi )
i
⊂
[
i
(y + Wxi ) ∩ (xi + C + Wxi ) = ∅.
Lemma 8.1.3. If V is a (not necessarily locally convex) topological vector space
and K, C, U ⊂ V with K compact, C closed, U open, and C ∩ K ⊂ U, then there
is a neighborhood of the origin W such that (C + W ) ∩ K ⊂ U.
Proof. Let L := K \ U. Our goal is to find a neighborhood of the origin W such
that (C + W ) ∩ L = ∅. Since C is closed, for each x ∈ L there is (by Lemma
6.3.2) a neighborhood Wx of the origin such that (x + Wx + Wx ) ∩ C = ∅. We can
replace Wx with −Wx ∩ Wx , so we may insist that −Wx = Wx . As a closed subset
of K, L is compact, so there are x1 , . . . , xk such that the sets xi + Wxi cover L. Let
W := Wx1 ∩ . . . ∩ Wxk . Then W = −W , so if (C + W ) ∩ L is nonempty, then so is
C ∩ (L + W ), but
[
[
L+W ⊂
xi + Wxi + W ⊂
xi + Wxi + Wxi .
i
i
Proof of Theorem 8.1.1. Let U be a closed convex neighborhood of the origin.
(Lemma 6.3.4 implies that such a U exists.) Let FU : X → X be the correspondence FU (x) := (F (x) + U) ∩ X. Evidently FU (x) is nonempty and convex,
and the first of the two results above implies that it is a closed subset of X, so it is
compact.
To show that FU is upper semicontinuous we consider a particular x and a
neighborhood T of FU (x). The second of the two results above implies that there
is a neighborhood W of the origin such that (F (x) + U + W ) ∩ X ⊂ T . Since F is
upper semicontinuous there is a neighborhood A of x such that F (x′ ) ⊂ F (x) + W
for all x′ ∈ A, and for such an x′ we have
FU (x′ ) = (F (x′ ) + U) ∩ X ⊂ (F (x) + W + U) ∩ X ⊂ T.
Since X is compact, there are finitely many points x1 , . . . , xk ∈ X such that
x1 + U, . . . , xk + U is a cover of X. Let C be the convex hull of these points.
Define G : C → C by setting G(x) = FU (x) ∩ C; since G(x) contains some xi , it is
nonempty, and of course it is convex. Since C is the image of the continuous function
(α1 , . . . , αk ) 7→ α1 x1 +· · ·+αk xk from the (k−1)-dimensional simplex, it is compact,
110
CHAPTER 8. ESSENTIAL SETS OF FIXED POINTS
and consequently closed because V is Hausdorff. Since Gr(G) = Gr(FU ) ∩ (C × C)
is closed, G is upper semicontinuous. Therefore G satisfies the hypothesis of the
Kakutani fixed point theorem and has a nonempty set of fixed points. Any fixed
point of G is a fixed point of FU , so the set FU of fixed points of FU is nonempty.
Of course it is also closed in X, hence compact.
The collection of compact sets
{ FU : U is a closed convex neighborhood of the origin }
has the finite intersection property because FU1 ∩...Uk ⊂ FU1 ∩ . . . ∩ FUk , so its
intersection is nonempty. Suppose that x∗ is an element of this intersection. If x∗
was not an element of F (x∗ ) there would be a closed neighborhood U of the origin
such that (x∗ − U) ∩ F (x∗ ) = ∅, which contradicts x∗ ∈ FU , so x∗ is a fixed point
of F .
8.2
Convex Valued Correspondences
Let X be a topological space, and let Y be a subset of a topological vector space
V . Then Con(X, Y ) is the set of upper semicontinuous convex valued correspondences from X to Y . Let ConS (X, Y ) denote this set endowed with the relative
topology inherited from US (X, Y ), which was defined in Section 5.2. This section
treats two topological issues that are particular to convex valued correspondences:
a) approximation by continuous functions; b) the continuity of the process by which
they are recombined using convex combinations and partitions of unity.
The following result is a variant, for convex valued correspondences, of the
approximation theorem (Theorem 9.1.1) that is the subject of the next chapter.
Proposition 8.2.1. If X is a metric space, V is locally convex, and Y is either
open or convex, then C(X, Y ) is dense in ConS (X, Y ).
Proof. Fix F ∈ Con(X, Y ) and a neighborhood U ⊂ X × Y of Gr(F ). Our goal is
to produce a continuous function f : X → Y with Gr(f ) ⊂ U.
Consider a particular x ∈ X. For each y ∈ F (x) there is a neighborhood Tx,y of
x and (by Lemma 6.3.2) a neighborhood Wx,y of the origin in V such that
Tx,y × (y + Wx,y + Wx,y ) ⊂ U.
If Y is open we can also require that y + Wx,y + Wx,y ⊂ Y . The compactness of
F (x) T
implies that there T
are y1 , . . . , yk such that the yi + Wx,yi cover F (x). Setting
Tx = i Tx,yi and Wx = i Wx,yi , we have Tx ×(F (x)+Wx ) ⊂ U and F (x)+Wx ⊂ Y
if Y is open. Since V is locally convex, we may assume that Wx is convex because
we can replace it with a smaller convex neighborhood. Upper semicontinuity gives
a δx > 0 such that Uδx (x) ⊂ Tx and F (x′ ) ⊂ F (x) + Wx for all x′ ∈ Uδx (x).
Since metric spaces are paracompact there is a locally finite open cover {Tα }α∈A
of X that refines {Uδx /2 (x)}x∈X . For each α ∈ A choose xα such that Tα ⊂
Uδα /2 (xα ), where δα := δxα , and choose yα ∈ F (xα ). Since metric spaces are
8.2. CONVEX VALUED CORRESPONDENCES
111
normal, Theorem 6.2.2 gives a partition of unity {ψα } subordinate to {Tα }α∈A . Let
f : X → V be the function
X
f (x) :=
ψα (x)yα .
α∈A
Fixing x ∈ X, let α1 , . . . , αn be the α such that ψα (x) > 0. After renumbering
we may assume that δα1 ≥ δαi for all i = 2, . . . , n. For each such i we have
xαi ∈ Uδαi /2 (x) ⊂ Uδα1 (xα1 ), so that yαi ∈ F (xα1 ) + Wxα1 . Since F (xα1 ) + Wxα1 is
convex we have
(x, f (x)) ∈ Uδα1 (xα1 ) × (F (xα1 ) + Wxα1 ) ⊂ U.
Note that f (x) is contained in Y either because Y is convex or because F (xα1 ) +
Wxα1 ⊂ Y . Since x was arbitrary, we have shown that Gr(f ) ⊂ U.
We now study correspondences constructed from given correspondences by taking a convex combination, where the weights are given by a partition of unity. Let
X be a compact metric space and let V be a topological vector space. Since addition and scalar multiplication are continuous, Proposition 4.5.9 and Lemma 4.5.10
imply that the composition
(α, K) 7→ {α} × L 7→ αK = { αv : v ∈ K }
(∗)
and the Minkowski sum
(K, L) 7→ K × L 7→ K + L := { v + w : v ∈ K, w ∈ L }
(∗∗)
are continuous functions from R × K(V ) and K(V ) × K(V ) to K(V ).
These operations define continuous functions on the corresponding spaces of
functions and correspondences. Let CS (X) denote the space CS (X, R) defined in
Section 5.5.
Lemma 8.2.2. The function (ψ, F ) 7→ ψF from CS (X)×ConS (X, V ) to ConS (X, V )
is continuous.
Proof. To produce a contradiction suppose the assertion is false. Then there is
a directed set (D, <) and a convergent net, say {(ψ d , F d )}d∈D with limit (ψ, F ),
such that ψ d F d 6→ ψF . Failure of convergence means that there is a neighborhood
W ⊂ X × V of Gr(ψF ) such that (after choosing a subnet) for every d there are
points xd ∈ X and y d ∈ F d (xd ) such that (xd , ψ d (xd )y d) ∈
/ W.
Taking a further subnet, we may assume that xd → x and ψ d (xd ) → α. For each
y ∈ F (x) there are neighborhoods Ty and Uy of x and y such
T that Ty × UyS⊂ W .
Let Uy1 , . . . , Uym be a finite subcover of F (x), and set T := j Tyj and U := j Uyj .
Then T and U are neighborhoods of x and F (x) such that T × U ⊂ W .
The continuity of (∗) and (∗∗) implies that there are neighborhoods A of α and
U of F (x) such that α′ K ⊂ U whenever α′ ∈ A and K ⊂ U. By replacing T with a
smaller neighborhood of x if need be, we can insure that ψ(x′ ) ∈ A and F (x′ ) ⊂ U
for all x′ ∈ T . Then the set of (ψ ′ , F ′ ) such that ψ ′ (x′ ) ∈ A and F ′ (x′ ) ⊂ U for all
112
CHAPTER 8. ESSENTIAL SETS OF FIXED POINTS
x′ ∈ T is a neighborhood of (ψ, F ), so when d is “large” we will have (ψ d , F d ) in
this neighborhood and xd ∈ T , which implies that
{xd } × ψ d (xd )F d (xd ) ⊂ T × U ⊂ W.
This contradicts our supposition, so the proof is complete.
The proof of the following follows the same pattern, and is left to the reader.
Lemma 8.2.3. The function (F1 , F2 ) 7→ ψF1 + F2 from ConS (X, V ) × ConS (X, V )
to ConS (X, V ) is continuous.
If ψ1 , . . . , ψk is a partition of unity subordinate to this cover and F1 , . . . , Fk ∈
Con(X, V ), then each Fi may be regarded as a continuous function from X to K(V ),
so we may define a new continuous function from X to K(V ) by setting
(ψ1 F1 + · · · + ψk Fk )(x) := ψ1 (x)F1 (x) + · · · + ψk (x)Fk (x).
A continuous function from X to K(V ) is the same thing as an upper semicontinuous
compact valued correspondence, so we may regard ψ1 F1 + · · · + ψk Fk as an element
of Con(X, V ). Let PU k (X) be the space of k-element partitions of unity ψ1 , . . . , ψk
of X. We endow PU k (X) with the relative topology it inherits as a subspace of
CS (X)k . The last two results now imply:
Proposition 8.2.4. The function
(ψ1 , . . . , ψk , F1 , . . . , Fk ) 7→ ψ1 F1 + · · · + ψk Fk
from PU k (X) × ConS (X, V )k to ConS (X, V ) is continuous.
8.3
Kinoshita’s Theorem
Let X be a compact convex subset of a locally convex topological vector space,
and fix a particular F ∈ Con(X, X).
Definition 8.3.1. A set K ⊂ F P(F ) is an essential set of fixed points of F
if it is compact and for any open U ⊃ K there is a neighborhood V ⊂ ConS (X, X)
of F such that F P(F ′ ) ∩ U 6= ∅ for all F ′ ∈ V .
The following result from Kinoshita (1952) is a key element of the theory of
essential sets.
Theorem 8.3.2. (Kinoshita) If K ⊂ F P(F ) is essential and K1 , . . . , Kk is a
partition of K into disjoint compact sets, then some Kj is essential.
Proof. Suppose that no Kj is essential. Then for each j = 1, . . . , k there is a
neighborhood Uj of Kj such that for every neighborhood Vj ⊂ ConS (X, X) there is
an Fj ∈ Vj with no fixed points in Uj . Replacing the Uj with smaller neighborhoods
if need be, we can assume that they are pairwise disjoint. Let U be a neighborhood of
X \(U1 ∪. . .∪Uk ) whose closure does not intersect K. A compact Hausdorff space is
8.3. KINOSHITA’S THEOREM
113
normal, so Theorem 6.2.2 implies the existence of a partition of unity ϕ1 , . . . , ϕk , ϕ :
X → [0, 1] subordinate to the open cover U1 , . . . , Uk , U. Let V ⊂ ConS (X, X)
be a neighborhood of F . Proposition 8.2.4 implies that there are neighborhoods
V1 , . . . , Vk ⊂ ConS (X, X) of F such that ϕ1 F1 + · · · + ϕk Fk + ϕF ∈ V whenever
F1 ∈ V1 , . . . , Fk ∈ Vk . For each j we can choose a Fj ∈ Vj that has no fixed points
in Uj . Then ϕ1 F1 + · · · + ϕk Fk + ϕF has no fixed points in X \ U because on each
Uj \ U it agrees with Fj . Since X \ U is a neighborhood of K and V was arbitrary,
this contradicts the assumption that K is essential.
Recall that a topological space is connected if it is not the union of two disjoint
nonempty open sets. A subset of a topological space is connected if the relative
topology makes it a connected space.
Corollary 8.3.3. A minimal essential set is connected.
Proof. Let K be an essential set. If K is not connected, then there are disjoint
open sets U1 , U2 such that K ⊂ U1 ∪ U2 and K1 := K ∩ U1 and K2 := K ∩ U2
are both nonempty. Since K1 and K2 are closed subsets of K, they are compact,
so Kinoshita’s theorem implies that either K1 or K2 is essential. Consequently K
cannot be minimal.
Naturally we would like to know whether minimal essential sets exist. Because
of important applications in game theory, we will develop the analysis in the context
of a slightly more general concept.
Definition 8.3.4. A pointed space is a pair (A, a0 ) where A is a topological
space and a0 ∈ A. A pointed map f : (A, a0 ) → (B, b0 ) between pointed spaces is
a continuous function f : A → B with f (a0 ) = b0 .
Definition 8.3.5. Suppose (A, a0 ) is a pointed space and
Q : (A, a0 ) → (ConS (X, X), F )
is a pointed map. A nonempty compact set K ⊂ F P(F ) is Q-robust if, for every neighborhood V ⊂ X of K, there is a neighborhood U ⊂ A of a0 such that
F P(Q(a)) ∩ V 6= ∅ for all a ∈ U.
A set of fixed points is essential if and only if it is Id(ConS (X,X),F ) -robust. At the
other extreme, if Q is a constant function, so that Q(a) = F for all a, then any
nonempty compact K ⊂ F P(F ) is Q-robust. The weakening of the notion of an
essential set provided by this definition is useful when certain perturbations of F
are thought to be more relevant than others, or when the perturbations of F are
derived from perturbations of the parameter a in a neighborhood of a0 . Some of
the most important refinements of the Nash equilibrium concept have this form. In
particular, Jiang (1963) defines essential Nash equilibria, and essential sets of Nash
equilibria, in terms of perturbations of the game’s payoffs, while Kohlberg and
Mertens (1986) define stable sets of Nash equilibria in terms of those perturbations
of the payoffs that are induced by the trembles of Selten (1975).
Lemma 8.3.6. F P(F ) is Q-robust.
114
CHAPTER 8. ESSENTIAL SETS OF FIXED POINTS
Proof. The continuity of F P (Theorem 5.2.1) implies that for any neighborhood
V ⊂ X of F P(F ) there is a neighborhood U ⊂ A of a0 such that F P(Q(a)) ⊂ V
for all a ∈ U. The Fan-Glicksberg fixed point theorem implies that F P(Q(a)) is
nonempty.
This result shows that if our goal is to discriminate between some fixed points
and others, these concepts must be strengthened in some way. The two main
methods for doing this are to require either connectedness or minimality.
Definition 8.3.7. A nonempty compact set K ⊂ F P(F ) is a minimal Q-robust
set if it is Q-robust and minimal in the class of such sets: K is Q-robust and no
proper subset is Q-robust. A minimal connected Q-robust set is a connected
Q-robust set that does not contain a proper subset that is connected and Q-robust.
In general a minimal Q-robust set need not be connected. For example, if
(A, a0 ) = ((−1, 1), 0) and Q(a)(t) = argmaxt∈[0,1] at (so that F (t) = [0, 1] for all t)
then F P(Q(a)) is {0} if a < 0 and it is {1} if a > 0, so the only minimal Q-robust
set is {0, 1}. In view of this one must be careful to distinguish between a minimal
connected Q-robust set and a minimal Q-robust set that happens to be connected.
Theorem 8.3.8. If K ⊂ F P(F ) is a Q-robust set, then it contains a minimal
Q-robust set, and if K is a connected Q-robust set, then it contains a minimal
connected Q-robust set.
Proof. Let C be the set of Q-robust sets that are contained in K. We order this set
by reverse inclusion, so that our goal is to show that C has a maximal element. This
follows from Zorn’s lemma if we can show that any completely ordered subset O has
an upper bound in C. The finite intersection property implies that the intersection
of all elements of O is nonempty; let K∞ be this intersection. If K∞ is not Qrobust, then there is a neighborhood V of K∞ such that every neighborhood U of
a0 contains a point a such that Q(a) has no fixed points in V . If L ∈ O, we cannot
have L ⊂ V because L is Q-robust, but now { L \ V : L ∈ O } is a collection of
compact sets with the finite intersection property, so it has a nonempty intersection
that is contained in K∞ but disjoint from V . Of course this is absurd.
The argument for connected Q-robust sets follows the same lines, except that in
addition to showing that K∞ is Q-robust, we must also show that it is connected.
If not there are disjoint open sets V1 and V2 such that K∞ ⊂ V1 ∪ V2 and K∞ ∩ V1 6=
∅=
6 K∞ ∩ V2 . For each L ∈ O we have L ∩ V1 6= ∅ =
6 L ∩ V2 , so L \ (V1 ∪ V2 ) must
be nonempty because L is connected. As above, { L \ (V1 ∪ V2 ) : L ∈ O } has a
nonempty intersection that is contained in K∞ but disjoint from V1 ∪ V2 , which is
impossible.
Chapter 9
Approximation of
Correspondences
In extending fixed point theory from functions to correspondences, an important
method is to show that continuous functions are dense in the space of correspondences, so that any correspondence can be approximated by a function. In the
last chapter we saw such a result (Theorem 8.2.1) for convex valued correspondences, but much greater care and ingenuity is required by the arguments showing
that contractible valued correspondences have good approximations. This chapter states and proves the key result in this direction. This result was proved in
the Euclidean case by Mas-Colell (1974) and extended to ANR’s by the author in
McLennan (1991).
9.1
The Approximation Result
Our main result can be stated rather easily. We now fix ANR’s X and Y . We
assume throughout this chapter that X is separable, in order to be able to invoke
the domination theorem.
Theorem 9.1.1 (Approximation Theorem). Suppose that C and D are compact
subsets of X with C ⊂ int D. Let F : D → Y be an upper semicontinuous contractible valued correspondence. Then for any neighborhood U of Gr(F |C ) there
are:
(a) a continuous f : C → Z with Gr(f ) ⊂ U;
(b) a neighborhood U ′ of Gr(F ) such that, for any two continuous functions f0 , f1 :
D → Y with Gr(f0 ), Gr(f1 ) ⊂ U ′ , there is a homotopy h : C × [0, 1] → Y with
h0 = f0 |C , h1 = f1 |C , and Gr(ht ) ⊂ U for all 0 ≤ t ≤ 1.
Roughly, (a) is an existence result, while (b) is uniqueness up to effective equivalence.
Here, and later in the book, things would be much simpler if we could have
C = D. More precisely, it would be nice to drop the assumption that C ⊂ int D.
This may be possible (that is, I do not know a relevant counterexample) but a proof
would certainly involve quite different methods.
115
116
CHAPTER 9. APPROXIMATION OF CORRESPONDENCES
The following is an initial indication of the significance of this result.
Theorem 9.1.2. If X is a compact ANR with the fixed point property, then any
upper semicontinuous contractible valued correspondence F : X → X has a fixed
point.
Proof. In the last result let Y = X and C = D = X. Endow X with a metric dX .
For each j = 1, 2, . . . let
Uj := { (x′ , y ′ ) ∈ X × X : dX (x, x′ ) + dX (y, y ′) < 1/j }
for some (x, y) ∈ Gr(F ), let fj : X → X be a continuous function with Gr(fj ) ⊂ Uj ,
let zj be a fixed point of fj , and let (x′j , yj′ ) be a point in Gr(F ) with dX (x′j , zj ) +
dX (yj′ , zj ) < 1/j. Passing to convergent subsequences, we find that the common
limit of the sequences {x′j }, {yj′ }, and {zj } is a fixed point of F .
Much later, applying Theorem 9.1.1, we will show that a nonempty compact
contractible ANR has the fixed point property.
9.2
Extending from the Boundary of a Simplex
The proof of Theorem 9.1.1 begins with a concrete geometric construction that
is given in this section. In subsequent sections we will transport this result to
increasingly general settings, eventually arriving at our objective.
We now fix a locally convex topological vector space T and a convex Q ⊂ T . A
subset Z of a vector space is balanced if λz ∈ Z whenever z ∈ Z and |λ| ≤ 1. Since
T is locally convex, every neighborhood of the origin contains a convex neighborhood
U, and U ∩ −U is a neighborhood that is convex and balanced. Working with
balanced neighborhoods of the origin allows us to not keep track of the difference
between a neighborhood and its negation.
Proposition 9.2.1. Let A and B be convex balanced neighborhoods of the origin in
T with 2A ⊂ B. Suppose S ⊂ Q is compact and c : S×[0, 1] → S is a contraction for
which there is a δ > 0 such that c(s, t) − c(s′ , t′ ) ∈ B for all (s, t), (s′, t′ ) ∈ S × [0, 1]
with s − s′ ∈ 3A and |t − t′ | < δ. Let L be a simplex. Then any continuous
f ′ : ∂L → (S + A) ∩ Q has a continuous extension f : L → (S + B) ∩ Q.
Proof. Let β be the barycenter of L. We define “polar coordinate” functions
y : L \ {β} → ∂L and t : L \ {β} → [0, 1)
implicitly by requiring that
(1 − t(x))y(x) + t(x)β = x.
Let
L1 = t−1 ([0, 31 ]),
L2 = t−1 ([ 13 , 32 ]),
L3 = t−1 ([ 32 , 1)) ∪ {β}.
We first define f at points in L2 , then extend to L1 and L3 .
9.2. EXTENDING FROM THE BOUNDARY OF A SIMPLEX
117
Let d be a metric on L. Since f ′ , t(·), and y(·) are continuous, and L2 is compact,
for some sufficiently small λ > 0 it is the case that
f ′ (y(x)) − f ′ (y(x′)) ∈ A and |t(x) − t(x′ )| < 31 δ
for all x, x′ ∈ L2 such that d(x, x′ ) < λ. There is a polyhedral subdivision of L2
whose cells are the sets
y −1 (F ) ∩ t−1 ( 31 ),
y −1 (F ) ∩ L2 ,
y −1(F ) ∩ t−1 ( 32 )
for the various faces F of L. Proposition 2.5.2 implies that repeated barycentric
subdivision of this polyhedral complex results eventually in a simplicial subdivision
of L2 whose mesh is less than λ.
For each vertex v of this subdivision choose s(v) ∈ (f ′ (y(v)) + A) ∩ S, and set
f (v) := c(s(v), 3t(v) − 1).
Let ∆ be a simplex of the subdivision of L2 with vertices v1 , . . . , vr . We define f
on ∆ by linear interpolation on ∆: if x = α1 v1 + · · · + αr vr , then
f (x) := α1 f (v1 ) + · · · + αr f (vr ).
This definition does not depend on the choice of ∆ if x is contained in more than
one simplex, it is continuous on each ∆, and the simplices are a finite closed cover
of L2 , so f is continuous.
Suppose that v and v ′ are two vertices of ∆, so they are the endpoints of an
edge. We have d(v, v ′) < λ, so f ′ (y(v)) − f ′ (y(v ′)) ∈ A and |t(v) − t(v ′ )| < 31 δ. In
addition, s(v) − f ′ (y(v)) and f ′ (y(v ′)) − s(v ′ ) are elements of A, so
s(v) − s(v ′ ) ∈ 3A and |(3t(v) − 1) − (3t(v ′ ) − 1)| < δ,
from which it follows, by hypothesis, that f (v) − f (v ′ ) ∈ B. Consider a point
x = α1 v1 + · · · + αr vr ∈ ∆. Since f (v1 ) ∈ S and
f (x) − f (v1 ) =
r
X
j=1
αj (f (vj ) − f (v1 ))
is a convex combination of the vectors f (vj ) − f (v1 ) for the vertices vj of ∆, we
have f (x) ∈ (f (v1 ) + B) ∩ Q ⊂ (S + B) ∩ Q. Thus f (L2 ) ⊂ (S + B) ∩ Q.
We now define f on L1 by setting
f (x) := (1 − 3t(x))f ′ (y(x)) + 3t(x)f ( 31 β + 32 y(x)).
Since f is continuous on L2 , this formula defines a continuous function. Suppose
that
2
y(x) + 13 β = α1 v1 + · · · + αr vr
3
as above. Consider a particular vj . Above we showed that
f ( 32 y(x) + 13 β) ∈ (s(vj ) + B) + Q.
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CHAPTER 9. APPROXIMATION OF CORRESPONDENCES
The point s(vj ) was chosen with f ′ (y(vj )) − s(vj ) ∈ A, and f ′ (y(x)) − f ′ (y(vj )) ∈ A
because d( 32 y(x) + 13 β, y(vj )) < λ, so
f ′ (y(x)) ∈ (s(vj ) + 2A) ∩ Q ⊂ (s(vj ) + B) ∩ Q.
Since f (x) is a convex combination of f ′ (y(x)) and f ( 32 y(x) + 13 β) we have
f (x) ∈ (s(vj ) + B) ∩ Q ⊂ (S + B) ∩ Q.
Thus f (L1 ) ⊂ (S + B) ∩ Q.
Let z be the point S is contracted to by c: c(S, 1) = {z}. We define f on L3 by
setting f (x) := z. Of course this is a continuous function whose image is contained
in S ⊂ (S + B) ∩ Q.
If x ∈ L1 ∩ L2 , then t(x) = 31 and 23 y(x) + 31 β = x, so the formula defining f on
L1 agrees with the definition of f for elements of L2 at x. If v is a vertex of the
subdivision of L2 contained in L2 ∩ L3 , then t(v) = 23 , so that the definition of f on
L2 gives f (v) = c(s(v), 3t(v) − 1) = z. If x ∈ L2 ∩ L3 , then L2 ∩ L3 contains any
simplex of the subdivision of L2 that has x as an element, and the definition of f
on L2 gives f (x) = z. Thus this definition agrees with the definition of f on L2 at
points in L2 ∩ L3 . Thus f is well defined and continuous.
9.3
Extending to All of a Simplicial Complex
As above, Q is a convex subset of T , and we now fix a relatively open Z ⊂ Q.
We also fix a simplicial complex K and a subcomplex J.
Proposition 9.3.1. Let F : K → Z be an upper semicontinuous contractible valued correspondence. Then for any neighborhood W ⊂ K × Z of Gr(F ) there is a
neighborhood W ′ of Gr(F |J ) such that any continuous f ′ : J → Z with Gr(f ′ ) ⊂ W ′
has a continuous extension f : K → Z with Gr(f ) ⊂ W .
The main argument will employ two technical results, the first of which will also
be applied in the next section. Recall that an ANR can be embedded in a normed
space (Proposition 7.4.3) so it is metrizable.
Lemma 9.3.2. Let X be an ANR, let F : X → Z be an upper semicontinuous
correspondence with metric d, and let V ⊂ X × Z be a neighborhood of Gr(F ).
For any x ∈ X there is δ > 0 and a neighborhood B of the origin in Z such that
Uδ (x) × ((F (x) + B) ∩ Z) ⊂ V .
Proof. By the definition of the product topology, for every z ∈ F (x) there exist
δz > 0 and an open neighborhood Az ⊂ Z of the origin in T such that
Uδz (x) × ((z + Az ) ∩ Z) ⊂ V,
and the continuity of addition in T implies that there is a neighborhood Bz of the
origin with Bz + Bz ⊂ Az . Since F (x) is compact there are z1 , . . . ,T
zK such that
z1 + Bz1 , . . . , zk + Bzk is a cover of F (x). Let δ := minj δzj and B := j Bzj .
9.3. EXTENDING TO ALL OF A SIMPLICIAL COMPLEX
119
Lemma 9.3.3. Let U1 , . . . , Un be a cover of a metric space X by open sets, none
of which are X itself. For each y ∈ X let
ry = max sup{ ε > 0 : Uε (y) ⊂ Ui },
i:y∈Ui
and let Vy be an open subset of U(√5−2)ry (y) that contains y. Then for all y, y ′ ∈ X,
if Vy ∩ Vy′ 6= ∅, then Vy′ ⊂ Ury (y).
√
√
Proof. Let α = 5−2 and β = 3− 5. Suppose Vy ∩Vy′ 6= ∅. The distance from y to
any point in Vy′ cannot exceed α(ry + 2ry′ ), so if Vy′ is not contained in Ury (y), then
α(ry +2ry′ ) > ry , which boils down to 2αry′ > βry . Let iy′ be one of the indices such
that Ury′ (y) ⊂ Uiy . We claim that x ∈ Uiy′ because ry′ > α(ry + ry′ ), which reduces
to βry′ > αry . A quick computation verifies that β/2α > α/β, so this follows
from the inequality above. Since y ∈ Uiy′ , and the distance from y to y ′ is less than
α(ry +ry′ ), we have ry > ry′ −α(ry +ry′ ), which reduces to (α−1)r
√ y > βry′ . Together
this inequality and the one above imply that 2α/β > (3 − 5)/(α − 1), but one
may easily compute that in fact these two quantities are equal. This contradiction
completes the proof.
Proof of Proposition 9.3.1. Let m be the largest dimension of any simplex in K that
is not in J. The main idea is to use induction on m, but one of the methods used in
the construction is subdivision of K, and the formulation of the induction hypothesis
must be sensitive to this. Precisely, we will show that for each k = 0, . . . , m there
is a neighborhood Wk ⊂ W of Gr(F ) and a simplicial subdivision of K such that
if Hk is the union of J with the k-skeleton of some further subdivision, then any
f ′ : J → Z with Gr(f ′ ) ⊂ Wk has an extension f : Hk → Z with Gr(f ) ⊂ W .
For k = 0 the claim is obvious: we can let W0 = W and take K itself without any
further subdivision. By induction we may assume that the claim has already been
established with k − 1 in place of k. That is, there is a neighborhood Wk−1 ⊂ W
of Gr(F ) and a simplicial subdivision of K such that if Hk−1 is the union of J
with the (k − 1)-skeleton of some further subdivision, then any f ′ : J → Z with
Gr(f ′ ) ⊂ Wk−1 has an extension f : Hk−1 → Z with Gr(f ) ⊂ W .
We now develop two open coverings of K. Consider a particular x ∈ K. Fix a
contraction cx : F (x) × [0, 1] → F (x). Lemma 9.3.2 allows us to choose a convex
balanced neighborhood Bx of the origin in T and δx > 0 such that
Ux × (F (x) + Bx ) ∩ Z ⊂ Wk−1
where Ux := Uδx (x). By choosing Bx sufficiently small we can also have
(F (x) + Bx ) ∩ Q ⊂ Z.
Since cx is continuous, we can choose a convex balanced neighborhood Ax of the origin in T and a number δx > 0 such that cx (z ′ , t′ ) ∈ cx (z, t)+Bx for all (z, t), (z ′ , t′ ) ∈
F (x) × [0, 1] such that z ′ − z ∈ 3Ax and |t′ − t| < δx . Replacing Ax with a smaller
convex neighborhood if need be, we may assume that 2Ax ⊂ Bx . Since F is upper semicontinuous and δx may be replaced by a smaller positive number, we can
120
CHAPTER 9. APPROXIMATION OF CORRESPONDENCES
′
insure that F (x′ ) ⊂ F (x) + 21 Ax whenever
Tnx ∈ Ux . Choose x1 , . . . , xn such that
Ux1 , . . . , Uxn is a covering of K. Let A := i=1 Axi .
The second open covering of K is finer. For each y ∈ K let
ry = max sup{ ε > 0 : Uε (y) ⊂ Ui }.
i:y∈Ui
The upper semicontinuity of F implies that each y has an open neighborhood Vy
such that F (y ′) ⊂ F (y) + 12 A for all y ′ ∈ Uεy (y). We can replace Vy with a smaller
neighborhood to bring about Vy ⊂ U(√5−2)ry (y). Choose y1 , . . . , yp ∈ K such that
Vy1 , . . . , Vyp cover K. Set
Wk :=
p
[
j=1
Vyj × ((F (yj ) + 12 A) ∩ Z).
Evidently Gr(F ) ⊂ Wk . We have Wk ⊂ Wk−1 because for each j there is some i
such that Vyj ⊂ Uxi and
(F (yj ) + 21 A) ∩ Z ⊂ ((F (xi ) + 21 Axi ) + 12 A) ∩ Z ⊂ (F (xi ) + Axi ) ∩ Z.
Starting with the subdivision of K obtained at stage k − 1, by Proposition 2.5.2
repeated barycentric subdivision leads eventually to a subdivision of K with each
simplex contained in some Vyj . Let Hk be the union of J with the k-skeleton of
some further subdivision, and fix a continuous f ′ : J → Z with Gr(f ′ ) ⊂ Wk . By
the induction hypothesis there is an extension f of f ′ to the (k − 1)-skeleton of the
further subdivision. Since extensions to each of the k-simplices that are in Hk but
not in J combine to give the desired sort of extension, it suffices to show that there
is an extension to a single such k-simplex L.
By construction there is a j such that L ⊂ Vyj . Let J be the set of j ′ such
that Vyj ∩ Vyj′ 6= ∅. There is some Xi with Vyj′ ⊂ Uxi for all j ′ ∈ J, either because
all of K is contained in a single Xi or as an application of the lemma above. The
conditions imposed on our construction imply that
[
[
F (yj ′ ) + 21 Axi ⊂ F (xi ) + Axi .
f (∂L) ⊂
F (yj ′ ) + 21 A ⊂
j ′ ∈J
j ′ ∈J
Now Lemma 9.2.1, with Ax , Bx , δx , F (x), and f |∂L in place of A, B, δ, S, and f ′ ,
gives a continuous extension f : L → Z with f (L) ⊂ (F (xi ) + Bxi ) ∩ Q, and by
construction this set is contained in Z. The proof is complete.
9.4
Completing the Argument
The next step is a result in which the domains are subsets of the ANR X.
Proposition 9.4.1. Suppose that C ⊂ D ⊂ X where C and D are compact with
C ⊂ int D. Let F : D → Z be an upper semicontinuous contractible valued correspondence. Then for any neighborhood V of Gr(F |C ) there exist:
(a) a continuous f : C → Z with Gr(f ) ⊂ V ;
9.4. COMPLETING THE ARGUMENT
121
(b) a neighborhood V ′ of Gr(F ) such that for any two functions f0 , f1 : D → Z
with Gr(f0 ), Gr(f1 ) ⊂ V ′ there is a homotopy h : C×[0, 1] → Z with h0 = f0 |C ,
h1 = f1 |C , and Gr(ht ) ⊂ V for all 0 ≤ t ≤ 1.
The passage from this to the main result is straightforward.
Proof of Theorem 9.1.1. Recall (Proposition 7.4.1) that an ANR is a retract of a
relatively open subset of a convex subset of a locally convex space. In particular,
we now fix a locally convex space T , an open subset Z of a convex subset of T , and
a retraction r : Z → Y . Let i : Y → Z be the inclusion. Let
V := (IdX × r)−1 (U).
Proposition 9.4.1(a) implies that there is a continuous f ′ : C → Z with Gr(f ′ ) ⊂ V ,
and setting f := r ◦ f ′ verifies (a) of Theorem 9.1.1.
Let V ′ ⊂ V be a neighborhood of Gr(i ◦ F ) with the property asserted by
Proposition 9.4.1(b). Let U ′ := (IdX × i)−1 (V ′ ). Suppose that f0 , f1 : D → Y with
Gr(f0 ), Gr(f1 ) ⊂ U ′ . Then there is a homotopy h : C × [0, 1] → Z with
h0 = i ◦ f0 |C ,
h1 = i ◦ f1 |C ,
and Gr(ht ) ⊂ V for all 0 ≤ t ≤ 1,
so that
r ◦ h0 = f0 |C ,
r ◦ h1 = f1 |C ,
and Gr(r ◦ ht ) ⊂ U for all 0 ≤ t ≤ 1.
This confirms (b) of Theorem 9.1.1.
The proof of Proposition 9.4.1 depends on two more technical lemmas. Below
d denotes a metric for X. For the two lemmas below an upper semicontinuous
correspondence F : X → Z is given.
Lemma 9.4.2. Suppose that C ⊂ X is compact, and V ⊂ C × Z is a neighborhood
of Gr(F |C ). Then there is ε > 0 and a neighborhood Ṽ of Gr(F ) such that
[
Uε (x) × {z} ⊂ V.
(x,z)∈Ṽ
Proof. For each x ∈ C Lemma 9.3.2 allows us to choose δx > 0 and a neighborhood
Ax of F (x) such that Uδx (x) × Ax ⊂ V . Replacing δx with a smaller number
if need be, we may assume without loss of generality that F (x′ ) ⊂ Ax for all
x′ ∈ Uδx (x). Choose x1 , . . . , xH such that Uδx1 /2 (x1 ), . . . , UδxH /2 (xH ) cover C. Let
ε := min{δxi /2}, and set
[
Ṽ :=
Uδxi /2 (xi ) × Axi .
i
Lemma 9.4.3. Suppose that f : S → X is a continuous function, where S is
a compact metric space. If U is a neighborhood of Gr(F ◦ f ), then there is a
neighborhood V of Gr(F ) such that (f × IdZ )−1 (V ) ⊂ U.
122
CHAPTER 9. APPROXIMATION OF CORRESPONDENCES
Proof. Consider a particular x ∈ X. Applying Lemma 9.3.2, for any s ∈ f −1 (x) we
can choose a neighborhood Ns of s and a neighborhood As ⊂ Y of F (x) such that
Ns × As ⊂ U. Since f −1 (s) is compact, there are s1 , . . . , sℓ such that Ns1 , . . . , Nsℓ
cover f −1 (s). Let A := As1 ∩ . . . ∩ Asℓ , and let W be a neighborhood of x small
that f −1 (W ) ⊂ Ns1 ∪ . . . ∪ Nsℓ and F (x′ ) ⊂ A for all x′ ∈ W . (Such a W must exist
because S is compact and F is upper semicontinuous.) Then
[
(f × IdY )−1 (W × A) ⊂
Nsi × A ⊂ U.
i
Since x was arbitrary, this establishes the claim.
Proof of Proposition 9.4.1. Lemma 9.4.2 gives a neighborhood V ′′ of Gr(F ) and
ε > 0 such that
[
Uε (x) × {z} ⊂ V.
(x,z)∈V ′′
After replacing ε with a smaller number, Uε (C) is contained in the interior of D.
Because X is separable, the domination theorem (Theorem 7.6.3) implies that there
is a simplicial complex K that ε-dominates D by virtue of the maps ϕ : D → K
and ψ : K → X. Let
W ′′ := (ψ × IdZ )−1 (V ′′ ).
Since ψ ◦ ϕ is ε-homotopic to IdD we have ϕ(C) ⊂ ψ −1 (Uε (C)). Since ϕ(C)
is compact and ψ −1 (Uε (C)) is open, Proposition 2.5.2 implies that after repeated
subdivisions of K the subcomplex H consisting of all simplices that intersect ϕ(C)
will satisfy ψ(H) ⊂ Uε (C). Since W ′′ is a neighborhood of Gr(F ◦ψ|H ), Proposition
9.3.1 implies the existence of a function f ′ : H → Z with Gr(f ′ ) ⊂ W ′′ . Let
f := f ′ ◦ ϕ|C . Then Gr(f ) ⊂ V , which verifies (a), because
[
(ϕ|C × IdZ )−1 (W ′′ ) = ((ψ ◦ ϕ|C ) × IdZ )−1 (V ′′ ) ⊂
Uε (x) × {z} ⊂ V. (∗)
(x,z)∈V ′′
Turning to (b), let G : H × [0, 1] → Z be the correspondence G(z, t) = F (ψ(z)).
We apply Proposition 9.3.1, with G, H × [0, 1], W ′′ × [0, 1], and H × {0, 1} in
place of F , K, W , and J respectively, obtaining neighborhoods W0′ , W1′ ⊂ W ′′ of
Gr(F ◦ψ|H ) such that for any continuous functions f0′ , f1′ : H → Z with Gr(f0′ ) ⊂ W0′
and Gr(f1′ ) ⊂ W1′ , there is a homotopy h′ : H × [0, 1] → Z with h′0 = f0′ , h′1 = f1′ ,
and Gr(h′t ) ⊂ W ′′ for all t. Let W ′ = W0′ ∩ W1′ .
Lemma 9.4.3 implies that there is a neighborhood V ′ of Gr(F ) such that
(ψ|H × IdZ )−1 (V ′ ) ⊂ W ′.
Replacing V ′ with V ′ ∩ V ′′ if need be, we may assume that V ′ ⊂ V ′′ .
Now consider continuous f0 , f1 : D → Z with Gr(f0 ), Gr(f1 ) ⊂ V ′ . We have
Gr(f0 ◦ ψ|H ), Gr(f1 ◦ ψ|H ) ⊂ W ′ .
Therefore there is a homotopy j : H×[0, 1] → Z with j0 = f0 ◦ψ|H , j1 = f1 ◦ψ|H , and
Gr(jt ) ⊂ W ′′ for all t. Let h′′ : C ×[0, 1] → Z be the homotopy h′′ (x, t) = j(ϕ(x), t).
In view of (∗) we have
Gr(h′′t ) ⊂ (ϕ|C × IdZ )−1 (W ′′ ) ⊂ V
9.4. COMPLETING THE ARGUMENT
123
for all t. Of course h′′0 = f0 ◦ ψ ◦ ϕ|C and h′′1 = f1 ◦ ψ ◦ ϕ|C .
We now construct a homotopy h′ : C×[0, 1] → Z with h′0 = f0 |C , h′1 = f0 ◦ψ◦ϕ|C ,
and Gr(h′t ) ⊂ V for all t. Let η : D × [0, 1] → X be an ε-homotopy with η0 = IdD
and η1 = ψ ◦ ϕ, and define h′ by h′ (x, t) := f0 (η(x, t)). Then h′ has the desired
endpoints, and for all (x, t) in the domain of h′ we have (x, h′t (x)) ∈ V because
d(x, η(x, t)) < ε and
(η(x, t), h′t (x)) = (η(x, t), f0 (η(x, t))) ∈ V ′ ⊂ V ′′ .
Similarly, there is a homotopy h′′′ : C × [0, 1] → Z with h′′′
0 = f0 ◦ ψ ◦ ϕ|C ,
′′′
′′′
h1 = f1 |C , and Gr(ht ) ⊂ V for all t. To complete the proof of (b) we construct a
homotopy h by setting ht = h′3t for 0 ≤ t ≤ 1/3, ht = h′′3t−1 for 1/3 ≤ t ≤ 2/3, and
ht = h′′′
3t−2 for 2/3 ≤ t ≤ 1.
Part II
Smooth Methods
124
Chapter 10
Differentiable Manifolds
This chapter introduces the basic concepts of differential topology: ‘manifold,’
‘tangent vector,’ ‘smooth map,’ ‘derivative.’ If these concepts are new to you,
you will probably be relieved to learn that these are just the basic concepts of
multivariate differential calculus, with a critical difference.
In multivariate calculus you are handed a coordinate system, and a geometry,
when you walk in the door, and everything is a calculation within that given Euclidean space. But many of the applications of multivariate calculus take place in
spaces like the sphere, or the physical universe, whose geometry is not Euclidean.
The theory of manifolds provides a language for the concepts of differential calculus
that is in many ways more natural, because it does not presume a Euclidean setting.
Roughly, this has two aspects:
• In differential topology spaces that are locally homeomorphic to Euclidean
spaces are defined, and we then impose structure that allows us to talk about
differentiation of functions between such spaces. The concepts of interest
to differential topology per se are those that are “invariant under diffeomorphism,” much as topology is sometimes defined as “rubber sheet geometry,”
namely the study of those properties of spaces that don’t change when the
space is bent or stretched.
• The second step is to impose local notions of angle and distance at each point
of a manifold. With this additional structure the entire range of geometric
issues can be addressed. This vast subject is called differential geometry.
For us differential topology will be primarily a tool that we will use to set up an
environment in which issues related to fixed points have a particularly simple and
tractable structure. We will only scratch its surface, and differential geometry will
not figure in our work at all.
The aim of this chapter is provide only as much information as we will need
later, in the simplest and most concrete manner possible. Thus our treatment of
the subject is in various ways terse and incomplete, even as an introduction to this
topic, which has had an important influence on economic theory. Milnor (1965) and
Guillemin and Pollack (1974) are recommended to those who would like to learn a
bit more, and at a somewhat higher level Hirsch (1976) is more comprehensive, but
still quite accessible.
125
126
10.1
CHAPTER 10. DIFFERENTIABLE MANIFOLDS
Review of Multivariate Calculus
We begin with a quick review of the most important facts of multivariate differential calculus. Let f : U → Rn be a function where U ⊂ Rm is open. Recall
that if r ≥ 1 is an integer, we say that f is C r if all partial derivatives of order
≤ r are defined and continuous. For reasons that will become evident in the next
paragraph, it can be useful to extend this notation to include r = 0, with C 0 interpreted as a synonym for “continuous.” We say that f is C ∞ if it is C r for all finite
r. An order of differentiability is either a nonnegative integer r or ∞, and we
write 2 ≤ r ≤ ∞, for example, to indicate that r is such an object, within the given
bounds.
If f is C 1 , then f is differentiable: for each x ∈ U and ε > 0 there is δ > 0
such that
kf (x′ ) − f (x) − Df (x)(x′ − x)k ≤ εkx′ − xk
for all x′ ∈ U with kx′ − xk < δ, where the derivative of f at x is the linear
function
Df (x) : Rm → Rn
given by the matrix of first partial derivatives at x. If f is C r , then the function
Df : U → L(Rm , Rn )
is C r−1 if we identify L(Rm , Rn ) with the space Rn×m of n × m matrices. The
reader is expected to know the standard facts of elementary calculus, especially
that addition and multiplication are C ∞ , so that functions built up from these
operations (e.g., linear functions and matrix multiplication) are known to be C ∞ .
There are three basic operations used to construct new C r functions from give
functions. The first is restriction of the function to an open subset of its domain,
which requires no comment because the derivative is unaffected. The second is
forming the cartesian product of two functions: if f1 : U → Rn1 and f2 : U → Rn2
are functions, we define f1 × f2 : U → Rn1 +n2 to be the function x 7→ (f1 (x), f2 (x)).
Evidently f1 × f2 is C r if and only if f1 and f2 are C r , and when this is the case
we have
D(f1 × f2 ) = Df1 × Df2 .
The third operation is composition. The most important theorem of multivariate
calculus is the chain rule: if U ⊂ Rm and V ⊂ Rn are open and f : U → V and
g : V → Rp are C 1 , then g ◦ f is C 1 and
D(g ◦ f )(x) = Dg(f (x)) ◦ Df (x)
for all x ∈ U. Of course the composition of two C 0 functions is C 0 . Arguing inductively, suppose we have already shown that the composition of two C r−1 functions
is C r−1 . If f and g are C r , then Dg ◦ f is C r−1 , and we can apply the result above
about cartesian products, then the chain rule, to the composition
x 7→ (Dg(f (x)), Df (x)) 7→ Dg(f (x)) ◦ Df (x)
to show that D(g ◦ f ) is C r−1 , so that g ◦ f is C r .
10.1. REVIEW OF MULTIVARIATE CALCULUS
127
Often the domain and range of the pertinent functions are presented to us as
vector spaces without a given or preferred coordinate system, so it is important to
observe that we can use the chain rule to achieve definitions that are independent of
the coordinate systems. Let X and Y be m- and n-dimensional vector spaces. (In
this chapter all vector spaces are finite dimensional, with R as the field of scalars.)
Let c : X → Rm and d : Y → Rn be linear isomorphisms. If U ⊂ X is open, we can
say that a function f : U → Y is C r , by definition, if d ◦ f ◦ c−1 : c(U) → Rk is C r ,
and if this is the case and x ∈ U, then we can define the derivative of f at x to be
Df (x) = d−1 ◦ D(d ◦ f ◦ c−1 )(c(x)) ◦ c ∈ L(X, Y ).
Using the chain rule, one can easily verify that these definitions do not depend
on the choice of c and d. In addition, the chain rule given above can be used to
show that this “coordinate free” definition also satisfies a chain rule. Let Z be a
third p-dimensional vector space. Then if V ⊂ Y is open, g : V → Z is C r , and
f (U) ⊂ V , then g ◦ f is C r and D(g ◦ f ) = Dg ◦ Df .
Sometimes we will deal with functions whose domains are not open, and we need
to define what it means for such a function to be C r . Let S be a subset of X of any
sort whatsoever. If Y is another vector space and f : S → Y is a function, then
f is C r by definition if there is an open U ⊂ X containing S and a C r function
F : U → Y such that f = F |S . Evidently being C r isn’t the same thing as having
a well defined derivative at each point in the domain!
Note that the identity function on S is always C r , and the chain rule implies
that compositions of C r functions are C r . Those who are familiar with the category
concept will recognize that there is a category of subsets of finite dimensional vector
spaces and C r maps between them. (If you haven’t heard of categories it would
certainly be a good idea to learn a bit about them, but what happens later won’t
depend on this language.)
We now state coordinate free versions of the inverse and implicit function theorems. Since you are expected to know the usual, coordinate dependent, formulations
of these results, and it is obvious that these imply the statements below, we give
no proofs.
Theorem 10.1.1 (Inverse Function Theorem). If n = m (that is, X and Y are
both m-dimensional) U ⊂ X is open, f : U → Y is C r , x ∈ U, and Df (x) is
nonsingular, then there is an open V ⊂ U containing x such that f |V is injective,
f (V ) is open in Y , and (f |V )−1 is C r .
Suppose that U ⊂ X × Y is open and f : U → Z is a function. If f is C 1 , then,
at a point (x, y) ∈ U, we can define “partial derivatives” Dx f (x, y) ∈ L(X, Z) and
Dy f (x, y) ∈ L(Y, Z) to be the derivatives of the functions
f (·, y) : { x ∈ X : (x, y) ∈ U } → Z
and f (x, ·) : { y ∈ Y : (x, y) ∈ U } → Z
at x and y respectively.
Theorem 10.1.2 (Implicit Function Theorem). Suppose that p = n. (That is Y
and Z have the same dimension.) If U ⊂ X × Y is open, f : U → Z is C r ,
128
CHAPTER 10. DIFFERENTIABLE MANIFOLDS
(x0 , y0 ) ∈ U, f (x0 , y0 ) = z0 , and Dy f (x0 , y0 ) is nonsingular, then there is an open
V ⊂ X containing x0 , an open W ⊂ U containing (x0 , y0 ), and a C r function
g : V → Y such that g(x0 ) = y0 and
{ (x, g(x)) : x ∈ V } = { (x, y) ∈ W : f (x, y) = z0 }.
In addition
Dg(x0 ) = −Dy f (x0 , y0 )−1 ◦ Dx f (x0 , y0 ).
We will sometimes encounter settings in which the decomposition of the domain
into a cartesian product is not given. Suppose that T is a fourth vector space,
U ⊂ T is open, t0 ∈ U, f : U → Z is C r , and Df (t0 ) : T → Z is surjective.
Let Y be a linear subspace of T of the same dimension as Z such that Df (t0 )|Y
is surjective, and let X be a complementary linear subspace: X ∩ Y = {0} and
X + Y = T . If we identify T with X × Y , then the assumptions of the result above
hold. We will understand the implicit function theorem as extending in the obvious
way to this setting.
10.2
Smooth Partitions of Unity
A common problem in differentiable topology is the passage from local to global.
That is, one is given or can prove the existence of objects that are defined locally
in a neighborhood of each point, and one wishes to construct a global object with
the same properties. A common and simple method of doing so is to take convex
combinations, where the weights in the convex combination vary smoothly. This
section develops the technology underlying this sort of argument, then develops
some illustrative and useful applications.
Fix a finite dimensional vector space X.
Definition
10.2.1. Suppose that {Uα }α∈A is a collection of open subsets of X,
S
U = α Uα , and 0 ≤ r ≤ ∞. A C r partition of unity for U subordinate to
{Uα } is a collection {ϕβ : X → [0, 1]}β∈B of C r functions such that:
(a) for each β the closure of Vβ = { x ∈ X : ϕβ (x) > 0 } is contained in some Uα ;
(b) {Vβ } is locally finite (as a cover of U);
P
(c)
β ϕβ (x) = 1 for each x ∈ U.
The first order of business is to show that such partitions of unity exist. The
key idea is the following ingenious construction.
Lemma 10.2.2. There is a C ∞ function γ : R → R with γ(t) = 0 for all t ≤ 0 and
γ(t) > 0 for all t > 0.
Proof. Let
γ(t) :=
(
0,
t ≤ 0,
−1/t
e
, t > 0.
129
10.2. SMOOTH PARTITIONS OF UNITY
Standard facts of elementary calculus can be combined inductively to show that for
each r ≥ 1 there is a polynomial Pr such that γ (r) (t) is Pr (1/t)e−1/t if t > 0. Since
the exponential function dominates any polynomial, it follows that γ (r) (t)/t → 0 as
t → 0, so that each γ (r) is differentiable at 0 with γ (r+1) (0) = 0. Thus γ is C ∞ .
Note that for any open rectangle
x 7→
Y
i
Qm
i=1 (ai , bi )
⊂ Rm the function
γ(xi − ai )γ(bi − xi )
is C ∞ , positive everywhere in the rectangle, and zero everywhere else.
S
Lemma 10.2.3. If {Uα } is a collection of open subsets of Rm and U = α Uα ,
then U has a locally finite (relative to U) covering by open rectangles, each of whose
closures in contained in some Uα .
Proof. For any integer j ≥ 0 and vector k = (k1 , . . . , km ) with integer components
let
Qj,k =
m
Y
i=1
(ki − 1)/2j , (ki + 1)/2j
and Q′j,k =
m
Y
i=1
(ki − 2)/2j , (ki + 3)/2j .
The cover consists of those Qj,k such that the closure of Qj,k is contained in some
Uα and, if j > 0, there is no α such that the closure of Q′j,k is contained in Uα .
Consider a point x ∈ U. The last requirement implies that x has a neighborhood
that intersects only finitely many cubes in the collection, which is to say that the
collection is locally finite.
For any j the Qj,k cover Rm , so there is some k such that x ∈ Qj,k , and if j
is sufficiently small, then the closure of Qj,k is contained in some Uα . If Qj,k is
not in the collection, then the closure of Q′j,k is contained in some Uα . Define k ′
by letting ki′ be ki /2 or (ki + 1)/2 according to whether ki is even or odd. Then
Qj,k ⊂ Qj−1,k′ ⊂ Q′j,k . Repeating this leads eventually to an element of the collection
that contains x, so the collection is indeed a cover of U.
Imposing a coordinate system on X, then combining the observations above,
proves that:
∞
Theorem 10.2.4. For
S any collection {Uα }α∈A of open subsets of X there is a C
partition of unity for α Uα subordinate to {Uα }.
For future reference we mention a special case that comes up frequently:
Corollary 10.2.5. If U ⊂ X is open and C0 and C1 are disjoint closed subsets of
U, then there is a C ∞ function α : U → [0, 1] with α(x) = 0 for all x ∈ C0 and
α(x) = 1 for all x ∈ C1 .
Proof. Let {ϕ0 , ϕ1 } be a C ∞ partition of unity subordinate to the open cover {U \
C1 , U \ C0 }, and set α = ϕ1 .
130
CHAPTER 10. DIFFERENTIABLE MANIFOLDS
Now let Y be a second vector space. As a first application we consider a problem
that arises in connection with the definition in the last section of what it means for
a C r function f : S → Y on a general domain S ⊂ X to be C r . We say that f is
locally C r if each x ∈ S has a neighborhood Ux ⊂ X that is the domain of a C r
function Fx : Ux → Y with Fx |S∩Ux = f |S∩Ux . This seems like the “conceptually
correct” definition of what it means for a function to be C r , because this should be
a local property that can be checked by looking at a neighborhood of an arbitrary
point in the function’s domain. A C r function is locally C r , obviously. Fortunately
the converse holds, so that the definition we have given agrees with the one that
is conceptually correct. (In addition, it will often be pleasant to apply the given
definition because it is simpler!)
Proposition 10.2.6. If S ⊂ X and f : S → Y is locally C r , then f is C r .
∞
Proof. Let
S {Fx : Ux → Y }x∈S be as above. Let {ϕβ }β∈B be a C partition of unity
for U = x Ux subordinate to {Ux }. For each β chooseP
an xβ such that the closure
of { x : ϕβ (x) > 0 } is contained in Uxβ , and let F := β ϕβ · Fxβ : U → Y . Then
F is C r because each point in U has a neighborhood in which it is a finite sum of
C r functions. For x ∈ S we have
X
X
F (x) =
ϕβ (x) · Fxβ (x) =
ϕβ (x) · f (x) = f (x).
β
β
Here is another useful result applying a partition of unity.
Proposition 10.2.7. For any S ⊂ X, C ∞ (S, Y ) is dense in CS (S, Y ).
Proof. Fix a continuous f : S → Y and an open W ⊂ S × Y containing the graph
of f . Our goal is to find a C ∞ function from S to Y whose graph is also contained
in W .
For each p ∈ S choose a neighborhood Up of p and εp > 0 small enough that
f (Up ∩ S) ⊂ Uεp (f (p)) and (Up ∩ S) × U2εp (f (p)) ⊂ W.
S
Let U = p∈W Up . Let {ϕβ }β∈B be a C ∞ partition of unity for U subordinate to
{Up }p∈S . For each β let Vβ = { x : ϕβ (x) > 0 }, choose some pβ such that Vβ ⊂ Upβ ,
P
and let Uβ = Upβ and εβ = εpβ . Let f˜ : U → Y be the function x 7→ β ϕβ (x)·f (pβ ).
Since {Vβ } is locally finite, f˜ : U → Y is C ∞ , so f˜|S is C ∞ .
We still need to show that the graph of f˜|S is contained in W . Consider some
p ∈ S. Of those β with ϕβ (p) > 0, let α be one of those for which εβ is maximal. Of
course p ∈ Upα , and f˜(p) ∈ U2εα (f (pα )) because for any other β such that ϕβ (p) > 0
we have
kf (pβ ) − f (pα )k ≤ kf (pβ ) − f (p)k + kf (p) − f (pα )k < 2εα .
Therefore (p, f˜(p)) ∈ Upα × U2εα (f (pα )) ⊂ W .
10.3. MANIFOLDS
10.3
131
Manifolds
The maneuver we saw in Section 10.1—passing from a calculus of functions
between Euclidean spaces to a calculus of functions between vector spaces—was
accomplished not by fully “eliminating” the coordinate systems of the domain and
range, but instead by showing that the “real” meaning of the derivative would not
change if we replaced those coordinate systems by any others. The definition of a
C r manifold, and of a C r function between such manifolds, is a more radical and
far reaching application of this idea.
A manifold is an object like the sphere, the torus, and so forth, that “looks like”
a Euclidean space in a neighborhood of any point, but which may have different
sorts of large scale structure. We first of all need to specify what “looks like” means,
and this will depend on a degree of differentiability. Fix an m-dimensional vector
space X, an open U ⊂ X, and a degree of differentiability 0 ≤ r ≤ ∞.
Recall that if A and B are topological spaces, a function e : A → B is an
embedding if it is continuous and injective, and its inverse is continuous when
e(A) has the subspace topology. Concretely, e maps open sets of A to open subsets
of e(A). Note that the restriction of an embedding to any open subset of the domain
is also an embedding.
Lemma 10.3.1. If U ⊂ X is open and ϕ : U → Rk is a C r embedding such that
for all x ∈ U the rank of Dϕ(x) is m, then ϕ−1 is a C r function.
Proof. By Proposition 10.2.6 it suffices to show that ϕ−1 is locally C r . Fix a point p
in the image of ϕ, let x = ϕ−1 (p), let X ′ be the image of Dϕ(x), and let π : Rk → X ′
be the orthogonal projection. Since ϕ is an immersion, X ′ is m-dimensional, and
the rank of D(π ◦ ϕ)(x) = π ◦ Dϕ(x) is m. The inverse function theorem implies
that the restriction of π ◦ ϕ to some open subset of Ũ containing x has a C r inverse.
Now the chain rule implies that ϕ−1 |ϕ(Ũ ) = (π ◦ ϕ|Ũ )−1 ◦ π|ϕ(Ũ ) is C r .
Definition 10.3.2. A set M ⊂ Rk is an m-dimensional C r manifold if, for
each p ∈ M, there is a C r embedding ϕ : U → M, where U is an open subset of
an m-dimensional vector space, such that for all x ∈ U the rank of Dϕ(x) is m
and ϕ(M) is a relatively open subset of M that contains p. We say that ϕ is a C r
parameterization for M and ϕ−1 is a C r coordinate chart for M. A collection
{ϕi }i∈I of C r parameterizations for M whose images cover M is called a C r atlas
for M.
Although the definition above makes sense when r = 0, we will have no use for
this case because there are certain pathologies that we wish to avoid. Among other
things, the beautiful example known as the Alexander horned sphere (Alexander
(1924)) shows that a C 0 manifold may have what is known as a wild embedding
in a Euclidean space. From this point on we assume that r ≥ 1.
There are many “obvious” examples of C r manifolds such as spheres, the torus,
etc. In analytic work one should bear in mind the most basic examples:
(i) A set S ⊂ Rk is discrete if each p ∈ S has a neighborhood W such that
S ∩ W = {p}. A discrete set is a 0-dimensional C r manifold.
132
CHAPTER 10. DIFFERENTIABLE MANIFOLDS
(ii) Any open subset (including the empty set) of an m-dimensional affine subspace of Rk is an m-dimensional C r manifold. More generally, an open subset
of an m-dimensional C r manifold is itself an m-dimensional C r manifold.
(iii) If U ⊂ Rm is open and φ : U → Rk−m is C r , then the graph
Gr(φ) := { (x, φ(x)) : x ∈ U } ⊂ Rk
of φ is an m-dimensional C r manifold, because ϕ : x 7→ (x, φ(x)) is a C r
parameterization.
10.4
Smooth Maps
Let M ⊂ Rk be an m-dimensional C r manifold, and let N ⊂ Rℓ be an ndimensional C r manifold. We have already defined what it means for a function
f : M → N is C r to be C r : there is an open W ⊂ Rk that contains M and a C r
function F : W → Rℓ such that F |M = f . The following characterization of this
condition is technically useful and conceptually important.
Proposition 10.4.1. For a function f : M → N the following are equivalent:
(a) f is C r ;
(b) for each p ∈ M there are C r parameterizations ϕ : U → M and ψ : V → N
such that p ∈ ϕ(U), f (ϕ(U)) ⊂ ψ(V ), and ψ −1 ◦ f ◦ ϕ is a C r function;
(c) ψ −1 ◦ f ◦ ϕ is a C r function whenever ϕ : U → M and ψ : V → N are C r
parameterizations such that f (ϕ(U)) ⊂ ψ(V ).
Proof. Because compositions of C r functions are C r , (a) implies (c), and since each
point in a manifold is contained in the image of a C r parameterization, it is clear
that (c) implies (b). Fix a point p ∈ M and C r parameterizations ϕ : U → M and
ψ : V → N with p ∈ ϕ(U) and f (ϕ(U)) ⊂ ψ(V ). Lemma 10.3.1 implies that ϕ−1
and ψ −1 are C r , so ψ ◦ (ψ −1 ◦ f ◦ ϕ) ◦ ψ −1 is C r on its domain of definition. Since
p was arbitrary, we have shown that f is locally C r , and Proposition 10.2.6 implies
that f is C r . Thus (b) implies (a).
There is a more abstract approach to differential topology (which is followed in
Hirsch (1976)) in which an m-dimensional C r manifold is a topological space M
together with a collection { ϕα : Uα → M }α∈A , where each ϕα is a homeomorphism
betweenS an open subset Uα of an m-dimensional vector space and an open subset
r
of M, α ϕα (Uα ) = M, and for any α, α′ ∈ A, ϕ−1
α′ ◦ ϕα is C on its domain of
definition. If N with collection { ψβ : Vβ :→ N } is an n-dimensional C r manifold,
a function f : M → N is C r by definition if, for all α and β, ψβ−1 ◦ f ◦ ϕα is a C r
function on its domain of definition.
The abstract approach is preferable from a conceptual point of view; for example, we can’t see some Rk that contains the physical universe, so our physical
theories should avoid reference to such an Rk if possible. (Sometimes Rk is called
10.5. TANGENT VECTORS AND DERIVATIVES
133
the ambient space.) However, in the abstract approach there are certain technical
difficulties that must be overcome just to get acceptable definitions. In addition,
the Whitney embedding theorems (cf. Hirsch (1976)) show that, under assumptions that are satisfied in almost all applications, a manifold satisfying the
abstract definition can be embedded in some Rk , so our approach is not less general
in any important sense. From a technical point of view, the assumed embedding
of M in Rk is extremely useful because it automatically imposing conditions such
as metrizability and thus paracompactness, and it allows certain constructions that
simplify many proofs.
There is a category of C r manifolds and C r maps between them. (This can
be proved from the definitions, or we can just observe that this category can be
obtained from the category of subsets of finite dimensional vector spaces and C r
maps between them by restricting the objects.) The notion of isomorphism for this
category is:
Definition 10.4.2. A function f : M → N is a C r -diffeomorphism if f is a
bijection and f and f −1 are both C r . If such an f exists we say that M and N are
C r diffeomorphic.
If M and N are C r diffeomorphic we will, for the most part, regard them as
two different “realizations” of “the same” object. In this sense the spirit of the
definition of a C r manifold is that the particular embedding of M in Rk is of no
importance, and k itself is immaterial.
10.5
Tangent Vectors and Derivatives
There are many notions of “derivative” in mathematics, but invariably the term
refers to a linear approximation of a function that is accurate “up to first order.”
The first step in defining the derivative of a C r map between manifolds is to specify
the vector spaces that serve as the linear approximation’s domain and range.
Fix an m-dimensional C r manifold M ⊂ Rk . Throughout this section, when we
refer to a C r parameterization ϕ : U → M, it will be understood that U is an open
subset of the m-dimensional vector space X.
Definition 10.5.1. If ϕ : U → M is a C 1 parameterization and p = ϕ(x), then the
tangent space of M at p is the image of this linear transformation Dϕ(x) : X →
Rk .
We should check that this does not depend on the choice of ϕ. If ϕ′ : U ′ → M is
a second C 1 parameterization with ϕ′ (x′ ) = p, then the chain rule gives Dϕ′ (x′ ) =
Dϕ(x)◦D(ϕ−1 ◦ϕ′ )(x′ ), so the image of Dϕ′ (x′ ) is contained in the image of Dϕ(x).
We can combine the tangent spaces at the various points of M:
Definition 10.5.2. The tangent bundle of M is
[
T M :=
{p} × Tp M ⊂ Rk × Rk .
p∈M
134
CHAPTER 10. DIFFERENTIABLE MANIFOLDS
For a C r parameterization ϕ : U → M for M we define
Tϕ : U × X → { (p, v) ∈ T M : p ∈ ϕ(U) } ⊂ T M
by setting
Tϕ (x, w) := (ϕ(x), Dϕ(x)w).
Lemma 10.5.3. If r ≥ 2, then Tϕ is a C r−1 parameterization for T M.
Proof. It is easy to see that Tϕ is a C r−1 immersion, and that it is injective. The
inverse function theorem implies that its inverse is continuous.
Every p ∈ M is contained in the image of some C r parameterization ϕ, and for
every v ∈ Tp M, (p, v) is in the image of Tϕ , so the images of the Tϕ cover T M.
Thus:
Proposition 10.5.4. If r ≥ 2, then T M is a C r−1 manifold.
Fix a second C r manifold N ⊂ Rℓ , which we assume to be n-dimensional, and
a C r function f : M → N.
Definition 10.5.5. If F is a C 1 extension of f to a neighborhood of p, the derivative of f at p is the linear function
Df (p) = DF (p)|Tp M : Tp M → Tf (p) N.
We need to show that this definition does not depend on the choice of extension
F . Let ϕ : U → M be a C r parameterization whose image is a neighborhood of p,
let x = ϕ−1 (p), and observe that, for any v ∈ Tp M, there is some w ∈ Rm such that
v = Dϕ(x)w, so that
DF (p)v = DF (p)(Dϕ(x)w) = D(F ◦ ϕ)(x)w = D(f ◦ ϕ)(x)w.
We also need to show that the image of Df (p) is, in fact, contained in Tf (p) N.
Let ψ : V → N be a C r parameterization of a neighborhood of f (p). The last
equation shows that the image of Df (p) is contained in the image of
D(f ◦ ϕ)(x) = D(ψ ◦ ψ −1 ◦ f ◦ ϕ)(x) = Dψ(ψ −1 (f (p))) ◦ D(ψ −1 ◦ f ◦ ϕ),
so the image of Df (p) is contained in the image of Dψ −1 (ψ(f (p)), which is Tf (p) N.
Naturally the chain rule is the most important basic result about the derivative.
We expect that many readers have seen the following result, and at worst it is a
suitable exercise, following from the chain rule of multivariable calculus without
trickery, so we give no proof.
Proposition 10.5.6. If M ⊂ Rk , N ⊂ Rℓ , and P ⊂ Rm are C 1 manifolds, and
f : M → N and g : N → P are C 1 maps, then, at each p ∈ M,
D(g ◦ f )(p) = Dg(f (p)) ◦ Df (p).
We can combine the derivatives defined at the various points of M:
10.5. TANGENT VECTORS AND DERIVATIVES
135
Definition 10.5.7. The derivative of f is the function T f : T M → T N given by
T f (p, v) := (f (p), Df (p)v).
These objects have the expected properties:
Proposition 10.5.8. If r ≥ 2, then T f is a C r−1 function.
Proof. Each (p, v) ∈ T M is in the image of Tϕ for some C r parameterization ϕ
whose image contains p. The chain rule implies that
T f ◦ Tϕ : (x, w) 7→ f (ϕ(x)), D(f ◦ ϕ)(x)w ,
which is a C r−1 function. We have verified that T f satisfies (c) of Proposition
10.4.1.
Proposition 10.5.9. T IdM = IdT M .
Proof. Since IdRk is a C ∞ extension of IdM , we clearly have DIdM (p) = IdTp M for
each p ∈ M. The claim now follows directly from the definition of T IdM .
Proposition 10.5.10. If M, N, and P are C r manifolds and f : M → N and
g : N → P are C r functions, then T (g ◦ f ) = T g ◦ T f .
Proof. Using Proposition 10.5.6 we compute that
T g(T f (p, v)) = T g(f (p), Df (p)v) = (g(f (p)), Dg(f (p))Df (p)v)
= (g(f (p)), D(g ◦ f )(p)v) = T (g ◦ f )(p, v).
For the categorically minded we mention that Proposition 10.5.4 and the last
three results can be summarized very succinctly by saying that if r ≥ 2, then T
is a functor from the category of C r manifolds and C r maps between them to the
category of C r−1 manifolds and C r−1 maps between them. Again, we will not use
this language later, so in a sense you do not need to know what a functor is, but
categorical concepts and terminology are pervasive in modern mathematics, so it
would certainly be a good idea to learn the basic definitions.
Let’s relate the definitions above to more elementary notions of differentiation.
Consider a C 1 function f : (a, b) → M and a point t ∈ (a, b). Formally Df (t) is
a linear function from Tt (a, b) to Tf (t) M, but thinking about things in this way is
usually rather cumbersome. Of course Tt (a, b) is just a copy of R, and we define
f ′ (t) = Df (t)1 ∈ Tf (t) M, where 1 is the element of Tt (A, b) corresponding to 1 ∈ R.
When M is an open subset of R we simplify further by treating f ′ (t) as a number
under the identification of Tf (t) M with R. In this way we recover the concept of
the derivative as we first learned it in elementary calculus.
136
10.6
CHAPTER 10. DIFFERENTIABLE MANIFOLDS
Submanifolds
For almost any kind of mathematical object, we pay special attention to subsets,
or perhaps “substructures” of other sorts, that share the structural properties of the
object. One only has to imagine a smooth curve on the surface of a sphere to see
that such substructures of manifolds arise naturally. Fix a degree of differentiability
1 ≤ r ≤ ∞. If M ⊂ Rk is an m-dimensional C r manifold, N is an n-dimensional
that is also embedded in Rk , and N ⊂ M, then N is a C r submanifold of M. The
integer m − n is called the codimension of N in M.
The reader can certainly imagine a host of examples, so we only mention one
that might easily be overlooked because it is so trivial: any open subset of M
is a C r manifold. Conversely, any codimension zero submanifold of M is just an
open subset. Evidently submanifolds of codimension zero are not in themselves
particularly interesting, but of course they occur frequently.
Submanifolds arise naturally as images of smooth maps, and as solution sets of
systems of equations. We now discuss these two points of view at length, arriving
eventually at an important characterization result. Let M ⊂ Rk and N ⊂ Rℓ be C r
manifolds that are m- and n-dimensional respectively, and let f : M → N be a C r
function. We say that p ∈ M is:
(a) an immersion point of f if Df (p) : Tp M → Tf (p) N is injective;
(b) a submersion point of f if Df (p) is surjective;
(c) a diffeomorphism point of f is Df (p) is a bijection.
There are now a number of technical results. Collectively their proofs display the
inverse function and the implicit function theorem as the linchpins of the analysis
supporting this subject.
Proposition 10.6.1. If p is an immersion point of f , then there is a neighborhood
V of p such that f (V ) is an m-dimensional C r submanifold of N. In addition
Df (p) : Tp M → Tf (p) f (V ) is a linear isomorphism
Proof. Let ϕ : U → M be a C r parameterization for M whose image contains
p, and let x = ϕ−1 (p). The continuity of the derivative implies that there is a
neighborhood U ′ of x such that for all x′ ∈ U ′ the rank of D(f ◦ ϕ)(x′ ) is m. Let
X ⊂ Rℓ be the image of Df (p), and let π : Rℓ → X be the orthogonal projection.
Possibly after replacing U ′ with a suitable smaller neighborhood of x, the inverse
function theorem implies that π ◦ f ◦ ϕ|U ′ is invertible. Let V = ϕ(U ′ ). Now f ◦ ϕ|U ′
is an embedding because its inverse is (π ◦ f ◦ ϕ|U ′ )−1 ◦ π. Lemma 10.3.1 implies
that the inverse of f is also C r , so, for every x′ ∈ U ′ the rank of D(f ◦ ϕ)(x′ ) is m,
so f (V ) = f (ϕ(U ′ )) satisfies Definition 10.3.2.
The final assertion follows from Df (p) being injective while Tp M and Tf (p) (f (V )
are both m-dimensional.
Proposition 10.6.2. If p is a submersion point of f , then there is a neighborhood
U of p such that f −1 (f (p)) ∩ U is a (m − n)-dimensional C r submanifold of M. In
addition Tp f −1 (q) = ker Df (p).
10.6. SUBMANIFOLDS
137
Proof. Let ϕ : U → M be a C r parameterization whose image is an open neighborhood of p, let w0 = ϕ−1 (p), and let ψ : Z → Rn be a C r coordinate chart for an
open neighborhood Z ⊂ N of f (p). Without loss of generality we may assume that
f (ϕ(U)) ⊂ Z. Since Dϕ(w0 ) and Dψ(f (p)) are bijections,
D(ψ ◦ f ◦ ϕ)(w0 ) = Dψ(f (p)) ◦ Df (p) ◦ Dϕ(w0)
is surjective, and the vector space containing U can be decomposed as X × Y where
Y is n dimensional and Dy (ψ ◦ f ◦ ϕ)(w0 ) is nonsingular. Let w0 = (x0 , y0 ). The
implicit function theorem gives an open neighborhood V ⊂ X containing x0 , an
open W ⊂ U containing w0 , and a C r function g : V → Y such that g(x0 ) = y0 and
{ (x, g(x)) : x ∈ V } = { w ∈ W : f (ϕ(w)) = f (p) }.
Then
{ ϕ(x, g(x)) : x ∈ V } = f −1 (f (p)) ∩ ϕ(W )
is a neighborhood of p in f −1 (f (p)), and x 7→ ϕ(x, g(x)) is a C r embedding because
its inverse is the composition of ϕ−1 with the projection (x, y) 7→ x.
We obviously have Tp f −1 (q) ⊂ ker Df (p), and the two vector spaces have the
same dimension.
Proposition 10.6.3. If p is a diffeomorphism point of f , then there is a neighborhood W of p such that f (W ) is a neighborhood of f (p) and f |W : W → f (W ) is a
C r diffeomorphism.
Proof. Let ϕ : U → M be a C r parameterization of a neighborhood of p, let
x = ϕ−1 (p), and let ψ : V → N be a C r parameterization of a neighborhood of
f (p). Then
D(ψ −1 ◦ f ◦ ϕ)(x) = Dψ −1 (f (p)) ◦ Df (p) ◦ Dϕ(x)
is nonsingular, so the inverse function theorem implies that, after replacing U and
V with smaller open sets containing x and ψ −1 (f (p)), ψ −1 ◦ f ◦ ϕ is invertible with
C r inverse. Let W = ϕ(U). We now have
(f |W )−1 = ϕ ◦ (ψ −1 ◦ f ◦ ϕ)−1 ◦ ψ −1 ,
which is C r .
Now let P be a p-dimensional C r submanifold of N. The following is the technical basis of the subsequent characterization theorem.
Lemma 10.6.4. If q ∈ P then:
(a) There is a neighborhood V ⊂ P , a p-dimensional C r manifold M, a C r function f : M → P , a p ∈ f −1 (q) that is an immersion point of f , and a
neighborhood U of P , such that f (U) = V .
(b) There is a neighborhood Z ⊂ N of q, an (n − p)-dimensional C r manifold
M, and a C r function f : Z → M such q is a submersion point of f and
f −1 (f (q)) = P ∩ Z.
138
CHAPTER 10. DIFFERENTIABLE MANIFOLDS
Proof. Let ϕ : U → P be a C r parameterization for P whose image contains q.
Taking f = ϕ verifies (a).
Let w = ϕ−1 (q). Let ψ : V → N be a C r parameterization for N whose image
contains q. Then the rank of D(ψ −1 ◦ ϕ)(w) is p, so the vector space containing
V can be decomposed as X × Y where X is the image of D(ψ −1 ◦ ϕ)(w). Let
πX : X × Y → X and πY : X × X → Y be the projections (x, y) 7→ x and
(x, y) 7→ y respectively. The inverse function implies that, after replacing U with
a smaller neighborhood of w, πX ◦ ψ −1 ◦ ϕ is a C r diffeomorphism between U and
−1
an open W ⊂ X. Since we can replace V with V ∩ πX
(W ), we may assume that
πX (V ) ⊂ W . Let Z = ψ(V ), and let
f = πY ◦ ψ −1 − πY ◦ ψ −1 ◦ ϕ ◦ (πX ◦ ψ −1 ◦ ϕ)−1 ◦ πX ◦ ψ −1 : Z → Y.
Evidently every point of V is a submersion point of
πY − ψ −1 ◦ ϕ ◦ (πX ◦ ψ −1 ◦ ϕ)−1 ◦ πX ,
so every point of Z is a submersion point of f . If q ′ ∈ P ∩ Z, then q ′ = ϕ(w ′ )
for some w ′ ∈ U, so f (q ′ ) = 0. On the other hand, suppose f (q ′) = 0, and
let q ′′ be the image of q ′ under the map ϕ ◦ (πX ◦ ψ −1 ◦ ϕ)−1 ◦ πX ◦ ψ −1 . Then
πX (ψ −1 (q ′ )) = πX (ψ −1 (q ′′ )) and πY (ψ −1 (q ′ )) = πY (ψ −1 (q ′′ )), so q ′′ = q ′ and thus
q ′ ∈ P . Thus f −1 (f (q)) = P ∩ Z.
Theorem 10.6.5. Let N be a C r manifold. For P ⊂ N the following are equivalent:
(a) P is a p-dimensional C r submanifold of M.
(b) For every q ∈ P there is a relatively open neighborhood V ⊂ P , a p-dimensional
C r manifold M, a C r function f : M → P , a p ∈ f −1 (q) that is an immersion
point of f , and a neighborhood U of P , such that f (U) = V .
(c) For every q ∈ P there is a neighborhood Z ⊂ N of q, an (n − p)-dimensional
C r manifold M, and a C r function f : Z → M such q is a submersion point
of f and f −1 (f (q)) = P ∩ Z.
Proof. The last result asserts that (a) implies (b) and (c), Proposition 10.6.1 implies
that (b) implies (a), and Proposition 10.6.2 implies that (c) implies (a).
Let M ⊂ Rk and N ⊂ Rℓ be an m-dimensional and an n-dimensional C r manifold, and let f : M → N be a C r function. We say that f is an immersion if
every p ∈ M is an immersion point of f . It is a submersion if every p ∈ M is a
submersion point, and it is a local diffeomorphism if every p ∈ M is a diffeomorphism point. There are now some important results that derive submanifolds from
functions.
Theorem 10.6.6. If f : M → N is a C r immersion, and an embedding, then f (M)
is an m-dimensional C r submanifold of N.
Proof. We need to show that any q ∈ f (M) has a neighborhood in f (M) that is an
(n − m)-dimensional C r manifold. Proposition 10.6.1 implies that any p ∈ M has
an open neighborhood V such that f (V ) is a C r (n − m)-dimensional submanifold
of N. Since f is an embedding, f (V ) is a neighborhood of f (p) in f (M).
139
10.6. SUBMANIFOLDS
A submersion point of f is also said to be a regular point of f . If p is not a
regular point of f , then it is a critical point of f . A point q ∈ N is a critical value
of f if some preimage of q is a critical point, and if q is not a critical value, then it
is a regular value. Note the following paradoxical aspect of this terminology: if q
is not a value of f , in the sense that f −1 (q) = ∅, then q is automatically a regular
value of f .
Theorem 10.6.7 (Regular Value Theorem). If q is a regular value of f , then f −1 (q)
is an (m − n)-dimensional submanifold of M.
Proof. This is an immediate consequence of Proposition 10.6.2.
This result has an important generalization. Let P ⊂ N be a p-dimensional C r
submanifold.
Definition 10.6.8. The function f is transversal to P along S ⊂ M if, for all
p ∈ f −1 (P ) ∩ S,
im Df (p) + Tf (p) P = Tf (p) N.
We write f ⋔S P to indicate that this is the case, and when S = M we simply write
f ⋔ P.
Theorem 10.6.9 (Transversality Theorem). If f ⋔ P , then f −1 (P ) is an (m −
n + p)-dimensional C r submanifold of M. For each p ∈ f −1 (P ), Tp f −1 (P ) =
Df (p)−1 (Tf (p) P ).
Proof. Fix p ∈ f −1 (P ). (If f −1 (P ) = ∅, then all claims hold trivially.) We use
the characterization of a C r submanifold given by Theorem 10.6.5: since P is a
submanifold of N, there is a neighborhood W ⊂ N of f (p) and a C r function
Ψ : W → Rn−p such that DΨ(f (p)) has rank n − p and P ∩ W = Ψ−1 (0).
Let V = f −1 (W ) and Φ = Ψ ◦ f |V . Of course V is open, Φ is C r , and f −1 (P ) ∩
V = Φ−1 0). We compute that
im DΦ(p) = DΨ(f (p)) im Df (p) = DΨ(f (p)) im Df (p) + ker DΨ(f (p))
= DΨ(f (p)) im Df (p) + Tf (p) P = DΨ(f (p))(Tf (p) N) = Rn−s .
(The third equality follows from the final assertion of Proposition 10.6.2, and the
fourth is the transversality assumption.) Thus p is a submersion point of Φ. Since
p is an arbitrary point of f −1 (P ) the claim follows from Theorem 10.6.5.
We now have
Tp f −1 (P ) = ker DΦ(p) = ker(DΨ(f (p)) ◦ Df (p))
= Df (p)−1 (ker DΨ(p)) = Df (p)−1 (Tf (p) P )
where the first and last equalities are from Proposition 10.6.2.
140
10.7
CHAPTER 10. DIFFERENTIABLE MANIFOLDS
Tubular Neighborhoods
Fix a degree of differentiability r ≥ 2 and an n dimensional C r manifold N ⊂ Rℓ .
For each q ∈ N let νq N be the orthogonal complement of Tq N. The normal bundle
of N is
[
νN =
{q} × νq N.
q∈N
Proposition 10.7.1. νN is an ℓ-dimensional C r−1 submanifold of N × Rℓ .
Proof. Let ϕ : U → Rℓ be a C r parameterization for N. Let Z : U × Rℓ → Rn be
the function
Z(x, w) = hDϕ(x)e1 , wi, . . . , hDϕ(x)en , wi
where e1 , . . . , em is the standard basis for Rn . Clearly Z is C r−1 , and for every
(x, w) in its domain the rank of DZ(x, w) is n. Therefore the regular value theorem
implies that
Z −1 (0) = { (x, w) ∈ U × Rℓ : (ϕ(x), w) ∈ νN }
is a ℓ-dimensional C r−1 manifold. Since (x, w) 7→ (ϕ(x), w) and (q, w) 7→ (ϕ−1 (q), w)
are inverse C r−1 bijections between Z −1 (0) and νN ∩ (ϕ(U) × Rℓ ), the first of these
maps is a C r−1 embedding, which implies (Theorem 10.6.6) that the latter set is a
C r−1 manifold. Of course these sets cover νN because the images of C r parameterizations cover N.
Like the tangent bundle, the normal bundle attaches a vector space of a certain
dimension to each point of N. (The general term for such a construct is a vector
bundle.) The zero section of νN is { (q, 0) : q ∈ N }. There are maps
π : (q, v) 7→ q
and σ : (q, v) 7→ q + v
from N × Rℓ to N and Rℓ respectively. Let π T = Π|T M , π ν = π|νN , σ T = σ|T M ,
and σ ν = σ|νN .
For a continuous function ρ : N → (0, ∞) let
Uρ = { (q, v) ∈ νN : kvk < ρ(q) },
and let σρν = σ ν |Uρ . The main topic of this section is the following result and its
many applications.
Theorem 10.7.2 (Tubular Neighborhood Theorem). There is a continuous ρ :
N → (0, ∞) such that σρν is a C r−1 diffeomorphism onto its image, which is a
neighborhood of N.
The inverse function theorem implies that each (q, 0) in the zero section has a
neighborhood that is mapped C r−1 diffeomorphically by σ onto a neighborhood of
q in Rℓ . The methods used to produce a suitable neighborhood of the zero section
with this property are topological and quite technical, in spite of their elementary
character.
10.7. TUBULAR NEIGHBORHOODS
141
Lemma 10.7.3. If (X, d) and (Y, e) are metric spaces, f : X → Y is continuous,
S is a subset of X such that f |S is an embedding, and for each s ∈ S the restriction
of f to some neighborhood
Ns of s is an embedding, then there is an open U such
S
that S ⊂ U ⊂ s Ns and f |U is an embedding.
Proof. For s ∈ S let δ(s) be one half of the supremum of the set of ε > 0 such that
Uε (s) ⊂ Ns and f |Uε (s) is an embedding. The restriction of an embedding to any
subset of its domain is an embedding, which implies that δ is continuous.
Since f |S is an invertible, its inverse is continuous. In conjunction with the
continuity of δ and d, this implies that for each s ∈ S there is a ζs > 0 such that
d(s, s′ ) < min{δ(s) − 21 δ(s′ ), δ(s) − 12 δ(s′ )}
(∗)
for all s′ ∈ S with e(f (s), f (s′)) ≤ ζs . For each s chooseSan open Us ⊂ X such that
s ∈ Us ⊂ Uδ(s)/2 (s) and f (Us ) ⊂ Uζs /3 (f (s)). Let U = s∈S Us . We will show that
f |U is injective with continuous inverse.
Consider s, s′ ∈ S and y, y ′ ∈ Y with e(f (s), y) < ζs /3 and e(f (s′ ), y ′) < ζs′ /3.
We claim that if y = y ′ , then (∗) holds: otherwise e(f (s), f (s′)) > ζs , ζs′ , so that
e(y, y ′) ≥ e(f (s), f (s′)) − e(f (s), y) − e(f (s′ ), y ′)
> ( 21 e(f (s), f (s′)) − ζs /3) + ( 21 e(f (s), f (s′)) − ζs′ /3) ≥ 16 (ζs + ζs′ ).
In particular, if f (x) = y = y ′ = f (x′ ) for some x ∈ Us and x′ ∈ Us′ , then
1
δ(s′ ) + d(s, s′ ) ≤ δ(s) and thus
2
Us′ ⊂ Uδ(s′ )/2 (s′ ) ⊂ Uδ(s′ )/2+d(s,s′ ) (s) ⊂ Uδ(s) (s).
We have x ∈ Us , x′ ∈ Us′ , and Us , Us′ ⊂ Uδ(s) (s), and f |Uδ(s) (s) is injective, so it
follows that x = x′ . We have shown that f |U is injective.
We now need to show that the image of any open subset of U is open in the
relative topology of f (U). Fix a particular s ∈ S. In view of the definition of
U, it suffices to show that if Vs ⊂ Us is open, then f (Vs ) is relatively open. The
restriction of f to Uδ(s) (s) is an embedding, so there is an open Zs ⊂ Y such that
f (Vs ) = f (Uδ(s) (s)) ∩ Zs . Since f (Vs ) ⊂ f (Us ) ⊂ Uζs /3 (f (s)) we have
f (Vs ) = f (U) ∩ Uζs /3 (f (s)) ∩ Zs ∩ f (Uδ(s) (s)).
Above we showed that if Uζs /3 (f (s)) ∩ Uζs′ /3 (f (s′ )) is nonempty, then (∗) holds.
Therefore f (U) ∩ Uζs /3 (f (s)) is contained in the union of the f (Us′ ) for those s′
such that 12 δ(s′ ) + d(s, s′ ) < δ(s), and for each such s′ we have Us′ ⊂ Uδ(s′ )/2 (s′ ) ⊂
Uδ(s) (s). Therefore f (U) ∩ Uζs /3 (f (s)) ⊂ f (Uδ(s) (s)), and consequently
f (Vs ) = f (U) ∩ Uζs /3 (f (s)) ∩ Zs ,
so f (Vs ) is relatively open in f (U).
Lemma 10.7.4. If (X, d) is a metric space, S ⊂ X, and U is an open set containing
S, then there is a continuous δ : S → (0, ∞) such that for all s ∈ S, Uδ(s) (s) ⊂ U.
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CHAPTER 10. DIFFERENTIABLE MANIFOLDS
Proof. For each s ∈ S let βs = sup{ ε > 0 : Uε (s) ⊂ U }. Since X is paracompact
(Theorem 6.1.1) there is a locally finite refinement {Vα }α∈A of {Uβs (s)}s∈S . Theorem 6.2.2 gives a partition of unity {ϕα } subordinate to {Vα }. The claim holds
trivially if there is some α with Vα = X; otherwise for each α let δα : S → [0, ∞) be
the function δα (s) P
= inf x∈X\Vα d(s, x), which is of course continuous, and define δ
by setting δ(s) := α ϕα (s)δα (s). If s ∈ S, s ∈ Vα , and δα′ (s) ≤ δα (s) for all other
α′ such that s ∈ Vα′ , then
Uδ(s) (s) ⊂ Uδα (s) (s) ⊂ Vα ⊂ Uβs′ (s′ ) ⊂ U
for some s′ , so Uδ(s) (s) ⊂ U.
The two lemmas above combine to imply that:
Proposition 10.7.5. If (X, d) and (Y, e) are metric spaces, f : X → Y is continuous, S is a subset of X such that f |S is an embedding, and for each s ∈ S
the restriction of f to some neighborhood Ns of s is an embedding, then there is a
continuous
ρ : S → (0, ∞) such that Uρ(s) (s) ⊂ Ns for all s and the restriction of
S
f to s∈S Uρ(s) (s) is an embedding.
Proof of the Tubular Neighborhood Theorem. The inverse function theorem implies
that each point (q, 0) in the zero section of νN has a neighborhood Nq such that
then σρν is an embedding,
σ ν |Ns is a C r−1 diffeomorphism. If ρ is in the last result,
S
r−1
and its inverse is C
differentiable because Uρ ⊂ q Nq .
We now develop several applications of the tubular neighborhood theorem. Let
M be an m-dimensional C r ∂-manifold.
Theorem 10.7.6. For any S ⊂ M, C r−1 (S, N) is dense in CS (S, N).
Proof. Proposition 10.2.7 implies that C r−1 (S, Vρ ) is dense in CS (S, Vρ ), and Proposition 5.5.3 implies that f 7→ π ν ◦ σρ−1 ◦ f is continuous.
Recall that a topological space X is locally path connected if, for each x ∈ X,
each neighborhood U of x contains a neighborhood V such that for any x0 , x1 ∈ V
there is a continuous path γ : [0, 1] → U with γ(0) = x0 and γ(1) = x1 . For an
open subset of a locally convex topological vector space, local path connectedness
is automatic: any neighborhood of a point contains a convex neighborhood.
Theorem 10.7.7. For any S ⊂ M, CS (S, N) is locally path connected.
Proof. Fix a neighborhood U ⊂ CS (S, N) of a continuous f : S → N. The definition
of the strong topology implies that there is an open W ⊂ S ×N such that f ∈ { f ′ ∈
C(S, N) : Gr(f ′ ) ⊂ W } ⊂ U. Lemma 10.7.4 implies that there is a continuous
λ : N → (0, ∞) such that Uλ(y) (y) ⊂ Vρ for all y ∈ N and (x, π(σρ−1 (z))) ∈ W for
all x ∈ S and z ∈ Uλ(f (x)) (f (x)). Let W ′ = { (x, y) ∈ W : y ∈ Uλ(f (x)) (f (x)) }. For
f0 , f1 ∈ C(S, N) with Gr(f0 ), Gr(f1 ) ⊂ W ′ we define h by setting
h(x, t) = π ν σρ−1 ((1 − t)f0 (x) + tf1 (x)) .
If f0 and f1 are C r , so that they are the restrictions to S of C r functions defined
on open supersets of S, then this formula defines a C r extension of h to an open
superset of S × [0, 1], so that h is C r .
143
10.8. MANIFOLDS WITH BOUNDARY
Proposition 10.7.8. There is a continuous function λ : N → (0, ∞) and a C r−1
function κ : Vλ → N, where Vλ = { (q, v) ∈ T N : kvk < λ(q) }, such that the
function κ̃ : (q, v) 7→ (q, κ(q, v)) is a C r−1 diffeomorphism between Vλ and a neighborhood of the diagonal in N × N.
Proof. Let ρ and Uρ be as in the tubular neighborhood theorem. For each q ∈ N
there is a neighborhood Nq of (q, 0) ∈ Tq N such that σ T (Nq ) is contained in σρν (Uρ ).
Let
[
[
Nq → N and κ̃ = π T × κ :
Nq → N × N.
κ = π ν ◦ (σρν )−1 ◦ σ T :
q
q
It is easy to see (and not hard to compute formally using the chain rule) that
Dκ̃(q, 0) = IdTq N × IdTq N under the natural identification of T(q,0) (T N) with Tq N ×
Tq N. The inverse function theorem implies that after replacing Nq with a smaller
neighborhood of (q, 0), the restriction of κ̃ to Nq is a diffeomorphism onto its image.
We can now proceed as in the proof of the tubular neighborhood theorem.
The following construction simulates convex combination.
Proposition 10.7.9. There is a neighborhood W of the diagonal in N × N and a
continuous function c : W × [0, 1] → N such that:
(a) c((q, q ′), 0) = q for all (q, q ′) ∈ W ;
(b) c((q, q ′), 1) = q ′ for all (q, q ′ ) ∈ W ;
(c) c((q, q), t) = q for all q ∈ N and all t.
Proof. The tubular neighborhood gives an open neighborhood U of the zero section
in νN such that if σ : νN → Rk is the map σ(q, v) = q + v, then σ|U is a homeomorphism between U and σ(U). Let π : νN → N be the projection for the normal
bundle. Let
W = { (q, q ′) ∈ N × N : (1 − t)q + tq ′ ∈ σ(U) for all 0 ≤ t ≤ 1 },
and for (q, q ′) ∈ W and 0 ≤ t ≤ 1 let
c((q, q ′), t) = π (σ|U )−1 ((1 − t)q + tq ′ ) .
10.8
Manifolds with Boundary
Let X be an m-dimensional vector space, and let H be a closed half space of X.
In the same way that manifolds were “modeled” on open subsets of X, manifolds
with boundary are “modeled” on open subsets of H. Examples of ∂-manifolds
include the m-dimensional unit disk
D m := { x ∈ Rm : kxk ≤ 1 },
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CHAPTER 10. DIFFERENTIABLE MANIFOLDS
the annulus { x ∈ R2 : 1 ≤ x ≤ 2 }, and of course H itself. Since we will be very
concerned with homotopies, a particularly important example is M × [0, 1] where
M is a manifold (without boundary). Thus it is not surprising that we need to
extend our formalism in this direction. What actually seems more surprising is the
infrequency with which one needs to refer to “manifolds with corners,” which are
spaces that are “modeled” on the nonnegative orthant of Rm .
There is a technical point that we need to discuss. If U ⊂ H is open and
f : U → Y is C 1 , where Y is another vector space, then the derivative Df (x) is
defined at any x ∈ U, including those in the boundary of H, in the sense that all
C 1 extensions f˜ : Ũ → Y of f to open (in X) sets Ũ with Ũ ∩ H = U have the
same derivative at x. This is fairly easy to prove by showing that if w ∈ X and the
ray rw = { x + tw : t ≥ 0 } from x “goes into” H, then the derivative of f˜ along
rw is determined by f , and that the set of such w spans X. We won’t belabor the
point by formalizing this argument.
The following definitions parallel those of the last section. If U ⊂ H is open and
ϕ : U → Y is a function, we say that ϕ is a C r ∂-immersion if it is C r and the
rank of Dϕ(x) is m for all x ∈ U. If, in addition, ϕ is a homeomorphism between
U and ϕ(U), then we say that ϕ is a C r ∂-embedding.
Definition 10.8.1. If M ⊂ Rk , an m-dimensional C r ∂-parameterization for
M is a C r ∂-embedding ϕ : U → M, where U ⊂ H is open and ϕ(U) is a relatively
open subset of M. If each p ∈ M is contained in the image of a C r parameterization
for M, then M is an m-dimensional C r manifold with boundary.
We will often write “∂-manifold” in place of the cumbersome phrase “manifold with
boundary.”
Fix an m-dimensional C r ∂-manifold M ⊂ Rk . We say that p ∈ M is a boundary
point of M if there a C r ∂-parameterization of M that maps a point in the boundary
of H to p. If any C r parameterization of a neighborhood of p has this property, then
all do; this is best understood as a consequence of invariance of domain (Theorem
14.4.4) which is most commonly proved using algebraic topology. Invariance of
domain is quite intuitive, and eventually we will be able to establish it, but in the
meantime there arises the question of whether our avoidance of results derived from
algebraic topology is “pure.” One way of handling this is to read the definition
of a ∂-manifold as specifying which points are in the boundary. That is, a ∂manifold is defined to be a subset of Rk together with an atlas of m-dimensional
C r parameterizations {ϕi }i∈I such that each ϕ−1
j ◦ ϕi maps points in the boundary
of H to points in the boundary and points in the interior to points in the interior.
In order for this to be rigorous it is necessary to check that all the constructions in
our proofs preserve this feature, but this will be clear throughout. With this point
cleared up, the boundary of M is well defined; we denote this subset by ∂M. Note
that ∂M automatically inherits a system of coordinate systems that display it as
an (m − 1)-dimensional C r manifold (without boundary).
Naturally our analytic work will be facilitated by characterizations of ∂-manifolds
that are somewhat easier to verify than the definition.
Lemma 10.8.2. For M ⊂ Rk the following are equivalent:
10.8. MANIFOLDS WITH BOUNDARY
145
(a) M is an m-dimensional ∂-manifold;
(b) for each p ∈ M there is a neighborhood W ⊂ M, an m-dimensional C r
manifold (without boundary) W̃ , and a C r function h : W̃ → R such that
W = h−1 ([0, ∞)) and Dh(p) 6= 0.
Proof. Fix p ∈ M. If (a) holds then there is a C r ∂-embedding ϕ : U → M, where
U ⊂ H is open and ϕ(U) is a relatively open subset of M. After composing with
an affine function, we have assume that H = { x ∈ Rm : xm ≥ 0 }. Let ϕ̃ : Ũ → Rk
be a C r extension of ϕ to an open (in Rm ) superset of U. After replacing Ũ with a
smaller neighbohrood of ϕ−1 (p) it will be the case that ϕ is a C r embedding, and
we may replace U with its intersection with this smaller neighborhood. To verify
(b) we set W̃ = ϕ̃(Ũ ) and W = ϕ(U), and we let h be the last component function
of ϕ̃−1 .
Now suppose that W , W̃ , and h are as in (b). Let ψ̃ : Ṽ → W̃ be a C r
parameterization for W̃ whose image contains p, and let x̃ = ψ̃ −1 (p). Since Dh(p) 6=
ψ̃)
0 there is some i such that ∂(h◦
(x̃) 6= 0; after reindexing we may assume that
∂xi
m
i = m. Let η : W̃ → R be the function
η(x) = x1 , . . . , xm−1 , h(ψ̃(x)) .
Examination of the matrix of partial derivatives shows that Dη(x̃) is nonsingular, so,
by the inverse function, after replacing W̃ with a smaller neighborhood of x̃, we may
assume that η is a C r embedding. Let Ũ = η(Ṽ ), U = Ũ ∩H, ϕ̃ = ψ̃ ◦η −1 : Ũ → W̃ ,
and ϕ = ϕ̃|U : U → W . Evidently ϕ is a C r ∂-parameterization for M.
The following consequence is obvious, but is still worth mentioning because it
will have important applications.
Proposition 10.8.3. If M is an m-dimensional C r manifold, f : M → R is C r ,
and a is a regular value of f , then f −1 ([a, ∞)) is an m-dimensional C r ∂-manifold.
The definitions of tangent spaces, tangent manifolds, and derivatives, are only
slightly different from what we saw earlier. Suppose that M ⊂ Rk is an mdimensional C r ∂-manifold, ϕ : U → M is a C r ∂-parameterization, x ∈ U, and
ϕ(x) = p. The definition of a C r function gives a C r extension ϕ̃ : Ũ → Rk of ϕ to
an open (in Rm ) superset of U, and we define Tp M to be the image of D ϕ̃(x). (Of
course there is no difficulty showing that D ϕ̃(x) does not depend on the choice of
extension ϕ̃.) As before, the tangent manifold of M is
[
TM =
{p} × Tp M.
p∈M
Let πT M : T M → M be the natural projection π : (p, v) 7→ p.
We wish to show that T M is a C r−1 ∂-manifold. To this end define Tϕ : U ×
m
R → πT−1M (U) by setting Tϕ (x, w) = (ϕ(x), D ϕ̃(x)w). If r ≥ 2, then Tϕ is an
injective C r−1 ∂-immersion whose image is open in T M, so it is a C r ∂-embedding.
Since T M is covered by the images of maps such as Tϕ , it is indeed a C r−1 ∂manifold.
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CHAPTER 10. DIFFERENTIABLE MANIFOLDS
If N ⊂ Rℓ is an n-dimensional C r ∂-manifold and f : M → N is a C r map, then
the definitions of Df (p) : Tp M → Tf (p) N for p ∈ M and T f : T M → T N, and
the main properties, are what we saw earlier, with only technical differences in the
explanation. In particular, T extends to a functor from the category C r ∂-manifolds
and C r maps to the category of C r−1 ∂-manifolds and C r−1 maps.
We also need to reconsider the notion of a submanifold. One can of course define
a C r ∂-submanifold of M to be a C r ∂-manifold that happens to be contained in
M, but the submanifolds of interest to us satisfy additional conditions. Any point
in the submanifold that lies in ∂M should be a boundary point of the submanifold,
and we don’t want the submanifold to be tangent to ∂M at such a point.
Definition 10.8.4. If M is a C r ∂-manifold, a subset P is a neat C r ∂-submanifold
if it is a C r ∂-manifold, ∂P = P ∩∂M, and for each p ∈ ∂P we have Tp P +Tp ∂M =
Tp M.
The reason this is the relevant notion has to do with transversality. Suppose
that M is a C r ∂-manifold, N is a C r manifold, without boundary, P is a C r
submanifold of N, and f : M → N is C r . We say that f is transversal to P along
S ⊂ M, and write f ⋔S P , if f |M \∂M ⋔S\∂M P and f |∂M ⋔S∩∂M P . As above, when
S = M we write f ⋔ P .
The transversality theorem generalizes as follows:
Proposition 10.8.5. If f : M → N is a C r function that is transversal to P , then
f −1 (P ) is a neat C r submanifold of M with ∂f −1 (P ) = f −1 (P ) ∩ ∂M.
Proof. We need to show that a neighborhood of a point p ∈ f −1 (P ) has the required
properties. If p ∈ M \ ∂M, this follows from the Theorem 10.6.9, so suppose
that p ∈ ∂M. Lemma 10.8.2 implies that there is a neighborhood W ⊂ M of
p, an m-dimensional C r manifold W̃ , and a C r function h : W̃ → R such that
W = h−1 ([0, ∞)), h(p) = 0, and Dh(p) 6= 0. Let f˜ : W̃ → N be a C r extension
of f |W 1 . We may assume that f˜ is transverse to P , so the transversality theorem
implies that f˜−1 (P ) is a C r submanifold of W̃ .
Since f˜ and f |∂M are both tranverse to P , there must be a v ∈ Tp M \Tp ∂M such
that D f˜(p)v ∈ Tf (p) P . This implies two things. First, since v ∈
/ ker Dh(p) = Tp ∂M
−1
−1
−1
and f (P )∩W = f˜ (P )∩W ∩h ([0, ∞)), Lemma 10.8.2 implies that f −1 (P )∩W
is a C r ∂-manifold in a neighborhood of p. Second, the transversality theorem
implies that Tp f −1 (P ) includes v, so we have Tp f −1 (P ) + Tp ∂M = Tp M.
10.9
Classification of Compact 1-Manifolds
In order to study the behavior of fixed points under homotopy, we will need
to understand the structure of h−1 (q) when M and N are manifolds of the same
dimension,
h : M × [0, 1] → N
1
If ψ : V → N is a C r parameterization for N whose image contains f (W ), then ψ −1 has a C r
extension, because that is what it means for a function on a possibly nonopen domain to be C r ,
and this extension can be composed with ψ to give f˜.
10.9. CLASSIFICATION OF COMPACT 1-MANIFOLDS
147
is a C r homotopy, and q is a regular value of h. The transverality theorem implies
that h−1 (q) is a 1-dimensional C r ∂-manifolds, so our first step is the following
result.
Proposition 10.9.1. A nonempty compact connected 1-dimensional C r manifold is
C r diffeomorphic to the circle C = { (x, y) ∈ R2 : x2 +y 2 = 1 }. A compact connected
1-dimensional C r ∂-manifold with nonempty boundary is C r diffeomorphic to [0, 1].
Of course no one has any doubts about this being true. If there is anything to
learn from the following technical lemma and the subsequent argument, it can only
concern technique. Readers who skip this will not be at any disadvantage.
Lemma 10.9.2. Suppose that a < b and c < d, and that there is an increasing C r
diffeomorphism f : (a, b) → (c, d). Then for sufficiently large Q ∈ R there is an
increasing C r diffeomorphism λ : (a, b) → (a − Q, d) such that λ(s) = s − Q for all
s in some interval (a, a + δ) and λ(s) = f (s) for all s in some interval (b − ε, b).
Proof. Lemma 10.2.2 presented a C ∞ function γ : R → [0, ∞] with γ(t) = 0 for all
t ≤ 0 and γ ′ (t) > 0 for all t > 0. Setting
κ(s) =
γ(s − a − δ)
γ(s − a − δ) + γ(b − ε − s)
for sufficiently small δ, ε > 0 gives a C ∞ function κ : (a, b) → [0, 1] with κ(s) = 0
for all s ∈ (a, a + δ), κ(s) = 1 for all s ∈ (b − ε, b), and κ′ (s) > 0 for all s such that
0 < κ(s) < 1. For any real number Q we can define λ : (a, b) → R by setting
λ(s) = (1 − κ(s))(s − Q) + κ(s)f (s).
Clearly this will be satisfactory if λ′ (s) > 0 for all s. A brief calculation gives
λ′ (s) = 1 + κ(s)(f ′(s) − 1) + κ′ (s)(Q + f (s) − s)
= (1 − κ(s))(1 − f ′ (s)) + f ′ (s) + κ′ (s)(Q + f (s) − s).
If Q is larger than the upper bound for s − f (s), then λ′ (s) > 0 when κ(s) is close
to 0 or 1. Since those s for which this is not the case will be contained in a compact
interval on which κ′ positive and continuous, hence bounded below by a positive
constant, if Q is sufficiently large then λ′ (s) > 0 for all s.
Proof of Proposition 10.9.1. Let M be a nonempty compact connected 1-dimensional
C r manifold. We can pass from a C r atlas for M to a C r atlas whose elements all
have connected domains by taking the restrictions of each element of the atlas to
the connected components of its domain. To be concrete, we will assume that the
domains of the parameterizations are connected subsets of R, i.e., open intervals.
Since we can pass from a parameterization with unbounded domain to a countable
collection of restrictions to bounded domains, we may assume that all domains are
bounded. Since M is compact, any atlas has a finite subset that is also an atlas.
We now have an atlas of the form
{ ϕ1 : (a1 , b1 ) → M, . . . , ϕK : (aK , bK ) → M }.
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CHAPTER 10. DIFFERENTIABLE MANIFOLDS
Finally, we may assume that K is minimal. Since M is compact, K > 1.
Let p be a limit point of ϕ1 (s) as s → b1 . If p was in the image of ϕ1 , say
p = ϕ1 (s1 ), then the image of a neighborhood of s1 would be a neighborhood of
p, and points close to b1 would be mapped to this neighborhood, contradicting the
injectivity of ϕ1 . Therefore p is not in the image of ϕ1 . After reindexing, we may
assume that p is in the image of ϕ2 , say p = ϕ2 (t2 ).
Fix ε > 0 small enough that [t2 − ε, t2 + ε] ⊂ (a2 , b2 ). Since ϕ2 ((t2 − ε, t2 + ε))
and M \ ϕ2([t2 − ε, t2 + ε]) are open and disjoint, and there at most two s such that
ϕ1 (s) = ϕ2 (t2 ±ε), there is some δ > 0 such that ϕ1 ((b2 −δ, b1 )) ⊂ ϕ2 ((t2 −ε, t2 +ε)).
r
Then f = ϕ−1
2 ◦ ϕ1 |(b1 −δ,b1 ) is a C diffeomorphism. The intermediate value theorem
implies that it is monotonic. Without loss of generality (we could replace ϕ2 with
t 7→ ϕ2 (−t)) we may assume that it is increasing. Of course lims→b1 f (s) = t2 .
The last result implies that there is some real number Q and an increasing C r
diffeomorphism λ : (b1 − δ, b1 ) → (b1 − δ − c, t2 ) such that λ(s) = s − Q for all s near
b1 − δ and λ(s) = f (s) for all s near b1 . We can now define ϕ : (a1 − Q, b2 ) → M
by setting


ϕ1 (s + Q), s ≤ b1 − δ − Q,
ϕ(s) = ϕ1 (λ−1 (s)), b1 − δ − Q < s < t2 ,


ϕ2 (s),
s ≥ t2 .
We have λ−1 (s) = s + Q for all s in a neighborhood of b1 − δ − Q and ϕ(s) = ϕ2 (s)
for all s close to t2 . Therefore ϕ is a C r function. Each point in its domain
has a neighborhood such that the restriction of ϕ to that neighborhood is a C r
parameterization for M, which implies that if maps open sets to open sets. If it
was injective, it would be a C r coordinate chart whose image was the union of the
images of ϕ1 and ϕ2 , which would contradict the minimality of K.
Therefore ϕ is not injective. Since ϕ1 and ϕ2 are injective, there must be s <
b1 − δ − c such that ϕ(s) = ϕ(s′ ) for some s′ > t1 . Let s0 be the supremum of such
s. If ϕ(s0 ) = ϕ(s′ ) for some s′ > t1 , then the restrictions of ϕ to neighborhoods of
s0 and s′ would both map diffeomorphically onto some neighborhood of this point,
which would give a contradiction of the definition of s0 . Therefore ϕ(s0 ) is in the
closure of ϕ(((t1 , b2 )), but is not an element of this set, so it must be lims′ →b2 ϕ(s′ ).
Arguments similar to those given above imply that there are α, β > 0 such that the
images of ϕ|(b2 −α,b2 ) and ϕ|(s0 −β,s0 ) are the same, and the C r diffeomorphism
g = (ϕ|(s0 −β,s0 ) )−1 ◦ ϕ|(b2 −α,b2 )
is increasing. Applying the lemma above again, there is a real number R and an
increasing C r diffeomorphism λ : (b2 −α, b2 ) → (b2 −α−R, s0 ) such that λ(s) = s−R
for s near b2 − α and λ(s) = g(s) for s near b2 .
We now define ψ : [s0 , s0 + R) → M by setting
(
ϕ(s),
s0 ≤ s ≤ b2 − α,
ψ(s) =
−1
ϕ(λ (s − R)), b2 − α < s < s0 + R.
Then ψ agrees with ϕ near b2 − α, so it is C r , and it agrees with ϕ(s − R) near
s0 + R, so it can be construed as a C r function from the circle (thought of R modulo
10.9. CLASSIFICATION OF COMPACT 1-MANIFOLDS
149
R) to M. This function is easily seen to be injective, and it maps open sets to open
sets, so its image is open, but also compact, hence closed. Since M is connected,
its image must be all of M, so we have constructed te desired C r diffeomorphism
between the circle and M.
The argument for a compact connected one dimensional C r ∂-manifold with
nonempty boundary is similar, but somewhat simpler, so we leave it to the reader.
Although it will not figure in the work here, the reader should certainly be
aware that the analogous issues for higher dimensions are extremely important in
topology, and mathematical culture more generally. In general, a classification of
some type of mathematical object is a description of all the isomorphism classes
(for whatever is the appropriate notion of isomorphism) of the object in question.
The result above classifies compact connected 1-dimensional C r manifolds.
The problem of classifying oriented surfaces (2-dimensional manifolds) was first
considered in a paper of Möbius in 1870. The classification of all compact connected
surfaces was correctly stated by van Dyke in 1888. This result was proved for
surfaces that can be triangulated by Dehn and Heegaard in 1907, and in 1925 Rado
showed that any surface can be triangulated.
After some missteps, Poincaré formulated a fundamental problem for the the
classification of 3-manifolds: is a simply connected compact 3-manifold necessarily
homeomorphic to S 3 ? (A topological space X is simply connected if it is connected and any continuous function f : S 1 = { (x, y) ∈ R2 : x2 + y 2 = 1 } → X has
a continuous extension F : D 2 = { (x, y) ∈ R2 : x2 + y 2 ≤ 1 } → X.) Although
Poincaré did not express a strong view, this became known as the Poincaré conjecture, and over the course of the 20th century, as it resisted solution and the four
color theorem and Fermat’s last theorem were proved, it became perhaps the most
famous open problem in mathematics. Curiously, the analogous theorems for higher
dimensions were proved first, by Smale in 1961 for dimensions five and higher, and
by Freedman in 1982 for dimension four. Finally in late 2002 and 2003 Perelman
posted three papers on the internet that sketched a proof of the original conjecture.
Over the next three years three different teams of two mathematicians set about
filling in the details of the argument. In the middle of 2006 each of the teams posted
a (book length) paper giving a complete argument. Although Perelman’s papers
were quite terse, and many details needed to be filled in, all three teams agreed
that all gaps in his argument were minor.
Chapter 11
Sard’s Theorem
The results concerning existence and uniqueness of systems of linear equations
have been well established for a long time, of course. In the late 19th century Walras recognized that the system describing economic equilibria had (after recognizing
the redundant equation now known as Walras’ law) the same number of equations
and free variables, which suggested that “typically” economic equilibria should be
isolated and also robust, in the sense that the endogenous variables will vary continuously with the underlying parameters in some neighborhood of the initial point. It
was several decades before methods for making these ideas precise were established
in mathematics, and then several more decades elapsed before they were imported
into theoretical economics.
The original versions of what is now known as Sard’s theorem appeared during
the 1930’s. There followed a process of evolution, both in the generality of the
result and in the method of proof, that culminated in the version due to Federer
(see Section 11.3.) Our treatment here is primarily based on Milnor (1965), fleshed
out with some arguments from Sternberg (1983), which (in its first edition) seems
to have been Milnor’s primary source. While not completely general, this version of
the result is adequate for all of the applications in economic theory to date, many
of which are extremely important.
Suppose 1 ≤ r ≤ ∞, and let f : U → Rn be a C r function, where U ⊂ Rm
is open. If f (x) = y and Df (x) has rank n, then the implicit function theorem
(Theorem 10.1.2) implies that, in a neighborhood of x, f −1 (y) can be thought of
as the graph of a C r function. Intuition developed by looking at low dimensional
examples suggests that for “typical” values of y this pleasant situation will prevail
at all elements of f −1 (y), but even in the case m = n = 1 one can see that there can
be a countable infinity of exceptional y. Thus the difficulty in formulating this idea
precisely is that we need a suitable notion of a “small” subset of Rn . This problem
was solved by the theory of Lesbesgue measure, which explains the relatively late
date at which the result first appeared.
Measure theory has rather complex foundations, so it preferable that it not be
a prerequisite. Thus it is fortunate that only the notion of a set of measure zero
is required. Section 11.1 defines this notion and establishes its basic properties.
One of the most important results in measure theory is Fubini’s theorem, which,
roughly speaking, allows functions to be integrated one variable at a time. Section
150
151
11.1. SETS OF MEASURE ZERO
11.2 develops a Fubini-like result for sets of measure zero. With these elements
in place, it becomes possible to state and prove Sard’s theorem in Section 11.3.
Section 11.4 explains how to extend the result to maps between sufficiently smooth
manifolds.
The application of Sard’s theorem that is most important in the larger scheme of
this book is given in Section 11.5. The overall idea is to show that any map between
manifolds can be approximated by one that is transversal to a given submanifold
of the range.
11.1
Sets of Measure Zero
For each n there is a positive constant such that the volume of a ball in Rn is
that constant times r n , where r is the radius of the ball. Without knowing very
much about the constant, we can still say that sets satisfying the following definition
are “small.”
Definition 11.1.1. A set S ⊂ Rm has measure zero if, for any ε > 0, there is
k
a sequence {(xj , rj )}∞
j=1 in R × (0, 1) such that
S⊂
[
j
Urj (xj ) and
X
rjm < ε.
j
Of course we can use different sets, such as cubes, as a measure of whether
a set has measure
Specifically, if we can find a covering of S by balls of
P zero.
m
radiusPrj with
j rj < ε, then there is a covering by cubes of side length 2rj
with j (2rj )m < 2m ε, and if we can find a covering of S by cubes of side lengths
P
√
2ℓj with
(2ℓj )m < ε, then there is a covering by balls of radius mℓj with
j
Qm
P √
√
m
m
i=1 [ai , bi ] because we can
j ( mℓj ) < ( m/2) ε. We can also use rectangles
cover such a rectangle with a collection of cubes of almost the same total volume;
from the point of view of our methodology it is important to recognize that we
“know” this as a fact of arithmetic (and in particular the distributive law) rather
than as prior knowledge concerning volume.
The rest of this section develops a few basic facts. The following property of
sets of measure zero occurs frequently in proofs.
Lemma 11.1.2. If S1 , S2 , . . . ⊂ Rm are sets of measure zero, then S1 ∪ S2 ∪ . . . has
measure zero.
Proof. For given ε take the union of a countable cover of S1 by rectangles of total
volume < ε/2, a countable cover of S2 by rectangles of total volume < ε/4, etc.
It is intuitively obvious that a set of measure zero cannot have a nonempty
interior, but our methodology requires that we “forget” everything we know about
volume, using only arithmetic to prove it.
Lemma 11.1.3. If S has measure zero, its interior is empty, so its complement is
dense.
152
CHAPTER 11. SARD’S THEOREM
Proof. Suppose that, on the contrary, S has a nonempty interior. Then it contains a
closed cube C, say of side length
P 2ℓ. Fixing ε > 0, suppose that S has a covering by
cubes of side length 2ℓj with j (2ℓj )m < ε. Then it has a covering by open cubes
Cj of side length 3ℓj , and there is a finite subcover of C. For some large integer K,
Q
ij ij +1
consider all “standard cubes” of the form m
j=1 [ K , K ]. For each cube in our finite
subcover, let Dj be the union of all such standard cubes contained in Cj , and let nj
be the number of such cubes. Let D be the union of all standard cubes containing a
point in C, and let n be the number of them. Simply as a matter of counting (that
is to say, without reference to any theory of volume)Swe have nj /K m ≤ P
(3ℓj )m and
n/K m ≥ (2ℓ)m . If K is sufficiently large, then D ⊂ j Dj , so that n ≤ j nj and
(2ℓ)m ≤ n/K m ≤
X
j
nj /K m ≤
X
j
(3ℓj )m ≤ (3/2)m ε,
so that ε > (4ℓ/3)m cannot be arbitrarily small.
The next result implies that the notion of a set of measure zero is invariant
under C 1 changes of coordinates. In the proof of Theorem 11.3.1 we will use this
flexibility to choose coordinate systems with useful properties. In addition, this
fact is the key to the definition of sets of measure zero in manifolds. Recall that if
L : Rm → Rm is a linear transformation, then the operator norm of L is
kLk = max kL(v)k.
kvk=1
Lemma 11.1.4. If U ⊂ Rm is open, f : U → Rm is C 1 , and S ⊂ U has measure
zero, then f (S) has measure zero.
Proof. Let C ⊂ U be a closed cube. Since U can be covered by countably many such
cubes (e.g., all cubes contained in U with rational centers and rational side lengths)
it suffices to show that f (S ∩ C) has measure zero. Let B := maxx∈C kDf (x)k. For
any x, y ∈ C we have
Z 1
kf (x) − f (y)k = Df ((1 − t)x + ty)(y − x) dt
0
Z 1
≤
kDf ((1 − t)x + ty)k · ky − xk dt ≤ Bky − xk.
0
If {(xj , rj )}∞
j=1 is a sequence such that
S∩C ⊂
[
Urj (xj ) and
j
X
rjm < ε,
X
(Brj )m < B m ε.
j
then
f (S ∩ C) ⊂
[
j
UBrj (f (xj )) and
j
11.2. A WEAK FUBINI THEOREM
11.2
153
A Weak Fubini Theorem
For a set S ⊂ Rm and t ∈ R let
S(t) := { (x2 , . . . , xm ) ∈ Rm−1 : (t, x2 , . . . , xm ) ∈ S }
be the t-slice of S. Let P (S) be the set of t such that S(t) does not have (m − 1)dimensional measure zero. Certainly it seems natural to expect that if S is a set
of m-dimensional zero, then P (S) should be a set of 1-dimensional measure zero,
and conversely. This is true, by virtue of Fubini’s theorem, but we do not have the
means to prove it in full generality. Fortunately all we will need going forward is a
special case.
Proposition 11.2.1. If S ⊂ Rm is locally closed, then S has measure zero if and
only if P (S) has measure zero.
We will prove this in several steps.
Lemma 11.2.2. If C ⊂ Rm is compact, then C has measure zero if and only if
P (C) has measure zero.
Proof of Proposition 11.2.1. Suppose that S = C ∩ U where C is closed and U is
open. Let A1 , A2 , . . . be a countable collection of compact rectangles that cover U.
Then the following are equivalent:
(a) S has measure zero;
(b) each C ∩ Aj has measure zero;
(c) each P (C ∩ Aj ) has measure zero;
(d) P (S) has measure zero.
Specifically,
S Lemma 11.1.2 implies that (a) and (b) are equivalent, and also that
P (S) = j P (C ∩ Aj ), after which the equivalence of (c) and (d) follows from a
third application of the result. The equivalence of (b) and (c) follows from the
lemma above.
We now need to prove Lemma
Q 11.2.2. Fix a compact set C, which we assume
is contained in the rectangle m
i=1 [ai , bi ]. For each δ > 0 let Pδ (C) be the set of
t such that C(t) cannot be covered by a finite collection of open rectangles whose
total (m − 1)-dimensional volume is less than δ.
Lemma 11.2.3. For each δ > 0, Pδ (C) is closed.
Proof. If t is in the complement of Pδ (C), then any collection of open rectangles that
cover C(t) also covers C(t′ ) for t′ sufficiently close to t, because C is compact.
The next two results are the two implications of Lemma 11.2.2.
Lemma 11.2.4. If P (C) has measure zero, then C has measure zero.
154
CHAPTER 11. SARD’S THEOREM
Proof. Fix ε > 0, and choose δ < ε/2(b1 − a1 ). Since Pδ (C) ⊂ P (C), it has one
dimensional measure zero, and since it is closed, hence compact, it can be covered by
the union J of finitely many open intervals of total length ε/2(b2 − a2 ) · · · (bm − am ).
In this way { x ∈ C : x1 ∈ J } is covered by a union of open rectangles of total
volume ≤ ε/2.
For each t ∈
/ J we can choose a finite union of rectangles in Rm−1 of total volume
less than δ that covers C(t), and these will also cover C(t′ ) for all t′ in some open
interval around t. Since [a1 , b1 ] \ J is compact, it is covered by a finite collection of
such intervals, and it is evident that we can construct a cover of { x ∈ C : x1 ∈
/J}
of total volume less than ε/2.
Lemma 11.2.5. If C has measure zero, then P (C) has measure zero.
S
Proof. Since P (C) = n=1,2,... P1/n (C), it suffices to show that Pδ (C) has measure
zero for any δ > 0. For any ε > 0 there is a covering of C by finitely many rectangles
of total volume less than ε. For each t there is an induced covering C(t) be a finite
collection of rectangles, and there is an induced covering of [a1 , b1 ]. The total length
of intervals with induced coverings of total volume greater than δ cannot exceed
ε/δ.
11.3
Sard’s Theorem
We now come to this chapter’s central result. Recall that a critical point of a
C function is a point in the domain at which the rank of the derivative is less than
the dimension of the range, and a critical value is a point in the range that is the
image of a critical point.
1
Theorem 11.3.1. If U ⊂ Rm is open and f : U → Rn is a C r function, where
r > max{m − n, 0}, then the set of critical values of f has measure zero.
Proof. If n = 0, then f has no critical points and therefore no critical values. If
m = 0, then U is either a single point or the null set, and if n > 0 its image has
measure zero. Therefore we may assume that m, n > 0. Since r > m − n implies
both r > (m − 1) − (n − 1) and r > (m − 1) − n, by induction we may assume
that the claim has been established with (m, n) replaced by either (m − 1, n − 1) or
(m − 1, n).
Let C by the set of critical points of f . For i = 1, . . . , r let Ci be the set of
points in U at which all partial derivatives of f up to order i vanish. It suffices to
show that:
(a) f (C \ C1 ) has measure 0;
(b) f (Ci \ Ci+1 ) has measure zero for all i = 1, . . . , r − 1;
(c) f (Cr ) has measure zero.
Proof of (a): We will show that each x ∈ C \ C1 has a neighborhood V such that
f (V ∩ C) has measure zero. This suffices because C \ C1 is an open subset of a
155
11.3. SARD’S THEOREM
closed set, so it is covered by countably many compact sets, each of which is covered
by finitely many such neighborhoods, and consequently it has a countable cover by
such neighborhoods.
∂f1
After reindexing we may assume that ∂x
(x) 6= 0. Let V be a neighborhood of
1
∂f1
x in which ∂x1 does not vanish. Let h : V → Rm be the function
h(x) := (f1 (x), x2 , . . . , xm ).
The matrix of partial derivatives of h at x is
 ∂f
∂f1
1
(x) ∂x
(x) · · ·
∂x1
2
 0
1
···

 ..
.
..
 .
0
···
0
∂f1
(x)
∂xm
0
..
.
1



,

so the inverse function theorem implies that, after replacing V with a smaller neighborhood of x, h is a diffeomorphism onto its image. The chain rule implies that
the critical values of f are the critical values of g = f ◦ h−1 , so we can replace f
with g, and g has the additional property that g1 (z) = z1 for all z in its domain.
The upshot of this argument is that we may assume without loss of generality that
f1 (x) = x1 for all x ∈ V .
For each t ∈ R let V t := { w ∈ Rm−1 : (t, w) ∈ V }, let f t : V t → Rn−1 be the
function
f t (w) := (f2 (t, w), . . . , fn (t, w)),
and let C t be the set of critical points of f t . The matrix of partial derivatives of f
at x ∈ V is


1
0
···
0
 ∂f2 (x) ∂f2 (x) · · · ∂f2 (x)
∂x2
∂xm

 ∂x1
 ..
..
..  ,
 .
.
. 
∂fn
(x)
∂x1
∂fn
(x)
∂x2
···
∂fn
(x)
∂xm
so x is a critical point of f if and only if (x2 , . . . , xm ) is a critical point of f x1 , and
consequently
[
[
C∩V =
{t} × C t and f (C ∩ V ) =
{t} × f t (C t ).
t
t
Since the result is known to be true with (m, n) replaced by (m − 1, n − 1), each
f t (C t ) has (n − 1)-dimensional measure zero. In addition, the continuity of the
relevant partial derivatives implies that C \ C1 is locally closed, so Proposition
11.2.1 implies that f (C ∩ V ) has measure zero.
Proof of (b): As above, it is enough to show that an arbitrary x ∈ Ci \ Ci+1 has a
neighborhood V such that f (Ci ∩ V ) has measure zero. Choose a partial derivative
∂ i+1 f
that does not vanish at x. Define h : U → Rm by
∂xs ···∂xs ·∂xs
1
i
i+1
i
f
(x), x2 , . . . , xm ).
h(x) := ( ∂xs ∂···∂x
s
1
i
156
CHAPTER 11. SARD’S THEOREM
After reindexing we may assume that si+1 = 1, so that the matrix of partial derivatives of h at x is triangular with nonzero diagonal entries. By the inverse function
theorem the restriction of h to some neighborhood V of x is a C ∞ diffeomorphism.
Let g := f ◦ (h|V )−1 . Then h(V ∩ Ci ) ⊂ {0} × Rm−1 . Let
g0 : { y ∈ Rm−1 : (0, y) ∈ h(V ) } → Rn
be the map g0 (y) = g(0, y). Then f (V ∩(Ci \Ci+1 )) is contained in the set of critical
values of g0 , and the latter set has measure zero because the result is already known
when (m, n) is replaced by (m − 1, n).
Proof of (c): Since U can be covered by countably many compact cubes, it suffices
to show that f (Cr ∩I) has measure zero whenever I ⊂ U is a compact cube. Since I
is compact and the partials of f of order r are continuous, Taylor’s theorem implies
that for every ε > 0 there is δ > 0 such that
kf (x + h) − f (x)k ≤ εkhkr
whenever x, x + h ∈ I with x ∈ Cr and khk < δ. Let L be the side length of I. For
each integer d > 0√divide I into dm subcubes of side length L/d. The diameter of
such a subcube is mL/d. If this quantity is less than δ and the subcube
contains a
√
point x ∈ Cr , then its image is contained in a cube of sidelength 2ε( mL)r centered
at f (x). There are dm subcubes of I, each one of which may or may not contain a
point in Cr , so for large
r ∩ I) is contained in a finite union of cubes of total
√ d,r fn(C
n m−nr
volume at most 2( mL) ε d
. Now observe that nr ≥ m: either m < n and
r ≥ 1, or m ≥ n and
nr ≥ n(m − n + 1) = (n − 1)(m − n) + m ≥ m.
Therefore
f (Cr ∩ I) is contained in a finite union of cubes of total volume at most
√
r n n
2( mL) ε , and ε may be arbitrarily small.
Instead of worrying about just which degree of differentiability is the smallest
that allows all required applications of Sard’s theorem, in the remainder of the book
we will, for the most part, work with objects that are smooth, where smooth is a
synonym for C ∞ . This will result in no loss of generality, since for the most part
the arguments depend on the existence of smooth objects, which will follow from
Proposition 10.2.7. However, in Chapter 15 there will be given objects that may, in
applications, be only C 1 , but Sard’s theorem will be applicable because the domain
and range have the same dimension. It is perhaps worth mentioning that for this
particular case there is a simpler proof, which can be found on p. 72 of Spivak
(1965).
We briefly describe the most general and powerful version of Sard’s theorem,
which depends on a more general notion of dimension.
Definition 11.3.2. For α > 0, a set S ⊂ Rk has α-dimensional Hausdorff
measure zero if, for any ε > 0, there is a sequence {(xj , δj )}∞
j=1 such that
[
X
S⊂
Uδj (xj ) and
δjα < ε.
j
j
11.4. MEASURE ZERO SUBSETS OF MANIFOLDS
157
Note that this definition makes perfect sense even if α is not an integer! Let
U ⊂ Rm be open, and let f : U → Rn be a C r function. For 0 ≤ p < m let Rp
be the set of points x ∈ M such that the rank of Df (x) is less than or equal to
p. The most general and sophisticated version of Sard’s theorem, due to Federer,
states that f (Rp ) has α-dimensional measure zero for all α > p + m−p
. A beautiful
r
informal introduction to the circle of ideas surrounding these concepts, which is the
branch of analysis called geometric measure theory, is given by Morgan (1988). The
proof itself is in Section 3.4 of Federer (1969). This reference also gives a complete
set of counterexamples showing this result to be best possible.
11.4
Measure Zero Subsets of Manifolds
In most books Sard’s theorem is presented as a result concerning maps between
Euclidean spaces, as in the last section, with relatively little attention to the extension to maps between manifolds. Certainly this extension is intuitively obvious,
and there are no real surprises or subtleties in the details, which are laid out in this
section.
Definition 11.4.1. If M ⊂ Rk is an m-dimensional C 1 manifold, then S ⊂ M has
m-dimensional measure zero if ϕ−1 (S) has measure zero whenever U ⊂ Rm is
open and ϕ : U → M is a C 1 parameterization.
In order for this to be sensible, it should be the case that ϕ(S) has measure zero
whenever ϕ : U → M is a C 1 parameterization and S ⊂ U has measure zero. That
is, it must be the case that if ϕ′ : U ′ → M is another C 1 parameterization, then
ϕ′ −1 (ϕ(S)) has measure zero. This follows from the application of Lemma 11.1.4
to ϕ′ −1 ◦ ϕ.
Clearly the basic properties of sets of measure zero in Euclidean spaces—the
complement of a set of measure zero is dense, and countable unions of sets of measure zero have measure zero—extend, by straightforward verifications, to subsets of
manifolds of measure zero. Since uncountable unions of sets of measure zero need
not have measure zero, the following fact about manifolds (as we have defined them,
namely submanifolds of Euclidean spaces) is comforting, even if the definition above
makes it superfluous.
Lemma 11.4.2. If M ⊂ Rk is an m-dimensional C 1 manifold, then M is covered
by the images of a countable system of parameterizations {ϕj : Uj → M}j=1,2,....
Proof. If p ∈ M and ϕ : U → M is a C r parameterization with p ∈ ϕ(U), then
there is an open set W ⊂ Rk such that ϕ(U) = M ∩ W . Of course there is an open
ball B of rational radius whose center has rational coordinates with p ∈ B ⊂ W ,
and we may replace ϕ with its restriction to ϕ−1 (B). Now the claim follows from
the fact that there are countably many balls in Rk of rational radii centered at
points with rational coordinates.
The “conceptually correct” version of Sard’s theorem is an easy consequence of
the Euclidean special case.
158
CHAPTER 11. SARD’S THEOREM
Theorem 11.4.3. (Morse-Sard Theorem) If f : M → N is a smooth map, where
M and N are smooth manifolds, then the set of critical values of f has measure
zero.
Proof. Let C be the set of critical points of f . In view of the last result it suffices to
show that f (C ∩ϕ(U)) has measure zero whenever ϕ : U → M is a parameterization
for M. That is, we need to show that ψ −1 (f (C ∩ ϕ(U))) has measure zero whenever
ψ : V → N is a parameterization for N. But ψ −1 (f (C ∩ ϕ(U))) is the set of critical
values of ψ −1 ◦ f ◦ ϕ, so this follows from Theorem 11.3.1.
11.5
Genericity of Transversality
Intuitively, it should be unlikely that two smooth curves in 3-dimensional space
intersect. If they happen to, it should be possible to undo the intersection by
“perturbing” one of the curves slightly. Similarly, a smooth curve and a smooth
surface in 3-space should intersect at isolated points, and again one expects that
a small perturbation can bring about this situation if it is not the case initially.
Sard’s theorem can help us express this intuition precisely.
Let M and N be m- and n-dimensional smooth manifolds, and let P be a pdimensional smooth submanifold of N. Recall that a smooth function f : M → N
is transversal to P if
Df (p)(Tp M) + Tf (p) P = Tf (p) N
for every p ∈ f −1 (P ). The conceptual point studied in this section is that smooth
functions from M to N that are transversal to P are plentiful, in the sense that any
continuous function can be approximated by such a map. This is expressed more
precisely by the following result.
Proposition 11.5.1. If f : M → N is a continuous function and A ⊂ M × N
is a neighborhood of Gr(f ), then there is a smooth function f ′ : M → N that is
transverse to P with Gr(f ′ ) ⊂ A.
In some applications the approximation will be required to satisfy some restriction. A vector field on a set S ⊂ M is a continuous function ζ : S → T M such
that π ◦ ζ = IdM , where π : T M → M is the projection. We can write ζ(p) = (p, ζp )
where ζp ∈ Tp M. Thus a vector field on S attaches a tangent vector ζp to each
p ∈ S, in a continuous manner. The zero section of T M is M × {0} ⊂ T M. If we
like we can think of it as the image of the vector field that is identically zero, and
of course it is also an m-dimensional smooth submanifold of T M.
Proposition 11.5.2. If ζ is a vector field on M and A ⊂ T M is an open neighborhood of { (p, ζp ) : p ∈ M }, then there is a smooth vector field ζ ′ such that
{ (p, ζp′ ) : p ∈ M } ⊂ A and ζ ′ is transverse to the zero section of T M.
To obtain a framework that is sufficiently general we introduce an s-dimensional
smooth manifold S and a smooth submersion h : N → S. We now fix a continuous
function f : M → N such that h ◦ f is a smooth submersion. Our main result,
which has the two results above as corollaries, is:
11.5. GENERICITY OF TRANSVERSALITY
159
Theorem 11.5.3. If A ⊂ M × N is an open neighborhood of Gr(f ), then there is
a smooth function f ′ : M → N with Gr(f ′ ) ⊂ A, h ◦ f ′ = h ◦ f , and f ′ ⋔ P .
The first proposition above is the special case of this in which S is a point. To
obtain the second proposition we set S = M and let h be the natural projection
T M → M.
The rest of this section is devoted to the proof of Theorem 11.5.3. The proof
is a matter of repeated modifying f on sets that are small enough that we can
conduct the construction in a fully Euclidean setting. In the following result, which
describes the local modification, there is an open domain U ⊂ Rm and a compact
“target” set K ⊂ U. There is a closed set C ⊂ U and an open neighborhood Y of
C on which the desired tranversality already holds. We wish to modify the function
so that the desired transversality holds on a possibly smaller neighborhood of C,
and also on a neighborhood of K. However, when we apply this result U will be,
in effect, an open subset of M, and in order to preserve the properties of the given
function at points in the boundary of U in M it will need to be the case that the
function is unchanged outside of a given compact neighborhood of K. Collectively
these requirements create a significant burden of complexity.
Proposition 11.5.4. Let U be an open subset of Rm , suppose that C ⊂ W ⊂ U
with C relatively closed and W open, let K be a compact subset of U, and let Z be
an open neighborhood of K whose closure is a compact subset of U. Suppose that
g = (g s , g n−s) : U → Rn = Rs × Rn−s
is a continuous function, g|W is smooth, and g s is a smooth submersion. Let P be a
p-dimensional smooth submanifold of Rn , and suppose that g|C ⋔ P . Let A ⊂ U ×Rn
be a neighborhood of the graph of g. Then there is an open Z ′ ⊂ Z containing U and
a continuous function g̃ n−s : U → Rn−s such that, setting g̃ = (g s , g̃ n−s ) : U → Rn ,
we have:
(a) Gr(g̃) ⊂ A;
(b) g̃|U \Z = g|U \Z ;
(c) g̃|W ∪Z ′ is smooth;
(d) g̃ ⋔C∪K P .
We first explain how this result can be used to prove Theorem 11.5.3. The next
result describes how the local modification of f looks in context. Let ψ : V → N
be a smooth parameterization. We say that ψ is aligned with h if h(ψ(y)) is
independent of ys+1, . . . , yn .
Lemma 11.5.5. Suppose that C ⊂ W ⊂ M with C closed and W open, f |W is
smooth, and f ⋔C P . Let A ⊂ M × N be an open set that contains the graph of
f . Suppose that ϕ : U → M and ψ : V → N are smooth parameterizations with
f (ϕ(U)) ⊂ ψ(V ) and ψ aligned with h. Suppose that K ⊂ ϕ(U) is compact, and Z
is an open subset of ϕ(U) whose closure is compact and contained in ϕ(U). Then
there is an open Z ′ ⊂ Z containing K and a continuous function f˜ : M → N such
that:
160
CHAPTER 11. SARD’S THEOREM
(a) Gr(f˜) ⊂ A;
(b) f˜|M \Z = f |M \Z ;
(c) f˜|W ∪Z ′ is smooth;
(d) f˜ ⋔C∪K P .
Proof. Let
P̃ = ψ −1 (P ),
g = ψ −1 ◦ f ◦ ϕ,
C̃ = ϕ−1 (C),
à = { (x, y) ∈ U × V : (ϕ(x), ψ(y)) ∈ A },
W̃ = ϕ−1 (W ),
K̃ = ϕ−1 (K),
Z̃ = ϕ−1 (Z).
Let Z̃ ′ and g̃ be the set and function whose existence is guaranteed by the last result.
Set Z ′ = ϕ(Z̃ ′ ), and define f˜ by specifying that f˜ agrees with f on M \ ϕ(U), and
f˜|ϕ(U ) = ψ ◦ g̃ ◦ ϕ−1 .
Clearly f˜ has all the desired properties.
In order to apply this we need to have an ample supply of smooth parameterizations for N that are aligned with h.
Lemma 11.5.6. Each point q ∈ f (M) is contained in the image of a smooth
parameterization that is aligned with h.
Proof. Let ψ̃ : Ṽ → N be any smooth parameterization whose image contains q,
and let y = ψ̃ −1 (q). Let σ : W → S be a smooth parameterization whose image
contains h(q); we can replace Ṽ with a smaller open set containing y, so we may
assume that h(ψ̃(Ṽ )) ⊂ σ(W ).
Since h ◦ f is a submersion, the rank of Dh(q) is s, and consequently the rank
of D(σ −1 ◦ h ◦ ψ̃)(y) is also s. After some reindexing, y is a regular point of
′
, . . . , yn′ ).
θ : y ′ 7→ (σ −1 (h(ψ̃(y ′ ))), ys+1
Applying the inverse function theorem, a smooth parameterization whose image
contains q that is aligned with h is given by letting ψ be the restriction of ψ̃ ◦ θ−1
to some neighborhood of (σ −1 (h(q)), ys+1, . . . , yn ).
Proof of Theorem 11.5.3. Any open subset of M is a union of open subsets whose
closures are compact. In view of this fact and the last result, M is covered by the
sets ϕ(U) where ϕ : U → M is a smooth parameterization with f (ϕ(U)) ⊂ ψ(V ) for
some smooth parameterization ψ : V → N that is aligned with h, and the closure
of ϕ(U) is compact. Since M is paracompact, there is a locally finite cover by the
images of such parameterizations, and since M is separable, this cover is countable.
That is, there is a sequence ϕ1 : U1 → M, ϕ2 : U2 → M, . . . whose images cover
M, such that for each i, the closure of ϕi (Ui ) is compact and there is a smooth
parameterization ψi : Vi → N that is aligned with h such that f (ϕi (Ui )) ⊂ ψi (Vi ).
We claim that there is a sequence K1 , K2 , . . . of compact subsets of M that cover
M, with Ki ⊂ ϕi (Ui ) for each i. For p ∈ M let δ(p) be the maximum ε such that
11.5. GENERICITY OF TRANSVERSALITY
161
Uε (p) ⊂ ϕi (Ui ) for some i, and let ip be an integer that attains the maximum. Then
δ : M → (0, ∞) is a continuous function. For each i let δi := minp∈ϕi (Ui ) δ(p), and
let
Ki = { p ∈ ϕi (Ui ) : Uδi (p) ⊂ ϕi (Ui ) }.
Clearly Ki is a closed subset of ϕi (Ui ), so it is compact. For any p ∈ M we have
p ∈ Kip , so the sets K1 , K2 , . . . cover M.
Let C0 = ∅, and for each positive i let Ci = K1 ∪ . . . ∪ Ki . Let f0 = f . Suppose
for some i = 1, 2, . . . that we have already constructed a neighborhood Wi−1 of Ci−1
and a continuous function fi−1 : M → N with Gr(fi−1 ) ⊂ A such that f |Wi−1 is
smooth and f ⋔Ci−1 P . Let Zi be an open subset of ϕi (Ui ) that contains Ki , and
whose closure is compact and contained in ϕi (Ui ). Now Lemma 11.5.5 gives an open
Zi′ ⊂ Zi containing Ki and a continuous function fi : M → N with Gr(fi ) ⊂ A such
that fWi−1 ∪Zi′ is smooth, fi |M \Zi = fi−1 |M \Zi , and fi ⋔Ci P . Set Wi = Wi−1 ∪ Zi′ .
Evidently this constructive process can be extended to all i.
For each i, ϕi (Ui ) intersects only finitely many ϕj (Uj ), so the sequence
f1 |ϕi (Ui ) , f2 |ϕi (Ui ) , . . .
is unchanging after some point. Thus the sequence f1 , f2 , . . . has a well defined limit
that is smooth and transversal to P , and whose graph is contained in A.
We now turn to the proof of Proposition 11.5.4. The main idea is to select a
suitable member from a family of perturbations of g. The following lemma isolates
the step in the argument that uses Sard’s theorem.
Lemma 11.5.7. If U ⊂ Rm and B ⊂ Rn−s are open, P is a p-dimensional smooth
submanifold of Rn , and G : U × B → Rn is smooth and transversal to P , then for
almost every b ∈ B the functions gb = G(·, b) : U → N is transversal to P .
Proof. Let Q = G−1 (P ). By the transversality theorem, Q is a smooth manifold,
of dimension (m + (n − s)) − (n − p) = m + p − s. Let π be the natural projection
U × B → B. Sard’s theorem implies that almost every b ∈ B is a regular value of
π|Q . Fix such a b. We will show that gb is transversal to P .
Fix x ∈ gb−1 (P ), set q = gb (x), and choose some y ∈ Tq N. Since G is transversal
to P there is a u ∈ T(x,b) (U ×B) such that y is the sum of DG(x, b)u and an element
of Tq P . Let u = (v, w) where v ∈ Rm and w ∈ Rn−s . Since (x, b) is a regular point
of π|Q , there is a u′ ∈ T(x,b) Q such that Dπ|Q (x, b)u′ = −w. Let u′ = (v ′ , −w).
Then Tq P contains DG(x, b)u′ , so it contains
DG(x, b)u − y + DG(x, b)u′ = DG(x, b)(v + v ′ , 0) − y = Dgb(x)(v + v ′ ) − y.
Thus y is the sum of Dgb (x)(v + v ′ ) and an element of Tq P , as desired.
Proof of Proposition 11.5.4. For x ∈ U let α̃(x) be the supremum of the set of
αx > 0 such that (x, y) ∈ A whenever y ∈ Uαx (g(x′ )). Clearly α̃ is continuous
and positive, so (e.g., Proposition 10.2.7 applied to 21 α̃) there is a smooth function
α : U → (0, ∞) such that 0 < α(x) < α̃(x) for all x.
There is a neighborhood Y ⊂ U of C such that g|Y is smooth with g|Y ⋔ P .
′
Let Y ′ be an open subsets of U with C ⊂ Y ′ and Y ⊂ Y . Corollary 10.2.5 gives a
162
CHAPTER 11. SARD’S THEOREM
smooth function β : U → [0, 1] that vanishes on Y ′ and is identically equal to one
on U \ Y .
Let B be the open unit disk centered at the origin in Rn−s , and let G : U × B →
Rn be the smooth function
G(x, b) = g s (x), g n−s (x) + α(x)β(x)b .
For any (x, b) the image of DG(x, b) contains the image of Dg(x), so, since g|Y ⋔ P ,
we have G ⋔Y ×B P . Since g s is a submersion, at every (x, b) such that β(x) > 0 the
image of DG(x, b) is all of Rn , so G ⋔(U \Y )×B P . Therefore G ⋔ P . The last result
implies that for some b ∈ B, gb = G(·, b) is transversal to P . Evidently gb agrees
with g on Y ′ .
′
Let Z ′ be an open subset of U with K ⊂ Z ′ and Z ⊂ Z. Corollary 10.2.5 gives
a smooth γ : U → [0, 1] that is identically one on Z ′ and vanishes on U \ Z. Define
g̃ be setting
g̃(x) = g s (x), γ(x)gbn−s (x) + (1 − γ(x))g n−s (x) .
Clearly Gr(g̃) ⊂ A, g̃ is smooth on W ∪ Z ′ , and g̃ agrees with g on U \ Z. Moreover,
g̃ agrees with g on Y ′ and with gb on Z ′ , so g̃ ⋔Y ′ ∪Z ′ P .
Chapter 12
Degree Theory
Orientation is an intuitively familiar phenomenon, modelling, among other things,
the fact that there is no way to turn a left shoe into a right shoe by rotating it, but
the mirror image of a left shoe is a right shoe. Consider that when you look at a
mirror there is a coordinate system in which the map taking each point to its mirror
image is the linear transformation (x1 , x2 , x3 ) 7→ (−x1 , x2 , x3 ). It turns out that the
critical feature of this transformation is that its determinant is negative. Section
12.1 describes the formalism used to impose an orientation on a vector space and
consistently on the tangent spaces of the points of a manifold, when this is possible.
Section 12.2 discusses two senses in which an orientation on a given object
induces a derived orientation: a) an orientation on a ∂-manifold induces an orientation of its boundary; b) given a smooth map between two manifolds of the same
dimension, an orientation of the tangent space of a regular point in the domain
induces an orientation of the tangent space of that point’s image. If both manifolds
are oriented, we can define a sense in which the map is orientation preserving or
orientation reversing by comparing the induced orientation of the tangent space of
the image point with its given orientation.
In Section 12.3 we first define the smooth degree of a smooth (where “smooth”
now means C ∞ ) map over a regular value in the range to be the number of preimages
of the point at which the map is orientation preserving minus the number of points at
which it is orientation reversing. Although the degree for smooth functions provides
the correct geometric intuition, it is insufficiently general. The desired generalization is achieved by approximating a continuous function with smooth functions,
and showing that any two sufficiently accurate approximations are homotopic, so
that such approximations can be used to define the degree of the given continuous
function. However, instead of working directly with such a definition, it turns out
that an axiomatic characterization is more useful.
12.1
Orientation
The intuition underlying orientation is simple enough, but the formalism is a bit
heavy, with the main definitions expressed as equivalence classes of an equivalence
relation. We assume prior familiarity with the main facts about determinants of
matrices.
163
164
CHAPTER 12. DEGREE THEORY
No doubt most readers are well aware that a linear automorphism (that is, a
linear transformation from a vector space to itself) has a determinant. What we
mean by this is that the determinant of the matrix representing the transformation
does not depend on the choice of coordinate system. Concretely, if L and L′ are the
matrices of the transformation in two coordinate systems, then there is a matrix U
(expressing the change of coordinates) such that L′ = U −1 LU, so that
|L′ | = |U −1 LU| = |U|−1 |L| |U| = |L|.
Let X be an m-dimensional vector space. An ordered basis of X is an ordered
m-tuple (v1 , . . . , vm ) of linearly independent vectors in X. Mostly we will omit
the parentheses, writing v1 , . . . , vm when the interpretation is clear. If v1 , . . . , vm
′
and v1′ , . . . , vm
are ordered bases, we say that they have the same orientation if
′
the determinant |L| of the linear map L taking v1 7→ v1′ , . . . , vm 7→ vm
is positive,
and otherwise they have the opposite orientation. To verify that “has the same
orientation as” is an equivalence relation we observe that it is reflexive because the
determinant of the identity matrix is positive, symmetric because the determinant
of L−1 is 1/|L|, and transitive because the determinant of the composition of two
linear functions is the product of their determinants.
′
The last fact also implies that if v1 , . . . , vm and v1′ , . . . , vm
have the opposite
′′
′′
′
′
orientation, and v1 , . . . , vm and v1 , . . . , vm also have the opposite orientation, then
′′
v1 , . . . , vm and v1′′ , . . . , vm
must have the same orientation, so there are precisely
two equivalence classes. An orientation for X is one of these equivalence classes.
An oriented vector space is a vector space for which one of the two orientations
has been specified. An ordered basis of an oriented vector space is said to be
positively oriented (negatively oriented) if it is (not) an element of the specified
orientation.
Since the determinant is continuous, each orientation is an open subset of the
set of ordered bases of X. The two orientations are disjoint, and their union is the
entire set of ordered bases, so each path component of the space of ordered bases
is contained in one of the two orientations. If the space of ordered bases had more
than two path components, it would be possible to develop an invariant that was
more sophisticated than orientation. But this is not the case.
Proposition 12.1.1. Each orientation of X is path connected.
Proof. Fix a “standard” basis e1 , . . . , em and some ordered basis v1 , . . . , vm . We
will show that there is a path in the space of ordered bases from v1 , . . . , vm to
either e1 , e1 , . . . , em or −e1 , e1 , . . . , em . Thus the space of ordered bases has at
most two path components, and since each orientation is a nonempty union of path
components, each must be a path component.
If i 6= j, then for any t ∈ R the determinant of the linear transformation
taking v1 , . . . , vm to v1 , . . . , vi + tvj , . . . , vm is one, so varying t gives a continuous
path in the space of ordered bases. Combining
P such paths, we can find a path
from v1 , . . . , vm to w1 , . . . , wm where wi = j bij ej with bij 6= 0 for all i and j.
Beginning at w1 , . . . , wm , such paths can be combined to eliminate all off diagonal
coefficients, arriving at an ordered basis of the form c1 e1 , . . . , cm em . From here we
12.1. ORIENTATION
165
can continuously rescale the coefficients, arriving at an ordered basis d1 e1 , . . . , dm em
with di = ±1 for all i.
For any ordered basis v1 , . . . , vm and any i = 1, . . . , m − 1 there is a path
θ 7→ (v1 , . . . , vi−1 , cos θvi + sin θvi+1 , cos θvi+1 − sin θvi , vi+1 , . . . , vm )
from [0, π] to the space of ordered bases. Evidently such paths can be combined to
construct a path from d1 e1 , . . . , dm em to ±e1 , e2 , . . . , em .
This result has a second interpretation. The general linear group of X is
the group GL(X) of all nonsingular linear transformations L : X → X, with
composition as the group operation. The identity component of GL(X) is the
subgroup GL+ (X) of linear transformations with positive determinant. If we fix a
particular basis e1 , . . . , em there is a bijection L ↔ (Le1 , . . . , Lem ) between GL(X)
and the set of ordered bases of X, which gives the following version of the last
result.
Corollary 12.1.2. GL+ (X) is path connected.
We wish to extend the notion of orientation to ∂-manifolds. Let M ⊂ Rk be an
m-dimensional smooth ∂-manifold. Roughly, an orientation of M is a “continuous”
assignment of orientations to the tangent spaces at the various points of M. One
way to do this is to require that if ϕ : U → M is a smooth parameterization, where
U is a connected open subset of X, and (v1 , . . . , vm ) is an ordered basis of X, then
the bases (Dϕ(x)v1 , . . . , Dϕ(x)vm ) are all either positively oriented or negatively
oriented. The method we adopt is a bit more concrete, and its explanation is a bit
long winded, but the tools we obtain will be useful later.
A path in M is a continuous function γ : [a, b] → M, where a < b. Fix such a γ.
A vector field along γ is a continuous function from [a, b] to Rk that maps each t
to an element of Tγ(t) M. A moving frame along γ is an m-tuple v = (v1 , . . . , vm )
of vector fields along γ such that for each t, v(t) = (v1 (t), . . . , vm (t)) is a basis of
Tγ(t) M. More generally, for h = 0, . . . , m a moving h-frame along γ is an h-tuple
v = (v1 , . . . , vh ) of vector fields along γ such that for each t, v1 (t), . . . , vh (t) are
linearly independent.
We need to know that moving frames exist in a variety of circumstances.
Proposition 12.1.3. For any h = 0, . . . , m−1, any moving h-frame v = (v1 , . . . , vh )
along γ, and any vh+1 ∈ Tγ(a) M such that v1 (a), . . . , vh (a), vh+1 are linearly independent, there is a vector field vh+1 along γ such that vh+1 (a) = vh+1 and
(v1 , . . . , vh , vh+1 ) is a moving (h + 1)-frame for γ.
There are two parts to the argument, the first of which is concrete and geometric.
Lemma 12.1.4. If η : [a, b] → Rm is a path in Rm , then for any h = 0, . . . , m −
1, any moving h-frame u = (u1 , . . . , uh ) along η, and any uh+1 ∈ Rm such that
u1 (a), . . . , uh (a), uh+1 are linearly independent, there is a vector field uh+1 along η
such that uh+1 (a) = uh+1 and (u1 , . . . , uh , uh+1 ) is a moving (h + 1)-frame for η.
166
CHAPTER 12. DEGREE THEORY
Proof. If v1 , . . . , vh , w ∈ Rm and v1 , . . . , vh are linearly independent, let
X
π(v1 , . . . , vh , w) = w −
βi vi
i
be the projection of w onto the orthogonal complement
P of the span of v1 , . . . , vh .
The numbers βi are the solution of the linear system h i βi vi , vj i = hw, vj i, so the
continuity of matrix inversion implies that π is a continuous function.
First suppose that uh+1 is a unit vector that is orthogonal to u1 (a), . . . , uh (a).
Let s be the least upper bound of the set of s in [a, b] such that there is a continuous
uh+1 : [a, s] → Rm with uh+1 (t) orthogonal to u1 (t), . . . , uh (t) and kuh+1 (t)k = 1
for all t. The set of pairs (s, uh+1 (s)) for such functions has a point of the form
(s, vh+1 ) in its closure. The continuity of the inner product implies that vh+1 is a
unit vector that is orthogonal to u1 (s), . . . , uh (s). The continuity π implies that
there is an ε > 0 and a neighborhood U of vh+1 , which we may insist is convex,
such that π(u1 (t), . . . , uh (t), u) 6= 0 for all t ∈ [s − ε, s + ε] ∩ [a, b] and all u ∈ U. We
can choose s ∈ [s − ε, s) and a function uh+1 : [a, s] → Rm satisfying the conditions
above with uh+1 (s) ∈ U. We extend uh+1 to all of [a, min{s + ε, b}] by setting
uh+1 (t) = ũh+1 (t)/kũh+1 (t)k where
(
t−s
s−t
uh+1 (s) + s−s
vt+1 ), s ≤ t ≤ s,
π(u1 (t), . . . , uh (t), s−s
ũh+1 (t) =
π(u1 (t), . . . , uh (t), vt+1 ),
s ≤ t.
Then uh+1 contradicts the definition of s if s < b, and for s = b it provides a
satisfactory function.
P
′
′
To prove the general case we write uh+1 =
i αi ui (a) + βuh+1 where uh+1
′
is a unit vector that is orthogonal to u1 (a), . . . , uh (a). If uh+1 is the function
constructed
in the last paragraph with u′h+1 in place of uh+1, then we can let uh+1 =
P
′
i αi ui + βuh+1 .
The general result is obtained by applying this in the context of finite collection
of parameterizations that cover γ.
Proof of Proposition 12.1.3. There are a = t0 < t1 < · · · < tJ−1 < tJ = b such
that for each j = 1, . . . , J, the image of γ|[tj−1 ,tj ] is contained in the image of a
smooth parameterization. We may assume that J = 1 because the general case
can obviously be obtained from J applications of this special case. Thus there is
a smooth parameterization ϕ : U → M whose image contains the image of γ. Let
ψ = ϕ−1 , let η := ψ ◦ γ, let uh+1 = Dψ(γ(a))vh+1 (a), and define a moving h-frame
u along η by setting
u1 (t) := Dψ(γ(t))v1 (t), . . . , uh (t) := Dψ(γ(t))vh (t).
The last result gives a uh+1 : [a, b] → Rm such that uh+1 (a) = uh+1 and (u1 , . . . , uh , uh+1 )
is a moving (h + 1)-frame along η. We define vh+1 to [a, b] by setting
vh+1 (t) = Dϕ(η(t))uh+1(t).
167
12.1. ORIENTATION
Corollary 12.1.5. For any basis v1 , . . . , vm of Tγ(a) M there is a moving frame v
′
along γ such that vh (a) = vh for all h. If the ordered basis v1′ , . . . , vm
of Tγ(b) M has
the same orientation as v1 (b), . . . , vm (b), then there is a moving frame v′ along γ
such that vh′ (a) = vh and vh′ (b) = vh′ for all h.
Proof. The first assertion is obtained by applying the Proposition m times. To
prove the second we regard GL(Rm ) as a group of matrices and let ρ : [a, b] →
+
m
GL
with ρ(a) the identity matrix and ρ(b) the matrix such that
P (R ) be a path
′
′
j ρij (b)vj (b) = vi for all i, as per Corollary 12.1.2. Define v by setting
vi′ (t) =
X
ρij (t)vj (t).
j
Given a moving frame v and an orientation of Tγ(a) M, there is an induced
orientation of Tγ(b) M defined by requiring that v(b) is positively oriented if and
only if v(a) is positively oriented. The last result implies that it is always possible
to induce an orientation in this way, because a moving frame always exists, and the
next result asserts that the induced orientation does not depend on the choice of
moving frame, so there is a well defined orientation of Tγ(b) M that is induced by γ
and an orientation of Tγ(a) M.
Lemma 12.1.6. If v and ṽ are two moving frames along a path γ : [a, b] → M,
then v(a) and ṽ(a) have the same orientation if and only if v(b) and ṽ(b) have the
same orientation.
P
Proof. For a ≤ t ≤ b let A(t) = (aij (t)) be the matrix such that ṽi (t) = m
j=1 aij (t)vj (t).
Then A is continuous, and the determinant is continuous, so t 7→ |A(t)| is a continuous function that never vanishes, and consequently |A(a)| > 0 if and only if
|A(b)| > 0.
If γ(b) = γ(a) and a given orientation of Tγ(a) M = Tγ(b) M differs from the
one induced by the given orientation and γ, then we say that γ is an orientation
reversing loop. Suppose that M has no orientation reversing loops. For any
choice of a “base point” p0 in each path component of M and any specification
of an orientation of each Tp0 M, there is an induced orientation of Tp M for each
p ∈ M defined by requiring that whenever γ : [a, b] → M is a continuous path with
γ(a) = p0 , the orientation of Tγ(b) M is the one induced by γ and the given orientation
of Tp0 M. If γ ′ : [a′ , b′ ] → M is a second path with γ ′ (a′ ) = γ(a) and γ ′ (b′ ) = γ(b),
then for any given orientation of Tγ(a) the orientations of Tγ(b) induced by γ and γ ′
must be the same because otherwise following γ, then backtracking along γ ′ would
be an orientation reversing loop. Thus, in the absense of orientation reversing loops,
an orientation of Tp0 M induces an orientation at every p in the path component of
p0 .
We have arrived at the following collection of concepts.
Definition 12.1.7. An orientation for M is a assignment of an orientation to
each tangent space Tp M such that for every moving frame v along a path γ : [a, b] →
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CHAPTER 12. DEGREE THEORY
M, v(a) is a positively oriented basis of Tγ(a) M if and only if v(b) is a positively
oriented basis of Tγ(b) M. We say that M is orientable if it has an orientation. An
oriented ∂-manifold is a ∂-manifold with a specified orientation. If p is a point
in an oriented ∂-manifold M, we say that an ordered basis (x1 , . . . , xm ) of Tp M is
positively oriented if it is a member of the orientation of Tp M specified by the
orientation of M; otherwise it is negatively oriented. For any orientation of M
there is an opposite orientation obtained by reversing the orientation to each
Tp M.
Our discussion above has the following summary:
Proposition 12.1.8. Exactly one of the two situations occurs:
(a) M has an orientation reversing loop.
(b) Each path component of M has two orientations, and any specification of an
orientation for each path component of M determines an orientation of M.
Probably you already know that the Moëbius strip is the best known example of
a ∂-manifold that is not orientable, while the Klein bottle is the best known example
of a manifold that is not orientable. From several points of view two dimensional
projective space is a more fundamental example of a manifold that is not orientable,
but it is more difficult to visualize. (If you are unfamiliar with any of these spaces
you should do a quick web search.)
12.2
Induced Orientation
An orientation on a manifold induces an orientation on an open subset, obviously. More interesting is the orientation induced on ∂M by an orientation on the
a ∂-manifold M. We are also interested in how an orientation on a point in the
domain of a smooth map between manifolds of equal dimension induces an orientation on the tangent space of the image point. As we will see, this generalizes to the
image point being in an oriented submanifold whose codimension is the dimension
of the domain.
As before we work with an m-dimensional smooth ∂-manifold M with a given orientation. Consider a point p ∈ ∂M and a basis v1 , . . . , vm of Tp M with v2 , . . . , vm ∈
Tp ∂M. Of course v2 , . . . , vm is a basis of Tp ∂M. There is a visually obvious sense
in which v1 is either “inward pointing” or “outward pointing” that is made precise
by using a parameterization ϕ : U → M (where U ⊂ H is open) to determine
whether the first component of Dϕ−1 (p)v1 is positive or negative. Note that the
sets of inward and outward point vectors are both convex. Our convention will be
that an orientation of Tp M induces an orientation of Tp ∂M according to the rule
that if v1 , . . . , vm is positively oriented and v1 is outward pointing, then v1 , . . . , vm−1
is positively oriented.
Does our definition of the induced orientation make sense? There are two issues
to address.
′
First, we need to show that if v1 , . . . , vm and v1′ , . . . , vm
are two bases of Tp M
with the properties described in the definition above, so that either could be used to
12.2. INDUCED ORIENTATION
169
define the induced orientation of Tp ∂M, then they give the same induced orientation.
Suppose that v1 and v1′ are both outward pointing. Since the set of outward pointing
vectors is convex,
′
t 7→ (1 − t)v1′ + tv1 , v2′ , . . . , vm
′
′
is a path in the space of bases of Tp M, so v1′ , . . . , vm
and v1 , v2′ , . . . , vm
have the
same orientation. The first row of the matrix A of the
linear
transformation
P
′
taking v1 , v2 , . . . , vm to v1 , v2′ . . . , vm
(concretely, vi′ =
j aij vj ) is (1, 0, . . . , 0),
so the determinant of A is the same as the determinant of its lower right hand
(m − 1) × (m − 1) submatrix, which is the matrix of the linear transformation
′
taking v2 , . . . , vm to v2′ , . . . , vm
. Therefore v1 , . . . , vm has the same orientation as
′
′
′
′
v1 , v2 . . . , vm and v1 , . . . , vm if and only if v2 , . . . , vm has the same orientation as
′
v2′ , . . . , vm
.
We also need to check that what we have defined as the induced orientation of
∂M is, in fact, an orientation. Consider a path γ : [a, b] → ∂M. Corollary 12.1.5
gives a moving frame (v2 , . . . , vm ) for ∂M along γ, and Proposition 12.1.3 implies
that it extends to a moving frame (v1 , . . . , vm ) for M along γ. Suppose that v1 (a) is
outward pointing. By continuity, it must be the case that for all t, v1 (t) is outward
pointing. If we assume that v1 (a), . . . , vm (a) is positively oriented, for the given
orientation, then v2 (a), . . . , vm (a) is positively oriented, for the induced orientation.
In addition, v1 (b), . . . , vm (b) is positively oriented, for the given orientation, so, as
desired, v2 (b), . . . , vm (b) is positively oriented, both with respect to the induced
orientation and with respect to the orientation induced by γ and v2 (a), . . . , vm (a).
Now suppose that M and N are two m-dimensional oriented smooth manifolds,
now without boundary, and that f : M → N is a smooth function. If p is a
regular point of f , we say that f is orientation preserving at p if Df (p) maps
positively oriented bases of Tp M to positively oriented bases of Tf (p) N; otherwise f is
′
orientation reversing at p. This makes sense because if v1 , . . . , vm and v1′ , . . . , vm
are two bases of Tp M, then the matrix of the linear transformation taking each vi
to vi′ is the same as the matrix of the linear transformation taking each Df (p)vi to
Df (p)vi′ .
We can generalize this in a way that does not play a very large role in later
developments, but does provide some additional illumination at little cost. Suppose that M is an oriented m-dimensional smooth ∂-manifold, N is an oriented
n-dimensional boundaryless manifold, P is an oriented (n − m)-dimensional submanifold of N, and f : M → N is a smooth map that is transversal to P . We say
that f is positively oriented relative to P at a point p ∈ f −1 (P ) if
Df (p)v1 , . . . , Df (p)vm, w1 , . . . , wn−m
is a positively oriented ordered basis of Tf (p) N whenever v1 , . . . , vm is a positively
oriented ordered basis of Tp M and w1 , . . . , wn−m is a positively oriented ordered
basis of Tf (p) P . It is easily checked that whether or not this is the case does not
depend on the choice of positively oriented ordered bases v1 , . . . , vm and Tp M and
w1 , . . . , wn−m . When this is not the case we say that f is negatively oriented
relative to P at p.
Now, in addition, suppose that f −1 (P ) is finite. The oriented intersection
number I(f, P ) is the number of points in f −1 (P ) at which f is positively oriented
170
CHAPTER 12. DEGREE THEORY
relative to P minus the number of points at which f is negatively oriented relative
to P . An idea of critical importance for the entire project is that under natural
and relevant conditions this number is a homotopy invariant. This corresponds to
the special case of the following result in which M is the cartesian product of a
boundaryless manifold and [0, 1].
Theorem 12.2.1. Suppose that M is an (m+1)-dimensional oriented smooth manifold, N is an n-dimensional smooth manifold, P is a compact (n − m)-dimensional
smooth submanifold of N and f : M → N is a smooth function that is transverse
to P with f −1 (P ) compact. Then
I(f |∂M , P ) = 0.
Proof. Proposition 10.8.5 implies that f −1 (P ) is a neat smooth ∂-submanifold of
M. Since f −1 (P ) is compact, it has finitely many connected components, and
Proposition 10.9.1 implies that each of these is either a loop or a line segment.
Recalling the definition of neatness, we see that the elements of f −1 (P ) ∩ ∂M are
the endpoints of the line segments. Fix one of the line segments. It suffices to
show that f |∂M is positively oriented relative to P at one endpoint and negatively
oriented relative to P at the other.
The line segment is a smooth ∂-manifold, and by gluing together smooth parameterizations of open subsets, using a partition of unity, we can construct a smooth
path γ : [a, b] → M that traverses it, with nonzero derivative everywhere. Let
v1 (t) = Dγ(t)1 for all t. (Here 1 is thought of as an element of Tt [a, b] under the
identification of this space with R.) Neatness implies that v1 (a) is inward pointing
and v1 (b) is outward pointing.
Let v2 , . . . , vm+1 be a basis of Tγ(a) ∂M. Proposition 12.1.3 implies that v1 extends to a moving frame v1 , . . . , vm+1 along γ with v2 (a) = v2 , . . . , vm+1 (a) = vm+1 .
We have
vj (b) = vj′ + αj v1 (b)
(j = 2, . . . , m + 1)
′
of Tγ(b) ∂M and scalars α2 , . . . , αm+1 . We can replace
for some basis v2′ , . . . , vm+1
v with the moving frame given by Corollary 12.1.5 applied to the ordered basis
′
v1 (b), v2′ , . . . , vm+1
of Tγ(b) M, so we may assume that v2 (b), . . . , vm+1 (b) ∈ Tγ(b) ∂M.
Then v1 (a), . . . , vm+1 (a) is a positively oriented basis of Tγ(a) M if and only if
v1 (b), . . . , vm+1 (b) is a positively oriented basis of Tγ(b) M. Since v1 (a) is inward
pointing and v1 (b) is outward pointing, v2 (a), . . . , vm+1 (a) is a positively oriented
basis of Tγ(a) ∂M if and only if v2 (b), . . . , vm+1 (b) is a negatively oriented basis of
Tγ(b) M.
Proposition 12.1.3 implies that there is a moving frame w1 , . . . , wn−m along
f ◦γ : [a, b] → P . As we have defined orientation, w1 (a), . . . , wn−m (a) is a positively
oriented basis of Tf (γ(a)) P if and only if w1 (b), . . . , wn−m (b) is a positively oriented
basis of Tf (γ(b)) P , and
Df (p)v2 (a), . . . , Df (p)vm+1 (a), w1 (a), . . . , wn−m (a)
is a positively oriented basis of Tf (γ(a)) N if and only if
Df (p)v2 (b), . . . , Df (p)vm+1 (b), w1 (b), . . . , wn−m (b)
171
12.3. THE DEGREE
is a positively oriented basis of Tf (γ(b)) N. Combining all this, we conclude that f |∂M
is positively oriented relative to P at γ(a) if and only if it is and negatively oriented
relative to P at γ(b), which is the desired result.
12.3
The Degree
Let M and N be m-dimensional smooth manifolds. We can restrict a smooth f :
M → N to a subset of the domain and consider the degree of the restricted function
over some point in the range. The axioms characterizing the degree express relations
between the degrees of the various restrictions. In order to get a “clean” theory
we need to consider subdomains that are compact, and which have no preimages in
their topological boundaries. (Intuitively, a preimage that is in the boundary of a
compact subset is neither clearly inside the domain nor unambiguously outside it.)
For a compact C ⊂ M let ∂C = C \ (M \ C) be the topological boundary of C.
Definition 12.3.1. A continuous function f : C → N with compact domain C ⊂
M is degree admissible over q ∈ N if
f −1 (q) ∩ ∂C = ∅.
If, in addition, f is smooth and q is a regular value of f , then f is smoothly degree
admissible over q. Let D(M, N) be the set of pairs (f, q) in which f : C → N is
a continuous function with compact domain C ⊂ M that is degree admissible over
q ∈ N. Let D ∞ (M, N) be the set of (f, q) ∈ D(M, N) such that f is smoothly degree
admissible over q.
Definition 12.3.2. If C ⊂ M is compact, a homotopy h : C × [0, 1] → N is degree
admissible over q if , for each t, ht is degree admissible over q. We say that h is
smoothly degree admissible over q if, in addition, h is smooth and h0 and h1
are smoothly degree admissible over q.
Proposition 12.3.3. There is a unique function deg∞ : D ∞ (M, N) → Z, taking
(f, q) to deg∞
q (f ), such that:
∞
−1
(∆1) deg∞
(q) is a singleton {p}
q (f ) = 1 for all (f, q) ∈ D (M, N) such that f
and f is orientation preserving at p.
Pr
∞
∞
(∆2) deg∞
q (f ) =
i=1 degq (f |Ci ) whenever (f, q) ∈ D (M, N), the domain of f
is C, and C1 , . . . , Cr are pairwise disjoint compact subsets of C such that
f −1 (q) ⊂ C1 ∪ . . . ∪ Cr \ (∂C1 ∪ . . . ∪ ∂Cr ).
∞
(∆3) deg∞
q (h0 ) = deg q (h1 ) whenever C ⊂ M is compact and the homotopy h :
C × [0, 1] → N is smoothly degree admissible over q.
−1
Concretely, deg∞
(q) at which f is orientation preservq (f ) is the number of p ∈ f
−1
ing minus the number of p ∈ f (q) at which f is orientation reversing.
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CHAPTER 12. DEGREE THEORY
Proof. For (f, q) ∈ D(M, N) the inverse function theorem implies that each p ∈
f −1 (q) has a neighborhood that contains no other element of f −1 (q), and since U is
−1
compact it follows that f −1 (q) is finite. Let deg∞
(q)
q (f ) be the number of p ∈ f
−1
at which f is orientation preserving minus the number of p ∈ f (q) at which f is
orientation reversing.
Clearly deg∞ satisfies (∆1) and (∆2). Suppose that h : C × [0, 1] → N is
smoothly degree admissible over q. Let V be a neighborhood of q such that for all
q′ ∈ V :
(a) h−1 (q ′ ) ⊂ U × [0, 1];
(b) q ′ is a regular value of h0 and h1 ;
∞
∞
∞
(c) deg∞
q ′ (h0 ) = deg q (h0 ) and deg q ′ (h1 ) = deg q (h1 ).
Sard’s theorem implies that some q ′ ∈ V is a regular value of h. In view of (a) we
can apply Theorem 12.2.1, concluding that the degree of h|∂(U ×[0,1]) = h|U ×{0,1} over
q ′ is zero. Since the orientation of M × {0} induced by M × [0, 1] is the opposite
∞
of the induced orientation of M × {1}, this implies that deg∞
q ′ (h0 ) − degq ′ (h1 ) = 0,
∞
from which it follows that deg∞
q (h0 ) = degq (h1 ). We have verified (∆3).
It remains to demonstrate uniqueness. In view of (∆2), this reduces to showing
uniqueness for (f, q) ∈ D ∞ (M, N) such that f −1 (q) = {p} is a singleton. If f is
orientation preserving at p, this is a consequence of (∆1), so we assume that f is
orientation reversing at p.
The constructions in the remainder of the proof are easy to understand, but
tedious to elaborate in detail, so we only explain the main ideas. Using the path
connectedness of each orientation (Proposition 12.1.1) and an obvious homotopy
between an f that has p as a regular point and its linear approximation, with
respect to some coordinate systems for the domain and range, one can show that
(∆3) implies that deg∞
q (f ) does not depend on the particular orientation reversing
f . Using one of the bump functions constructed after Lemma 10.2.2, one can easily
construct a smooth homotopy j : M × [0, 1] → M such that j0 = IdM , each jt is a
smooth diffeomorphism, and j1 (p) is any point in some neighborhood of p. Applying
(∆3) to h = f ◦ j, we find that deg∞
q (f ) does not depend on which point (within
some neighborhood of p) is mapped to q. The final construction is a homotopy
between the given f and a function f ′ that has three preimages of q near p, with f ′
being orientation reversing at two of them and orientation preserving at the third.
In view of the other conclusions we have reached, (∆3) implies that
∞
deg∞
q (f ) = 2 degq (f ) + 1.
In preparation for the next result we show that deg∞ is continuous in a rather
strong sense.
Proposition 12.3.4. If C ⊂ M is compact, f : C → N is continuous, and q ∈
N \ f (∂C), then are neighborhoods Z ⊂ C(C, N) of f and V ⊂ N \ f (∂C) of q such
that
′
′′
deg∞
q ′ (f ) = deg q ′′ (f )
173
12.3. THE DEGREE
whenever f ′ , f ′′ ∈ Z ∩ C ∞ (C, N), q ′ , q ′′ ∈ V , q ′ is a regular value of f ′ , and q ′′ is a
regular value of f ′′ .
Proof. Let V be an open disk in N that contains q with V ⊂ N \ f (∂C). Then
Z ′ = { f ′ ∈ C(C, N) : f (∂C) ⊂ N \ V }
is an open subset of C(C, N), and Theorem 10.7.7 gives an open Z ⊂ Z ′ containing f
such that for any f ′ , f ′′ ∈ Z ∩C ∞ (C, N) there is a smooth homotopy h : C ×[0, 1] →
N with h0 = f ′ , h1 = f ′′ , and ht ∈ Z ′ for all t, which implies that h is a degree
∞
′′
′′′
′
admissible homotopy, so (∆3) implies that deg∞
q ′′′ (f ) = deg q ′′′ (f ) whenever q ∈ V
is a regular point of both f ′ and f ′′ .
Since Sard’s theorem implies that such a q ′′′ exists, it now suffices to show that
∞
′
′
∞
′ ′′
′
deg∞
q ′ (f ) = degq ′′ (f ) whenever f ∈ Z ∩C (C, N) and q , q ∈ V are regular values
of f ′ . Let j : N × [0, 1] → N be a smooth function with the following properties:
(a) j0 = IdN ;
(b) each jt is a smooth diffeomorphism;
(c) j(y, t) = y for all y ∈ N \ V and all t;
(d) j1 (q ′ ) = q ′′ .
(Construction of such a j, using the techniques of Section 10.2, is left as an exercise.)
Clearly jt (q ′ ) is a regular value of jt ◦ f for all t, so the concrete characterization of
′
deg∞ implies that deg∞
jt (q ′ ) (jt ◦ f ) is locally constant as a function of t. Since the
∞
′
′
unit interval is connected, it follows that deg∞
q ′ (f ) = deg q ′′ (j1 ◦ f ). On the other
hand jt ◦ f ′ ∈ Z ′ for all t, so the homotopy (y, t) 7→ j(f ′(y), t) is smoothly degree
∞
′
′
admissible over q ′′ , and (∆3) implies that deg∞
q ′′ (j1 ◦ f ) = deg q ′′ (f ).
The theory of the degree is completed by extending the degree to continuous
functions, dropping the regularity condition.
Theorem 12.3.5. There is a unique function deg : D(M, N) → Z, taking (f, q) to
degq (f ), such that:
(D1) degq (f ) = 1 for all (f, q) ∈ D(M, N) such that f is smooth, f −1 (q) is a
singleton {p}, and f is orientation preserving at p.
P
(D2) degq (f ) = ri=1 degq (f |Ci ) whenever (f, q) ∈ D(M, N), the domain of f is C,
and C1 , . . . , Cr are pairwise disjoint compact subsets of U such that
f −1 (q) ⊂ C1 ∪ . . . ∪ Cr \ (∂C1 ∪ . . . ∪ ∂Cr ).
(D3) If (f, q) ∈ D(M, N) and C is the domain of f , there is a neighborhood A ⊂
C(C, N) × N of (f, q) such that degq′ (f ′ ) = degq (f ) for all (f ′ , q ′ ) ∈ A.
174
CHAPTER 12. DEGREE THEORY
Proof. We claim that if deg : D(M, N) → Z satisfies (D1)-(D3), then its restriction
to D ∞ (M, N) satisfies (∆1)-(∆3). For (∆1) and (∆2) this is automatic. Suppose
that C ⊂ M is compact and h : U × [0, 1] → N is a smoothly degree admissible
homotopy over q. Such a homotopy may be regarded as a continuous function from
[0, 1] to C(U , N). Therefore (D3) implies that degq (ht ) is a locally constant function
of t, and since [0, 1] is connected, it must be constant. Thus (∆3) holds.
Proposition 11.5.1 implies that for any (f, q) ∈ D(M, N) the set of smooth
f ′ : M → N that have q as a regular value is dense at f . In conjunction with
Proposition 12.3.4, this implies that the only possibility consistent with (D3) is to
′
′ ′
∞
′
′
set degq (f ) = deg∞
q ′ (f ) for (f , q ) ∈ D (M, N) with f and q close to f and q.
This establishes uniqueness, and Proposition 12.3.4 also implies that the definition
is unambiguous. It is easy to see that (D1) and (D2) follow from (∆1) and (∆2),
and (D3) is automatic.
Since (D2) implies that the degree of f over q is the sum of the degrees of the
restrictions of f to the various connected components of the domain of f , it makes
sense to study the degree of the restriction of f to a single component. For this
reason, when studying the degree one almost always assumes that M is connected.
(In applications of the degree this may fail to be the case, of course.) The image
of a connected set under a continuous mapping is connected, so if M is connected
and f : M → N is continuous, its image is contained in one of the connected
components of N. Therefore it also makes sense to assume that N is connected.
So, assume that N is connected, and that f : M → N is continuous. We have
(M, f, q) ∈ D(M, N) for all q ∈ N, and (D3) asserts that degq (f ) is continuous as
a function of q. Since Z has the discrete topology, this means that it is a locally
constant function, and since N is connected, it is in fact constant. That is, degq (f )
does not depend on q; when this is the case we will simply write deg(f ), and we
speak of the degree of f without any mention of a point in N.
12.4
Composition and Cartesian Product
In Chapter 5 we emphasized restriction to a subdomain, composition, and cartesian products, as the basic set theoretic methods for constructing new functions from
ones that are given. The bahevior of the degree under restriction to a subdomain is
already expressed by (D3), and in this section we study the behavior of the degree
under composition and products. In both cases the result is given by multiplication,
reflecting basic properties of the determinant.
Proposition 12.4.1. If M, N, and P are oriented m-dimensional smooth manifolds, C ⊂ M and D ⊂ N are compact, f : C → N and g : D → P are continuous,
g is degree admissible over r ∈ P , and g −1 (r) is contained in one of the connected
components of N \ f (∂C), then for any q ∈ g −1(r) we have
degr (g ◦ f ) = degq (f ) · degr (g).
Proof. Since C ∞ (C, N) and C ∞ (D, P ) are dense in C(C, N) and C(D, P ) (Theorem
10.7.6) and composition is a continuous operation (Proposition 5.3.6) the continuity
175
12.4. COMPOSITION AND CARTESIAN PRODUCT
property (D3) of the degree implies that is suffices to prove the claim when f and
g are smooth. Sard’s theorem implies that there are points r arbitrarily near r
that are regular values of both g and g ◦ f , and Proposition 12.3.4 implies that the
relevant degrees are unaffected if r is replaced by such a point, so we may assume
that r has these regularity properties.
For q ∈ g −1 (r) let sg (q) be 1 or −1 according to whether g is orientation preserving or orientation reversing at q. For p ∈ (g ◦ f )−1 (q) define sf (p) and sg◦f (p)
similarly. In view of the chain rule and the definition of orientation preservation
and reversal, sg◦f (p) = sg (f (p))sf (p). Therefore
X
X
X
deg(g ◦ f ) =
sg (f (p))sf (p) =
sg (q)
sf (p)
p∈(g◦f )−1 (r)
=
q∈g −1 (r)
X
p∈g −1 (q)
sg (q) degq (f ).
q∈g −1 (r)
Since g −1 (r) is contained in a single connected component of N \f (∂C),
P Proposition
−1
12.3.4 implies that degq (f ) is the same for all q ∈ g (r), and q∈g−1 (r) sg (q) =
degr (g).
The hypotheses of the last result are rather stringent, which makes it rather
artificial. For topologists the following special case is the main point of interest.
Corollary 12.4.2. If M, N, and P are compact oriented m-dimensional smooth
manifolds, N is connected, and f : M → N and g : N → P are continuous, then
deg(g ◦ f ) = deg(f ) · deg(g).
For cartesian products the situation is much simpler.
Proposition 12.4.3. Suppose that M and N are oriented m-dimensional smooth
manifolds, M ′ and N ′ are oriented m′ -dimensional smooth manifolds, C ⊂ M and
C ′ ⊂ M are compact, and f : C → N and f ′ : C ′ → N ′ are index admissible over
q and q ′ respectively. Then
deg(q,q′ ) (f × f ′ ) = degq (f ) · degq′ (f ′ ).
Proof. For reasons explained in other proofs above, we may assume that f and f ′
are smooth and that q and q ′ are regular values of f and f ′ . For p ∈ f −1 (r) let
sf (p) be 1 or −1 according to whether f is orientation preserving or orientation
reversing at p, and define sf ′ (p′ ) for p′ ∈ f ′ −1 (q ′ ) similarly. Since the determinant
of a block diagonal matrix is the product of the determinants of the blocks, f × f ′
is orientation preserving or orientation reversing at (p, p′ ) according to whether
sp (f )sp′ (f ′ ) is positive or negative, so
X
sp (f )sp′ (f ′ )
deg(q,q′ ) (f × f ′ ) =
(p,p′ )∈(f ×f ′ )−1 (q,q ′ )
=
X
p∈f −1 (q)
sp (f ) ·
X
p′ ∈f ′ −1 (q ′ )
sp′ (f ′ ) = degq (f ) · degq′ (f ′ ).
Chapter 13
The Fixed Point Index
We now take up the theory of the fixed point index. For continuous functions defined on compact subsets of Euclidean spaces it is no more than a different rendering
of the theory of the degree; this perspective is developed in Section 13.1.
But we will see that it extends to a much higher level of generality, because the
domain and the range of the function or correspondence have the same topology.
Concretely, there is a property called Commutativity that relates the indices of the
two compositions ĝ ◦ g and g ◦ ĝ where g : C → X̂ and ĝ : Ĉ → X are continuous,
and other natural restrictions on this data (that will give rise to a quite cumbersome
definition) are satisfied. This property requires that we extend our framework to
allow comparison across spaces. Section 13.2 introduces the necessary abstractions
and verifies that Commutativity is indeed satisfied in the smooth case. It turns out
that this boils down to a fact of linear algebra that came as a surprise when this
theory was developed.
When we extended the degree from smooth to continuous functions, we showed
that continuous functions could be approximated by smooth ones, and that this gave
a definition of the degree for continuous functions that made sense and was uniquely
characterized by certain axioms. In somewhat the same way Commutativity will
be used, in Section 13.4, to extend from Euclidean spaces to separable ANR’s, as
per the ideas developed in Section 7.6. The argument is lengthy, technically dense,
and in several ways the culmination of our work to this point.
The Continuity axiom is then used in Section 13.5 to extend the index to contractible valued correspondences. The underlying idea is the one used to extend
from smooth to continuous functions: approximate and show that the resulting definition is consistent and satisfies all properties. Again, there are many verifications,
and the argument is rather dense.
Multiplication is an additional property of the index that describe its behavior in connection with cartesian products. For continuous functions on subsets of
Euclidean spaces it is a direct consequence of Proposition 12.4.3. At higher levels
of generality it is, in principle, a consequence of the axioms, because those axioms
characterize the index uniquely, but an argument deriving Multiplication from the
other axioms is not known. Therefore we carry Multiplication along as an additional property that is extended from one level of generality to the next along with
everything else.
176
13.1. AXIOMS FOR AN INDEX ON A SINGLE SPACE
13.1
177
Axioms for an Index on a Single Space
The axiom system for the fixed point index is introduced in two stages. This
section presents the first group of axioms, which describe the properties of the
index that concern a single space. Fix a metric space X. For a compact C ⊂ X
let int C = C \ ∂C be the interior of C, and let ∂C = C \ int C be its topological
boundary.
Definition 13.1.1. An index admissible correspondence for X is an upper
semicontinuous correspondence F : C → X, where C ⊂ X is compact, that has no
fixed points in ∂C.
There will be various indices, according to which sorts of correspondences are
considered. The next definition expresses the common properties of their domains.
Definition 13.1.2. An index base for X is a set of index admissible correspondences F : C → X such that:
(a) f ∈ I whenever C ⊂ X is compact and f : C → X is an index admissible
continuous function;
(b) F |D ∈ I whenever F : C → X is an element of I, D ⊂ C is compact, and
F |D is index admissible.
For each integer m ≥ 0 an index base for Rm is given by letting I m be the set of
index admissible continuous functions f : C → Rm .
We can now state the first batch of axioms.
Definition 13.1.3. Let I be an index base for X. An index for I is a function
ΛX : I → Z satisfying:
(I1) (Normalization1) If c : C → X is a constant function whose value is an
element of int C, then
ΛX (c) = 1.
(I2) (Additivity) If F : C → X is an element of I, C1 , . . . , Cr are pairwise disjoint
compact subsets of C, and F P(F ) ⊂ int C1 ∪ . . . ∪ int Cr , then
ΛX (F ) =
X
i
ΛX (F |Ci ).
(I3) (Continuity) For each element F : C → X of I there is a neighborhood
A ⊂ U(C, X) of F such that ΛX (F̂ ) = ΛX (F ) for all F̂ ∈ A ∩ I.
A proper appreciation of Continuity depends on the following immediate consequence of Theorem 5.2.1.
1
In the literature this condition is sometimes described as “Weak Normalization,” in contrast
with a stronger condition defined in terms of homology.
178
CHAPTER 13. THE FIXED POINT INDEX
Proposition 13.1.4. If C ⊂ X is compact, then
{ F ∈ U(C, X) : F is index admissible }
is an open subset of U(C, X).
Proposition 13.1.5. For each m = 1, 2, . . . there is a unique index ΛRm for I m
given by
ΛRm (f ) = deg0 (IdC − f ).
Proof. Observe that if C ⊂ Rm is compact, then f : C → Rm is index admissible if
and only if IdC −f is degree admissible over the origin, Now (I1)-(I3) follow directly
from (D1)-(D3).
To prove uniqueness suppose that Λ̃m is an index for I m . For (g, q) ∈ D(Rm , Rm )
let
dq (g) = Λ̃m (IdC − g − q),
where C is the domain of g. It is straightforward to show that d satisfies (D1)-(D3),
so it must be the degree, and consequently Λ̃m = Λm .
As we explain now, invariance under homotopy is subsumed by Continuity.
However, homotopies will still be important in our work, so we have the following
definition and result.
Definition 13.1.6. For a compact C ⊂ X a homotopy h : C × [0, 1] → X is index
admissible if each ht is index admissible.
Proposition 13.1.7. If I is an index base for X and ΛX is an index for I, then
ΛX (h0 ) = ΛX (h1 ) whenever h : [0, 1] → C(C, X) is an index admissible homotopy.
Proof. Continuity implies that ΛX (ht ) is a locally constant function of t, and [0, 1]
is connected.
We will refer to this result as the homotopy!principle.
13.2
Multiple Spaces
We now introduced two properties of the index that involve comparison across
different spaces. When we define an abstract notion of an index satisfying these
conditions, we need to require that the set of spaces is closed under the operations
that are involved in these conditions, so we require that the sets of spaces and
correspondences are closed under cartesian products.
Definition 13.2.1. An index scope S consists of a class of metric spaces SS and
an index base IS (X) for each X ∈ SS such that
(a) SS contains X × X ′ whenever X, X ′ ∈ SS ;
(b) F × F ′ ∈ IS (X × X ′ ) whenever X, X ′ ∈ SS , F ∈ IS (X), and F ′ ∈ IS (X ′ ).
179
13.2. MULTIPLE SPACES
Our first index scope S 0 has the collection of spaces SS 0 = {R0 , R1 , R2 , . . .} with
IS 0 (Rm ) = I m for each m. Of course (b) is satisfied by identifying Rm × Rn with
Rm+n .
To understand the motivation for the following definition, first suppose that
X, X̂ ∈ SS , and that g : X → X̂ and g : X̂ → X are continuous. In this
circumstance it will be the case that ĝ ◦ g and g ◦ ĝ have the same index. We
would like to develop this idea in greater generality, for functions g : C → X̂ and
ĝ : Ĉ → X, but for our purposes it is too restrictive to require that g(C) ⊂ Ĉ and
ĝ(Ĉ) ⊂ C. In this way we arrive at the following definition.
Definition 13.2.2. A commutativity configuration is a tuple
(X, D, E, g, X̂, D̂, Ê, ĝ)
where X and X̂ are metric spaces and:
(a) E ⊂ D ⊂ X, Ê ⊂ D̂ ⊂ X̂, and D, D̂, E, and Ê are compact;
(b) g ∈ C(D, X̂) and ĝ ∈ C(D̂, X) with g(E) ⊂ int D̂ and ĝ(Ê) ⊂ int D;
(c) ĝ ◦ g|E and g ◦ ĝ|Ê are index admissible;
(d) g(F P(ĝ ◦ g|E )) = F P(g ◦ ĝ|Ê ).
Before going forward, we should think through the details of what (d) means.
If x is a fixed point of ĝ ◦ g|E , then g(x) is certainly a fixed point of ĝ ◦ g, so it is a
fixed point of g ◦ ĝ|Ê if and only if g(x) ∈ Ê. Thus the inclusion
g(F P(ĝ ◦ g|E )) ⊂ F P(g ◦ ĝ|Ê )
holds if and only if
g(F P(ĝ ◦ g|E )) ⊂ Ê.
(∗)
On the other hand, if x̂ is a fixed point of g ◦ ĝ|Ê , then it is in the image of g
and ĝ(x̂) is a fixed point of ĝ ◦ g that is mapped to x̂ by g, so it is contained in
g(F P(ĝ ◦ g|E )) if and only if ĝ(x̂) ∈ E. Therefore the inclusion
F P(g ◦ ĝ|Ê ) ⊂ g(F P(ĝ ◦ g|E ))
holds if and only if
ĝ(F P(g ◦ ĝ|Ê )) ⊂ E.
(∗∗)
Thus (d) holds if and only if both (∗) and (∗∗) hold, and by symmetry this is the
case if and only if ĝ(F P(g ◦ ĝ|Ê )) = F P(ĝ ◦ g|E )).
Definition 13.2.3. An index for an index scope S is a specification of an index
ΛX for each X ∈ SS such that:
(I4) (Commutativity) If (X, D, E, g, X̂, D̂, Ê, ĝ) is a commutativity configuration
with X, X̂ ∈ SS , (E, ĝ ◦ g|E ) ∈ IS (X), and (Ê, g ◦ ĝ|Ê ) ∈ IS (X̂), then
ΛX (ĝ ◦ g|E ) = ΛX̂ (g ◦ ĝ|Ê ).
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CHAPTER 13. THE FIXED POINT INDEX
The index is said to be multiplicative if:
(M) (Multiplication) If X, X ′ ∈ SS , F ∈ IS (X), and F ′ ∈ IS (X ′ ), then
ΛX×X ′ (F × F ′ ) = ΛX (F ) · Λ′X (F ′ ).
We can now state the result that has been the main objective of all our work.
Let SS Ctr be the class of separable absolute neighborhood retracts, and for each
X ∈ SS Ctr let IS Ctr (X) be the union over compact C ⊂ X of the sets of index
admissible upper semicontinuous contractible valued correspondences F : C → X.
Since cartesian products of contractible valued correspondences are contractible
valued, we have defined an index scope S Ctr .
Theorem 13.2.4. There is a unique index ΛCtr for S Ctr , which is multiplicative.
13.3
The Index for Euclidean Spaces
The method of proof of Theorem 13.2.4 is to first establish an index in a quite
restricted setting, then show that it has unique extensions, first to an intermediate
index scope, and then to S Ctr . Our goal in the remainder of this section is to prove:
Theorem 13.3.1. There is a unique index Λ0 for the index scope S 0 given by setting
Λ0Rm = ΛRm for each m, and Λ0 is multiplicative.
Insofar as continuous functions can be approximated by smooth ones with regular fixed points, and we can use Additivity to focus on a single fixed point, the
verification of Commutativity will boil down to the following fact of linear algebra,
which is not at all obvious, and was not known prior to the discovery of its relevance
to the theory of fixed points.
Proposition 13.3.2 (Jacobson (1953) pp. 103–106). Suppose K : V → W and
L : W → V are linear transformations, where V and W are vector spaces of
dimensions m and n respectively over an arbitrary field. Suppose m ≤ n. Then the
characteristic polynomials κKL and κLK of KL and LK are related by the equation
κKL (λ) = λn−m κLK (λ). In particular,
κLK (1) = |IdV − LK| = |IdW − KL| = κKL (1).
Proof. We can decompose V and W as direct sums V = V1 ⊕ V2 ⊕ V3 ⊕ V4 and
W = W1 ⊕ W2 ⊕ W3 ⊕ W4 where
V1 = ker K ∩ im L,
V1 ⊕ V2 = im L,
V1 ⊕ V3 = ker K,
and similarly for W . With suitably chosen bases the matrices of K and L have the
forms




0 L12 0 L14
0 K12 0 K14


 0 K22 0 K24 
 and  0 L22 0 L24 

 0 0 0 0 
 0 0 0 0 
0 0 0 0
0 0 0 0
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13.3. THE INDEX FOR EUCLIDEAN SPACES
Computing the product of these matrices, we find that
λI
−K12 L22
0 −K12 L24
0 λI − K22 L22 0 −K22 L24
κKL (λ) = 0
λI
0
0
0
0
0
λI
Using elementary facts about determinants, this reduces to κKL (λ) = λn−k |λI −
K22 L22 |, where k = dim V2 = dim W2 . In effect this reduces the proof to the special
case V2 = V and W2 = W , i.e. K and L are isomorphisms. But this case follows
from the computation
|λIdV − LK| = |L−1 | · |λIdV − LK| · |L| = |L−1 (λIdV − LK)L| = |λIdW − KL|.
Lemma 13.3.4 states that if we fix the sets X, D, E, X̂, D̂, Ê, then the set of pairs
(g, ĝ) giving a commutativity configuration is open. This is simple and unsurprising,
but without the spadework we did in Chapter 5 the proof would be a tedious slog.
We extract one piece of the argument in order to be able to refer to it later.
Lemma 13.3.3. If X and X̂ are metric spaces, E ⊂ D ⊂ X, and Ê ⊂ D̂ ⊂ X̂,
where D, D̂, E, and Ê are compact, then
(g, ĝ) 7→ ĝ ◦ g|E
is a continuous function from { g ∈ C(D, X̂) : g(E) ⊂ D̂ } × C(D̂, X) to C(E, X).
Proof. Lemma 5.3.1 implies that the function g 7→ g|E is continuous, after which
Proposition 5.3.6 implies that (g, ĝ) 7→ ĝ ◦ g|E is continuous.
Lemma 13.3.4. If X and X̂ are metric spaces, E ⊂ D ⊂ X, Ê ⊂ D̂ ⊂ X̂, and
D, D̂, E, and Ê are open with compact closure, then the set of
(g, ĝ) ∈ C(D, X̂) × C(D̂, X)
such that (X, D, E, g, X̂, D̂, Ê, ĝ) is a commutativity configuration is an open subset
of C(D, X̂) × C(D̂, X).
Proof. Lemma 4.5.10 implies that the set of (g, ĝ) such that g(E) ⊂ int D̂ and
ĝ(Ê) ⊂ int D is an open subset of C(D, X̂) × C(D̂, X). The lemma above implies
that the functions (g, ĝ) 7→ ĝ ◦ g|E and (g, ĝ) 7→ g ◦ ĝ|Ê are continuous, and Proposition 13.1.4 implies that the set of (g, ĝ) satisfying part (c) of Definition 13.2.2 is
open. In view of the discussion in the last section, a pair (g, ĝ) that satisfies (a)-(c)
of Definition 13.2.2 will also satisfy (d) if and only if
g(F P(ĝ ◦ g|E )) ⊂ int Ê
and ĝ(F P(g ◦ ĝ|Ê )) ⊂ int E.
Since (g, ĝ) 7→ ĝ ◦g|E and (g, ĝ) 7→ g ◦ ĝ|Ê are continuous, Theorem 5.2.1 and Lemma
4.5.10 imply that the set of such pairs is open.
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CHAPTER 13. THE FIXED POINT INDEX
Proof of Theorem 13.3.1. Uniqueness and (I1)-(I3) follow from Proposition 13.1.5,
so, we only need to prove that (I4) and (M) are satisfied.
Suppose that (Rm , D, E, g, Rm̂, D̂, Ê, ĝ) is a commutativity configuration. Lemma
13.3.4 states that it remains a commutativity configuration if g and ĝ are replaced
by functions in any sufficiently small neighborhood, and Lemma 13.3.3 implies that
ĝ ◦ g|E and g ◦ ĝ|Ê are continuous functions of (g, ĝ), so, since we already know that
(I3) holds, it suffices to prove the equation of (I4) after such a replacement. Since
the smooth functions are dense in C(D, Rm̂ ) and C(D̂, Rm ) (Proposition 10.2.7) we
may assume that g and ĝ are smooth. In addition, Sard’s theorem implies that
the regular values of IdE − ĝ ◦ g|E are dense, so after perturbing ĝ by adding an
arbitrarily small constant, we can make it the case that 0 is a regular value. In the
same way we can add a small constant to g to make 0 a regular value of IdÊ −g ◦ ĝ|Ê ,
and if the constant is small enough it will still be the case that 0 is a regular value
of IdE − ĝ ◦ g|E .
Let x1 , . . . , xr be the fixed points of ĝ ◦ g|E , and for each i let x̂i = g(xi ). Then
x̂1 , . . . , x̂r are the fixed points of g◦ĝ|Ê . Let D1 , . . . , Dr be pairwise disjoint compact
subsets of E with xi ∈ int Di , and let D̂1 , . . . , D̂r be pairwise disjoint open subsets
of Ê with x̂i ∈ int D̂i . For each i let Ei be a compact subset of g −1(int D̂i ) with
xi ∈ int Ei , and let Êi be a compact subset of ĝ −1(int Di ) with x̂i ∈ int Êi . It is easy
to check that each (Rm , Di , Ei , gi , Rm̂ , D̂i , Êi , ĝi ) is a commutativity configuration.
Recalling the relationship between the index and the degree, we have
ΛRm (ĝ ◦ g|Ei ) = ΛRm̂ (g ◦ ĝ|Êi )
because Proposition 13.3.2 gives
|I − D(ĝ ◦ g)(xi )| = |I − Dĝ(x̂i )Dg(xi )| = |I − Dg(xi )Dĝ(x̂i )| = |I − D(g ◦ ĝ)(x̂i )|.
Applying Additivity to sum over i gives the equality asserted by (I4).
′
Turning to (M), suppose that C ⊂ Rm and C ′ ⊂ Rm are compact and f : C →
′
Rm and f ′ : C ′ → Rm are index admissible. Then Proposition 12.4.3 gives
ΛRm+m′ (f × f ′ ) = deg(0,0) (IdC×C ′ − f × f ′ ) = deg(0,0) ((IdC − f ) × (IdC ′ − f ′ ))
= deg0 (IdC − f ) · deg0 (IdC ′ − f ′ ) = ΛRm (f ) · ΛRm′ (f ′ ).
13.4
Extension by Commutativity
The extension of the fixed point index to absolute neighborhood retracts was
first achieved by Felix Browder in his Ph.D. thesis Browder (1948), using the extension method described in this section. This extension method is, perhaps, the most
important application of Commutativity, but Commutativity is also sometimes useful in applications of the index, which should not be particularly surprising since
the underlying fact of linear algebra it embodies (Proposition 13.3.2) is already
nontrivial.
183
13.4. EXTENSION BY COMMUTATIVITY
ˆ We
Throughout this section we will work with two fixed index scopes S and S.
say that S subsumes Ŝ if, for every X ∈ SŜ , we have X ∈ SS and IŜ (X) ⊂ IS (X).
If this is the case, and Λ is an index for S, then the restriction (in the obvious sense)
of Λ to Ŝ is an index for Ŝ. (It is easy to check that this is an automatic consequence
of the definition of an index.) If Λ̂ is an index for Ŝ, then an extension to S is an
index for S whose restriction to Ŝ is Λ̂.
If f : C → X is in IS (X), a narrowing of focus for f is a pair (D, E) of
compact subsets of int C such that
F P(f ) ⊂ int E,
E ∪ f (E) ⊂ int D,
and D ∪ f (D) ⊂ int C.
For such a pair let ε(D,E) be the minimum of:
• d E ∪ f (E), X \ (int D) ,
• d D ∪ f (D), X \ (int C) ;
• the supremum of the set of ε > 0 such that d(x′ , f (x′ )) > 2ε whenever x ∈
C \ (int E), x′ ∈ C, and d(x, x′ ) < ε,
where d is the given metric for X. (Of course X has many metrics that give the
same topology. In contexts such as this we will implicitly assume that one has been
selected.)
Since f is continuous and admissible, narrowings of focus for f exist: continuity
implies the existence of an open neighborhood V of F P(f ) satisfying V ∪ f (V ) ⊂
int C. Repeating this observation gives an open neighborhood W of F P(f ) satisfying W ∪ f (W ) ⊂ V , and we can let D = V and E = W .
ˆ ε)-domination of C is
Let C be a compact subset of a metric space X. An (S,
a quadruple (X̂, Ĉ, ϕ, ψ) in which X̂ ∈ SŜ , Ĉ is an compact subset of X̂, ϕ : C → Ĉ
and ψ : Ĉ → X are continuous functions, and ψ ◦ ϕ is ε-homotopic to IdC . We say
that Ŝ dominates S if, for each X ∈ SS , each compact C ⊂ X, and each ε > 0,
there is an (Ŝ, ε)-domination of C. This section’s main result is:
Theorem 13.4.1. If Ŝ dominates S and Λ̂ is an index for Ŝ, then there is an index
Λ for S that is defined by setting
ΛX (f ) = Λ̂X̂ (ϕ ◦ f ◦ ψ|ψ−1 (E) )
(†)
whenever X ∈ SS , f : C → X is an element of IS (X), (D, E) is a narrowing of
ˆ ε)-domination of C. If, in addifocus for f , ε < ε(D,E) , and (X̂, Ĉ, ϕ, ψ) is an (S,
ˆ then Λ is the unique extension of Λ̂ to S. If Λ̂ is multiplicative,
tion, S subsumes S,
then so is Λ.
Let SS ANR be the class of compact absolute neighborhood retracts, and for each
X ∈ SS ANR let IS ANR (X) be the union over open C ⊂ X of the sets of index
admissible functions in C(C, X). These definitions specify an index scope S ANR
because SS ANR is closed under formation of finite cartesian products, and f × f ′ ∈
IS ANR (X × X̂) whenever X, X̂ ∈ SS ANR , f ∈ IS ANR (X), and f ′ ∈ IS ANR (X̂).
184
CHAPTER 13. THE FIXED POINT INDEX
Theorem 13.4.2. There is a unique index ΛANR for S ANR that extends Λ0 , and
ΛANR is multiplicative.
Proof. Theorem 7.6.4 implies that S 0 dominates S ANR , and S ANR evidently subsumes S 0 .
The rest of this section is devoted to the proof of Theorem 13.4.1. Before
proceeding, the reader should be warned that this is, perhaps, the most difficult
argument in this book. Certainly it is the most cumbersome, from the point of
view of the burden of notation, because the set up used to extend the index is
complex, and then several verifications are required in that setting. To make the
expressions somewhat more compact, from this point forward we will frequently
drop the symbol for composition, for instance writing ψϕ rather than ψ ◦ ϕ.
Lemma 13.4.3. Suppose X ∈ SŜ , f : C → X is in IŜ (X), (D, E) is a narrowing
of focus for f , 0 < ε < ε(D,E) , and (X̂, Ĉ, ϕ, ψ) is an (Ŝ, ε)-domination of U. Let
D̂ = ψ −1 (D) and Ê = ψ −1 (E). Then
(X, D, E, ϕf |D , X̂, D̂, Ê, ψ|D̂ )
is a commutativity configuration.
Proof. We need to verify (a)-(d) of Definition 13.2.2. We have E ⊂ D ⊂ X with
D and E compact, so Ê ⊂ D̂ ⊂ X̂. In addition D̂ and Ê are closed because ψ is
continuous, so they are compact because they are subsets of Ĉ. Thus (a) holds.
Of course ϕf |D and ψ|D̂ are continuous. We have
ψ(ϕ(f (E))) ⊂ Uε (f (E)) ⊂ int D,
so ϕ(f (E)) ⊂ ψ −1 (int D) ⊂ int D̂. In addition, ψ(Ê) ⊂ E ⊂ int D. Thus (b) holds.
If x ∈ D \ int (E), then d(x, f (x)) > 2ε(D,E) > 2ε and d(f (x), ψ(ϕ(f (x)))) < ε,
so x cannot be a fixed point of ψϕf . Thus F P(ψϕf |E ) ⊂ int E. If x̂ ∈ D̂ is a fixed
point of ϕf ψ|D̂ , then ψ(x̂) is a fixed point of ψϕf |D , so F P(ϕf ψ|D̂ ) ⊂ ψ −1 (int E) ⊂
int Ê. Thus (c) holds.
We now establish (∗) and (∗∗). We have
ψ(ϕ(f (F P(ψϕf |D )))) = F P(ψϕf |D ) ⊂ int E,
so
ϕ(f (F P(ψϕf |D ))) ⊂ ψ −1 (int E) ⊂ Ê,
and F P(ϕf ψ|Ê ) ⊂ int Ê, so
ψ(F P(ϕf ψ|D̂ )) ⊂ ψ(int Ê) ⊂ E.
Thus (d) holds.
From this point forward we assume that there is a given index Λ̂ for Ŝ. In
order for our proposed definition of Λ to be workable it needs to be the case that
the definition of the derived index does not depend on the choice of narrowing or
domination, and it turns out that proving this will be a substantial part of the
overall effort. The argument is divided into a harder part proving a special case
and a reduction of the general case to this.
185
13.4. EXTENSION BY COMMUTATIVITY
Lemma 13.4.4. Let X be an element of SŜ , and let f : C → X be an element
of IŜ (X). Suppose that (D, E) is a narrowing of focus for f , 0 < ε1 , ε2 < ε(D,E) ,
ˆ ε2 )and (X̂1 , Ĉ1 , ϕ1 , ψ1 ) and (X̂2 , Ĉ2 , ϕ2 , ψ2 ) are an (Ŝ, ε1 )-domination and an (S,
domination of C. Set
D̂1 = ψ1−1 (D),
Ê1 = ψ1−1 (E),
D̂2 = ψ2−1 (D),
Ê2 = ψ2−1 (E).
Then
Λ̂X̂1 (ϕ1 f ψ1 |Ê1 ) = Λ̂X̂2 (ϕ2 f ψ2 |Ê2 ).
Proof. The definition of domination gives an ε-homotopy h : C × [0, 1] → X with
h0 = IdC and h1 = ψ1 ϕ1 and a ε-homotopy j : C × [0, 1] → X be an ε-homotopy
with j0 = IdC and j1 = ψ2 ϕ2 . We will show that:
(a) the homotopy t 7→ ϕ1 jt f ψ1 |Ê1 is well defined and index admissible;
(b) the homotopy t 7→ ϕ2 f ht ψ2 |Ê2 is well defined and index admissible;
(c) (X̂1 , D̂1 , Ê1 , ϕ2 f ψ1 , X̂2 , D̂2 , Ê2 , ϕ1 ψ2 ) is a commutativity configuration.
The claim follows from the computation
Λ̂X̂1 (ϕ1 f ψ1 |Ê1 ) = Λ̂X̂1 (ϕ1 ψ2 ϕ2 f ψ1 |Ê1 ) = Λ̂X̂2 (ϕ2 f ψ1 ϕ1 ψ2 |Ê2 ) = Λ̂X̂2 (ϕ2 f ψ2 |Ê2 ).
Specifically, in view of (a) and (b) the first and third equalities follows from the
homotopy principle, while (c) permits an application of Commutativity that gives
the second equality.
For each t the composition ϕ1 jt f ψ1 |Ê1 is well defined because
ψ1 (Ê1 ) ⊂ E
and jt (f (E)) ⊂ Uε (f (E)) ⊂ D ⊂ C.
In order to show that this homotopy is index admissible, we assume (aiming at a
contradiction) that for some 0 ≤ t ≤ 1, y1 ∈ ∂ Ê1 is a fixed point of ϕ1 jt f ψ1 . Then
ψ1 (y1 ) is a fixed point of ψ1 ϕ1 jt f . The definition of Ê1 and the continuity of ψ1
imply that ψ1 (y1 ) ∈ ∂E, so that d f ψ1 (y1 ), ψ1 (y1 ) ≥ 2ε(D,E) > 2ε, but
d ψ1 ϕ1 jt f ψ1 (y1 ), f ψ1 (y1 ) = d h1 jt f ψ1 (y1 ), f ψ1 (y1 )
≤ d h1 jt f ψ1 (y1 ), jt f ψ1 (y1 ) + d jt f ψ1 (y1 ), f ψ1 (y1) < 2ε,
so this is impossible. We have established (a) and (by symmetry) (b).
To establish (c) we need to verify (a)-(d) of Definition 13.2.2. Evidently Ê1 ⊂
D̂1 ⊂ X̂1 and Ê2 ⊂ D̂2 ⊂ X̂2 with D̂1 , Ê1 , D̂2 , and Ê2 compact, so (a) holds.
We have f (ψ1 (D̂1 )) ⊂ f (D) ⊂ C, so ψ1 (D̂1 ) is contained in the domain of ϕ2 ,
and ψ2 (D̂2 ) ⊂ D ⊂ C, so ψ2 (D̂2 ) is contained in the domain of ϕ1 . Thus ϕ2 f ψ1
and ϕ1 ψ2 are well defined, and of course they are continuous. In addition,
ψ2 ϕ2 f ψ1 (Ê1 ) ⊂ ψ2 ϕ2 f (E) ⊂ Uε (f (E)) ⊂ int D
and
ψ1 ϕ1 ψ2 (Ê2 ) ⊂ Uε (ψ2 (Ê2 )) ⊂ Uε (E) ⊂ int D,
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CHAPTER 13. THE FIXED POINT INDEX
so
ϕ2 f ψ1 (Ê1 ) ⊂ ψ2−1 (D) ⊂ int D̂2
and ϕ1 (ψ2 (Ê2 )) ⊂ ψ1−1 (int D) = int D̂1 .
Thus (b) holds.
Above we showed that
ϕ1 ψ2 ϕ2 f ψ1 |Ê1 = ϕ1 j1 f ψ1 |Ê1
and ϕ2 f ψ1 ϕ1 ψ2 |Ê2 = ϕ2 f h1 ψ2 |Ê2
are index admissible. That is, (c) holds.
Suppose that y1 ∈ F P(ϕ1 ψ2 ϕ2 f ψ1 |Ê1 ) and y2 = ϕ2 f ψ1 (y1 ). Then ψ2 (y2 ) is a
fixed point of ψ2 ϕ2 f ψ1 ϕ1 . The definition of ε(D,E) implies that this is impossible
if ψ2 (y2 ) ∈
/ E, so y2 ∈ ψ2−1 (E) = Ê2 . Now suppose that y2 ∈ F P(ϕ2 f ψ1 ϕ1 ψ2 |Ê2 )
and y1 = ϕ1 ψ2 (y2 ). Then ψ1 (y1 ) is a fixed point of ψ1 ϕ1 ψ2 ϕ2 f , so ψ1 (y1 ) ∈ E and
y1 ∈ Ê1 . We have shown that
ϕ2 f ψ1 (F P(ϕ1 ψ2 ϕ2 f ψ1 |Ê1 )) ⊂ Ê2
and ϕ1 ψ2 (F P(ϕ2 f ψ1 ϕ1 ψ2 |Ê2 )) ⊂ Ê1 ,
which is to say that (∗) and (∗∗) hold, which implies (d), completing the proof.
The hypotheses of the next result are mostly somewhat more general, but we
now need to assume that Ŝ dominates S.
Lemma 13.4.5. Assume that Ŝ dominates S. Let X be an element of SS , and let
f : C → X be an element of IS (X). Suppose (D1 , E1 ) and (D2 , E2 ) are narrowings
of focus for f , 0 < ε1 < ε(D1 ,E1 ) , 0 < ε2 < ε(D2 ,E2 ) , and (X̂1 , Ĉ1 , ϕ1 , ψ1 ) and
ˆ ε2 )-domination of C. Set
(X̂2 , Ĉ2 , ϕ2 , ψ2 ) are an (Ŝ, ε1)-domination and an (S,
D̂1 = ψ1−1 (D1 ),
Ê1 = ψ1−1 (E1 ),
D̂2 = ψ2−1 (D2 ),
Ê2 = ψ2−1 (E2 ).
Then
Λ̂X̂1 (ϕ1 f ψ1 |Ê1 ) = Λ̂X̂2 (ϕ2 f ψ2 |Ê2 ).
Proof. It suffices to show this when D1 ⊂ D2 and E1 ⊂ E2 , because then the
general case follows from two applications in which first D1 and E1 , and then D2
and E2 , are replaced by D1 ∩ D2 and E1 and E2 with E1 ∩ E2 . The assumption
that Sˆ dominates S which guarantees the existence of an (Ŝ, ε′2) domination of U
for arbitrarily small ε′2 , and if we apply the lemma above to this domination and
the given one we find that it suffices to prove the result with the given domination
replaced by this one. This means that we may assume that ε2 is as small as need
be, and in particular we may assume that ε2 < ε(D1 ,E1 ) . Now Additivity implies
that
Λ̂X̂2 (ϕ2 f ψ2 |Ê2 ) = Λ̂X̂2 (ϕ2 f ◦ ψ2 |ψ2−1 (E1 ) ),
which means that it suffices to establish the result with D2 and E2 replaced by D1
and E1 , which is the case established in the lemma above.
Proof of Theorem 13.4.1. Since Sˆ dominates S, the objects used to define Λ exist,
and the last result implies that the definition of Λ does not depend on the choice
of (D, E), ε, and (X̂, Ĉ, ϕ, ψ). We now verify that Λ satisfies (I1)-(I4) and (M).
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13.4. EXTENSION BY COMMUTATIVITY
For the proofs of (I1)-(I3) we fix a particular X ∈ SS and an f : C → X in
IS (X), and we let (D, E), ε, and (X̂, Ĉ, ϕ, ψ) be as in the hypotheses.
Normalization:
If f is a constant function, then so is ϕf ψ, so Normalization for Λ̂ gives
ΛX (f ) = Λ̂X̂ (ϕf ψ) = 1.
Additivity:
Suppose that F P(f ) ⊂ int C1 ∪ . . . ∪ int Cr where C1 , . . . , Cr ⊂ C are compact
and pairwise disjoint. For each j = 1, . . . , r choose open sets Dj ⊂ D ∩ Cj and
Ej ⊂ E ∩ Cj such that (Dj , Ej ) is a narrowing of focus for (Cj , f |Cj ). In view of
Lemma 13.4.5 we may assume that ε < ε(Dj ,Ej ) for all j. It is easy to see that for
ˆ ε)-domination of Cj . For each j let E ′ = ψ −1 (Ej ).
each j, (X̂, Ĉ, ϕ|Cj , ψ) is an (S,
j
Additivity for Λ̂ gives
X
X
ΛX (f |Cj ).
Λ̂X̂ (ϕf ψ|Ej′ ) =
ΛX (f ) = Λ̂X̂ (ϕf ψ|Ê ) =
j
j
Continuity:
It is easy to see that if f ′ : C → X that are sufficiently close to f , then (D, E) is
a narrowing of focus for (C, f ′), and (X̂, Ĉ, ϕ, ψ) is a (Ŝ, ε)-domination of C. Since
f ′ 7→ ϕf ′ ψ is continuous (Propositions 5.5.2 and 5.5.3) Continuity for Λ̂ gives
ΛX (f ) = Λ̂X̂ (ϕf ψ) = Λ̂X̂ (ϕf ′ ψ) = ΛX (f ′ )
when f ′ is sufficiently close to f .
Commutativity:
Suppose that (X1 , C1 , D1 , g1 , X2 , C2 , D2 , g2 ) is a commutativity configuration
with X1 , X2 ∈ SS . Replacing D1 and D2 with smaller open neighborhoods of
F P(g2 g1 ) and F P(g1 g2 ) if need be, we may assume that
D1 ∪ g2 g1 (D1 ) ⊂ C1
and D2 ∪ g1 g2 (D2 ) ⊂ C2 .
Choose open sets E1 and E2 with
F P(g2 g1 ) ⊂ E1 , E1 ∪ g2 g1 (E1 ) ⊂ D1 , F P(g1 g2 ) ⊂ E2 , E2 ∪ g1 g2 (E2 ) ⊂ D2 .
For any positive ε1 < ε(D1 ,E2 ) and ε2 < ε(D2 ,E2) Lemma 13.4.5 implies that there is
a (S, ε1 )-domination (X̂1 , Ĉ1 , ϕ1 , ψ1 ) of C1 and a (S, ε2 )-domination (X̂2 , Ĉ2 , ϕ2 , ψ2 )
of C2 . Let
D̂1 = ψ1−1 (D1 ),
Ê1 = ψ1−1 (E1 ),
D̂2 = ψ2−1 (D2 ),
Ê2 = ψ2−1 (E2 ).
Let h : C1 × [0, 1] → X1 be a ε1 -homotopy with h0 = IdC1 and h1 = ψ1 ϕ1 , and let
j : C2 × [0, 1] → X2 be a ε2 -homotopy with j0 = IdC2 and j1 = ψ2 ϕ2 . The desired
result will follow from the calculation
ΛX1 (g2 g1 ) = Λ̂X̂1 (ϕ1 g2 g1 ψ1 |Ê1 ) = Λ̂X̂1 (ϕ1 g2 ψ2 ϕ2 g1 ψ1 |Ê1 )
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CHAPTER 13. THE FIXED POINT INDEX
= Λ̂X̂2 (ϕ2 g1 ψ1 ϕ1 g2 ψ2 |Ê2 ) = Λ̂X̂2 (ϕ2 g1 g2 ψ2 |Ê2 ) = ΛX2 (g1 g2 ).
Here the first and fifth equality are from the definition of Λ, the second and fourth
are implied by Continuity for Λ̂, and the third is from Commutativity for Λ̂. In
order for this to work it must be the case that all the compositions in this calculation
are well defined, in the sense that the image of the first function is contained in the
domain of the second function, the homotopies
t 7→ ϕ1 g2 jt g1 ψ1 |Ê1
and t 7→ ϕ2 g1 ht g2 ψ2 |Ê2
are index admissible, and
(X̂1 , D̂1 , Ê1 , ϕ2 g1 ψ1 |D1 , X̂2 , D̂2 , Ê2 , ϕ1 g2 ψ2 |D2 )
is a commutativity configuration. Clearly this will be the case when ε1 and ε2 are
sufficiently small.
Multiplication:
We now consider X1 , X2 ∈ SS , f1 : C1 → X1 in IS (X1 ), and f2 : C2 → X2 in
IS (X2 ). For each i = 1, 2 let (Di , Ei ) be a narrowing of focus for (Ci , fi ), and let
(X̂i , Ĉi , ϕi , ψi ) be an (Ŝ, εi )-domination of C, where ε < ε(Di ,Ei ) . The definition of
an index scope requires that X1 × X2 ∈ SS and (C1 × C2 , f1 × f2 ) ∈ IS (X1 × X2 ).
Clearly (D1 × D2 , E1 × E2 ) is a narrowing of focus for (C1 × C2 , f1 × f2 ). If d1 and
d2 are given metrics for X1 and X2 respectively, endow X1 × X2 with the metric
(x1 , x2 ), (y1 , y2 ) →
7 max{d1 (x1 , y1 ), d2 (x2 , y2)}.
Let ε = max{ε1 , ε2 }. Then (X̂1 × X̂2 , Ĉ1 × Ĉ2 , ϕ1 ×ϕ2 , ψ1 ×ψ2 ) is a (Ŝ, ε)-domination
of C1 × C2 . It is also easy to check that ε(D1 ×D2 ,E1 ×E2 ) ≥ max{ε(D1 ,E1 ) , ε(D2 ,E2 ) }, so
ε < ε(D1 ×D2 ,E1 ×E2 ) . Therefore Lemma 13.4.3 implies that the validity of the first
equality in
ΛX1 ×X2 (f1 × f2 ) = Λ̂X̂1 ×X̂2 ((ϕ1 f1 ψ1 × ϕ2 f2 ψ2 )|Ê1 ×Ê2 )
= Λ̂X̂1 (ϕ1 f1 ψ1 |Ê1 ) · Λ̂X̂2 (ϕ2 f2 ψ2 |Ê2 ) = ΛX1 (f1 ) · ΛX2 (f2 )
the second one is an application of Multiplication for Λ̂ and the third is the definition
of Λ.
We now prove that if S subsumes Ŝ, then Λ is the unique extension of Λ̂ to
S. Consider X ∈ SŜ and (C, f ) ∈ IŜ (X). For any ε > 0, (X, C, IdC , IdC ) is
ˆ ε)-domination of C. For any narrowing of focus (D, E) equation (†) gives
an (S,
ΛX (f ) = Λ̂X (f |E ) and Additivity for Λ̂ gives Λ̂X (f |E ) = Λ̂X (f ). Thus Λ extends
Λ̂.
Two indices for S that restrict to Λ̂ necessarily agree everywhere because, by
Continuity and Commutativity, (†) holds in the circumstances described in the
statement of Theorem 13.4.1.
13.5. EXTENSION BY CONTINUITY
13.5
189
Extension by Continuity
This section extends the index from continuous functions to upper semicontinuous contractible valued correspondences. As in the last section, we describe the
extension process abstractly, thereby emphasizing the aspects of the situation that
drive the argument.
Definition 13.5.1. If I and Î are index bases for a compact metric space X, we
say that Î approximates I if:
(E1) If C, D ⊂ X are open with D ⊂ C, then Î ∩ C(D, X) is dense in
{ F |D : F ∈ I ∩ U(C, X) and F |D is index admissable }.
(E2) If C, D ⊂ X are open with D ⊂ C, F ∈ I ∩ U(C, X), and A ⊂ C × X is a
neighborhood of Gr(F ), then there is a neighborhood B ⊂ D × X of Gr(F |D )
such that any two functions f, f ′ ∈ C(D, X) with Gr(f ), Gr(f ′ ) ⊂ B are the
endpoints of a homotopy h : [0, 1] → C(D, X) with Gr(ht ) ⊂ A for all t.
It would be simpler if, in (E1) and (E2), we could have V = U, but unfortunately
Theorem 9.1.1 is not strong enough to justify working with such a definition.
Definition 13.5.2. If S and Ŝ are two index scopes with SŜ = SS , then Sˆ approximates S) if, for each X ∈ SŜ , IŜ (X) approximates IS (X), and
(E3) If (X, C, D, g, X ′, C ′ , D ′ , g ′) is a commutativity configuration such that X, X ′ ∈
SS , g ′ ◦ g ∈ IS (X), and g ◦ g ′ ∈ IS (X ′ ), and S ⊂ C(C, X ′) and S ′ ⊂ C(C ′ , X)
are neighborhoods of g and g ′, then there exist γ ∈ S and γ ′ ∈ S ′ such that
γ ′ ◦ γ|D ∈ IŜ (X) and γ ◦ γ ′ |D′ ∈ IŜ (X ′ ).
Theorem 13.5.3. Suppose that Sˆ approximates S, and Λ̂ is an index for Ŝ. For
each X ∈ SŜ let ΛX be the extension of Λ̂X to IS (X) given by the last result. Then
the system Λ of maps ΛX is an index for S. If, in addition, S subsumes Ŝ, then Λ
is the unique extension of Λ̂ to S.
Evidently S Ctr subsumes S ANR . The constant functions in S ANR and S Ctr are
the same, of course, and Theorem 9.1.1 implies that (E1) and (E2) are satisfied
when S = S Ctr and Ŝ = S ANR . Therefore Theorem 13.2.4 follows from Theorem
13.4.2 and the last result.
The remainder of this section is devoted to the proof of Theorem 13.5.3. The
overall structure of our work here is similar to what we saw in the last section. We
are given an index Λ̂ for an index base Î, and we wish to use this to define an
index for another index base I. In this case Continuity is the axiom that does the
heavy lifting. Assumption (E1) states that every element of the second base can be
approximated by an element of the first base. Therefore we can define the index
of an element of the second base to be the index of sufficiently fine approximations
by elements of the first base, provided these all agree, and assumption (E2), in
conjunction with Continuity, implies that this is the case.
Having defined the index on the second base, we must verify that it satisfies the
axioms. This phase is broken down into two parts. The following result verifies
that the axioms pertaining to a single index base hold.
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CHAPTER 13. THE FIXED POINT INDEX
Proposition 13.5.4. Suppose I and Î are index bases for a compact metric space
X, and Î approximates I. Then for any index Λ̂X : Î → Z there is a unique
index ΛX : I → Z such that for each open C ⊂ X with compact closure, each F ∈
I ∩U(C, X), and each open D with F P(F ) ⊂ D and D ⊂ C, there is a neighborhood
E ⊂ U(D, X) of F |D such that ΛX (F ) = Λ̂X (f ) for all f ∈ E ∩ C(D, X) ∩ Î.
Proof. Fix C, F ∈ I ∩ U(C, X), and D as in the hypotheses. Then F |D is index
admissable, hence an element of I because I is an index base.
Applying (E2), let B ⊂ D × X be a neighborhood of Gr(F |D ) such that for
any f, f ′ ∈ Î ∩ C(D, X) with Gr(f ), Gr(f ′ ) ⊂ B there is a homotopy h : [0, 1] →
C(D, X) with h0 = f , h1 = f ′ , and
Gr(ht ) ⊂ (C × X) \ { (x, x) : x ∈ C \ D }
for all t. Since F has no fixed points in C \ D, the right hand side is a neighborhood
of Gr(FD ). We define ΛX by setting
ΛX (F ) := Λ̂X (f )
for any such f .
We first have to show that this definition makes sense. First, (E1) implies that
{ F ′ ∈ U(D, X) : Gr(F ′ ) ⊂ B } ∩ C(D, X) ∩ Î =
6 ∅,
and Continuity implies that this definition does not depend on the choice of f .
Since A and B can be replaced by smaller open sets, it does not depend on the
choice of A and B. We must also show that it does not depend on the choice of D.
So, let D̃ be another open set with D̃ ⊂ C and F P(F ) ∩ (C \ D̃) = ∅. Then
F P(F ) ⊂ D ∩ D̃. The desired result follows if we can show that it holds when D
and D̃ are replaced by D and D ∩ D̃ and also when D and D̃ are replaced by D ∩ D̃
and D̃. Therefore we may assume that D̃ ⊂ D.
Let B̃ ⊂ D̃ × X be a neighborhood of Gr(F |D̃ ) such that for any f, f ′ ∈ Î ∩
C(D̃, X) with Gr(f ), Gr(f ′ ) ⊂ B̃ there is a homotopy h : [0, 1] → C(D̃, X) with
h0 = f , h1 = f ′ , and
Gr(ht ) ⊂ (C × X) \ { (x, x) : x ∈ C \ D̃ }
for all t. Since restriction to a compact subdomain is a continuous operation
(Lemma 5.3.1) we may replace B with a smaller neighborhood of F |D to obtain
Gr(f |D̃ ) ⊂ B ′ whenever Gr(f ) ⊂ B. For such an f Additivity gives Λ̂X (f ) =
Λ̂X (f |D̃ ) as desired.
It remains to show that (I1)-(I3) are satisfied.
Normalization:
If c is a constant function, we can take c itself as the approximation used to
define ΛX (c), so Normalization for ΛX follows from Normalization for Λ̂X .
Additivity:
Consider F ∈ I with domain C. Let C1 , . . . , Cr be disjoint open subsets of C
whose union contains F P(F ). Let D1 , . . . , Dr be open subsets of C with D1 ⊂
191
13.5. EXTENSION BY CONTINUITY
C1 , . . . , Dr ⊂ Cr and F P(F ) ⊂ D1 , . . . , Dr . For each i = 1, . . . , r let Bi be a
neighborhood of Gr(F |Di ) such that ΛX (F |Ci ) = Λ̂X (fi ) whenever fi ∈ Î ∩C(Di , X)
with Gr(fi ) ⊂ Bi . Let D := D1 ∪ . . . ∪ Dr , and let B be a neighborhood of F |D
such that ΛX (F |C ) = Λ̂X (f ) whenever f ∈ Î ∩ C(D, X) with Gr(f ) ⊂ B. Since
restriction to a compact subdomain is a continuous operation (Lemma 5.3.1) B
may be chosen so that, for all i, Gr(f |Di ) ⊂ Bi whenever Gr(f ) ⊂ B. For any
f ∈ Î ∩ C(D, X) with Gr(f ) ⊂ B we now have
X
X
ΛX (F ) = Λ̂X (f |D ) =
Λ̂X (f |Di ) =
ΛX (F |Ci ).
i
i
Continuity: Suppose that C ⊂ X is open with compact closure, D ⊂ C is open
with D ⊂ C, F ∈ I with F P(F ) ⊂ D, and B is a neighborhood of Gr(F |D )
with ΛX (F ) = Λ̂X (f ) for all f ∈ Î ∩ C(D, X) with Gr(f ) ⊂ B. Then the set of
F ′ ∈ I ∩ U(C, X) such that F ′ |D ∈ B and F P(F ′ ) ⊂ D is a neighborhood of F ,
and for every such F ′ we have ΛX (F ′ ) = ΛX (F ).
The remainder of the argument shows that the extension procedure described
above results in an index satisfying (I4) and (M) when it is used to define extensions
for all spaces in an index scope.
Proof of Theorem 13.5.3. We begin by noting that when S subsumes Ŝ, Continuity
for Λ̂ implies both that Λ is an extension of Λ̂ and that any extension must satisfy
the condition used to define Λ, so Λ is the unique extension. In view of the last
result, it is only necessary to verify that Λ satisfies (I4) and (M). The argument is
based on the description of Λ given in the first paragraph of the proof of the last
result.
Commutativity:
Suppose that (X, C, D, g, X ′, C ′ , D ′ , g) is a commutativity configuration with
g ′ ◦ g ∈ IS (X) and g ◦ g ′ ∈ IS (X ′ ). Lemma 13.3.4 implies that there are neighborhoods S ⊂ C(C, X ′) and S ′ ⊂ C(C ′ , X) such that (X, C, D, γ, X ′, C ′ , D ′ , γ ′ )
is a commutativity configuration for all γ ∈ S and γ ′ ∈ S ′ . Let B ⊂ C(D, X)
and B ′ ⊂ C(D ′ , X ′) be neighborhoods of g ′ ◦ g|D and g ◦ g ′|D′ , respectively, such
that Λ(g ′ ◦ g|D ) = Λ̂(f ) and Λ(g ◦ g ′|D′′ ) = Λ̂(f ′ ) whenever f ∈ B ∩ IŜ (X) and
f ′ ∈ C ∩ IŜ (X ′ ). The continuity of restriction and composition (Lemma 5.3.1 and
Proposition 5.3.6) implies the existence of neighborhoods T ⊂ C(C, X ′) of g and
T ′ ⊂ C(D ′ , X) of g ′ such that γ ′ ◦ γ|D ∈ B and γ ◦ γ ′ |D′ ∈ B ′ whenever γ ∈ T
and γ ′ ∈ T ′ . Applying (E3), we may choose γ ∈ S ∩ T and γ ′ ∈ S ′ ∩ T ′ such that
γ ′ ◦ γ|D ∈ IŜ (X) and γ ◦ γ ′ |D′ ∈ IŜ (X ′ ). Now Commutativity for Λ̂ gives
ΛX (g ′ ◦ g|D ) = Λ̂X (γ ′ ◦ γ|D ) = Λ̂X ′ (γ ◦ γ ′ |D′ ) = ΛX ′ (g ◦ g ′ |D′ ).
Multiplication:
For spaces X, X ′ ∈ SS and open C ⊂ X and C ′ ⊂ X ′ with compact closure
consider
F ∈ IS (X) ∩ U(C, X) and F ′ ∈ IS (X ′ ) ∩ U(C ′ , X ′ ).
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CHAPTER 13. THE FIXED POINT INDEX
Then the definition of an index scope implies that F ×F ′ ∈ IS (X×X ′ ). Choose open
sets D and D ′ with F P(F ) ⊂ D, D ⊂ C, F P(F ′ ) ⊂ D ′ , and D ′ ⊂ C ′ . As above, we
can find neighborhoods B ⊂ U(D, X), B ′ ⊂ U(D ′ , X ′ ), and D ⊂ U(D×D ′ , X ×X ′ ),
of F |D , F ′ |D′ , and (F × F ′ )|D×D′ respectively, such that ΛX (F ) = Λ̂X (f ) for all
f ∈ B ∩ IŜ (X), ΛX ′ (F ′ ) = Λ̂X ′ (f ′ ) for all f ′ ∈ B ′ ∩ IŜ (X ′ ), and ΛX×X ′ (F × F ′ ) =
Λ̂X×X ′ (j) for all j ∈ D ∩ IŜ (X × X ′ ). Since the formation of cartesian products
of correspondences is a continuous operation (this is Lemma 5.3.4) we may replace
B and B ′ with smaller neighborhoods to obtain F̃ × F̃ ′ ∈ D for all F̃ ∈ B and
F̃ ′ ∈ B ′ . Assumption (E1) implies that there are
f ∈ B ∩ IŜ (X) ∩ C(D, X) and f ′ ∈ B ′ ∩ IŜ (X ′ ) ∩ C(D ′ , X ′ ).
The definition of an index scope implies that f ×f ′ ∈ IŜ (X ×X ′), and Multiplication
(I4) for Λ̂ now gives
ΛX×X ′ (F × F ′ ) = Λ̂X×X ′ (f × f ′ ) = Λ̂X (f ) · Λ̂X ′ (f ′ ) = ΛX (F ) · ΛX ′ (F ′ ).
Part III
Applications and Extensions
193
Chapter 14
Topological Consequences
This chapter is a relaxing and refreshing change of pace. Instead of working very
hard to slowly build up a toolbox of techniques and specific facts, we are going to
harvest the fruits of our earlier efforts, using the axiomatic description of the fixed
point index, and other major results, to quickly derive a number of quite famous
results. In Section 14.1 we define the Euler characteristic, relate it to the Lefschetz
fixed point theorem, and then describe the Eilenberg-Montgomery as a special case.
For two general compact manifolds, the degree of a map from one to the other
is a rather crude invariant, in comparison with many others that topologists have
defined. Nevertheless, when the range is the m-dimensional sphere, the degree
is already a “complete” invariant in the sense that it classifies functions up to
homotopy: if M is a compact m-dimensional manifold that is connected, and f
and f ′ are functions from M to the m-sphere of the same degree, then f and f ′
are homotopic. This famous theorem, due to Hopf, is the subject of Section 14.2.
Section 12.4 proves a simple result asserting that the degree of a composition of two
functions is the products of their degrees.
Section 14.3 presents several other results concerning fixed points and antipodal
maps of a map from a sphere to itself. Some of these are immediate consequences
of index theory and the Hopf theorem, but the Borsuk-Ulam theorem requires a
substantial proof, so it should be thought of as a significant independent fact of
topology. It has many consequences, including the fact that spheres of different
dimensions are not homeomorphic.
In Section 14.4 we state and prove the theorem known as invariance of domain.
It asserts that if U ⊂ Rm is open, and f : U → Rm is continuous and injective, then
the image of f is open, and the inverse is continuous. One may think of this as a
purely topological version of the inverse function theorem, but from the technical
point of view it is much deeper.
If a connected set of fixed points has a nonzero index, it is essential. This raises
the question of whether a connected set of fixed points of index zero is necessarily
inessential. Section 14.5 presents two results of this sort.
194
14.1. EULER, LEFSCHETZ, AND EILENBERG-MONTGOMERY
14.1
195
Euler, Lefschetz, and Eilenberg-Montgomery
The definition of the Euler characteristic, and Euler’s use of it in the analyses
of various problems, is often described as the historical starting point of topology
as a branch of mathematics. In popular expositions the Euler characteristic of a 2dimensional manifold M is usually defined by the formula χ(M) := V −E +F where
V , E, and F are the numbers of vertices, edges, and 2-simplices in a triangulation
of M. Our definition is:
Definition 14.1.1. The Euler characteristic χ(X) of a compact ANR X is
ΛX (IdX ).
Here is a sketch of a proof that our definition of χ(M) agrees with Euler’s
when M is a triangulated compact 2-manifold. We deform the identity function
slightly, achieving a function f : M → M defined as follows. Each vertex of the
triangulation is mapped to itself by f . Each barycenter of an edge is mapped to
itself, and the points on the edge between the barycenter and either of the vertices
of the edge are moved toward the barycenter. Each barycenter of a two dimensional
simplex is mapped to itself. If x is a point on the boundary of the 2-simplex, the
line segment between x and the barycenter is mapped to the line segment between
f (x) and the barycenter, with points on the interior of the line segment pushed
toward the barycenter, relative to the affine mapping. It is easy to see that the only
fixed points of f are the vertices and the barycenters of the edges and 2-simplices.
Euler’s formula follows once we show that the index of a vertex is +1, the index of
the barycenter of an edge is −1, and the index of the barycenter of a 2-simplex is +1.
We will not give a detailed argument to this effect; very roughly it corresponds to
the intuition that f is “expansive” at each vertex, “compressive” at the barycenter
of each 2-simplex, and expansive in one direction and compressive in another at the
barycenter of an edge.
Although Euler could not have expressed the idea in modern language, he certainly understood that the Euler characteristic is important because it is a topological invariant.
Theorem 14.1.2. If X and X ′ are homeomorphic compact ANR’s, then
χ(X) = χ(X ′ ).
Proof. For any homeomorphism h : X → X ′ , Commutativity implies that
χ(X) = ΛX (IdX ) = ΛX (IdX ◦ h−1 ◦ h) = ΛX ′ (h ◦ IdX ◦ h−1 ) = ΛX ′ (IdX ′ ) = χ(X ′ ).
The analytic method implicit in Euler’s definition—pass from a topological space
(e.g., a compact surface) to a discrete object (in this case a triangulation) that
can be analyzed combinatorically and quantitatively—has of course been extremely
fruitful. But as a method of proving that the Euler characteristic is a topological invariant, it fails in a spectacular manner. There is first of all the question of
196
CHAPTER 14. TOPOLOGICAL CONSEQUENCES
whether a triangulation exists. That a two dimensional compact manifold is triangulable was not proved until the 1920’s, by Rado. In the 1950’s Bing and Moise
proved that compact three dimensional manifolds are triangulable, and a stream
of research during this same general period showed that smooth manifolds are triangulable, but in general a compact manifold need not have a triangulation. For
simplicial complexes topological invariance would follow from invariance under subdivision, which can be proved combinatorically, and the Hauptvermutung, which
was the conjecture that any two simplicial complexes that are homeomorphic have
subdivisions that are combinatorically isomorphic. This conjecture was formulated
by Steinitz and Tietze in 1908, but in 1961 Milnor presented a counterexample, and
in the late 1960’s it was shown to be false even for triangulable manifolds.
The Lefschetz fixed point theorem is a generalization Brouwer’s theorem
that was developed by Lefschetz for compact manifolds in Lefschetz (1923, 1926)
and extended by him to manifolds with boundary in Lefschetz (1927). Using quite
different methods, Hopf extended the result to simplicial complexes in Hopf (1928).
Definition 14.1.3. If X is a compact ANR and F : X → X is an upper semicontinuous contractible valued correspondence, the Lefschetz number of F is ΛX (F ).
Theorem 14.1.4. If X is a compact ANR, F : X → X is an upper semicontinuous
contractible valued correspondence and ΛX (F ) 6= 0, then F P(F ) 6= ∅.
Proof. When F P(F ) = ∅ two applications of Additivity give
Λ(F |∅ ) = Λ(F ) = Λ(F |∅ ) + Λ(F |∅).
In Lefschetz’ originally formulation the Lefschetz number of a function was defined using algebraic topology. Thus one may view the Lefschetz fixed point theorem
as a combination of the result above and a formula expressing the Lefschetz number
in terms of homology.
In the Kakutani fixed point theorem, the hypothesis that the correspondence is
convex valued cries out for generalization, because convexity is not a topological concept that is preserved by homeomorphisms of the space. The Eilenberg-Montgomery
theorem asserts that if X is a compact acyclic ANR, and F : X → X is an upper
semicontinuous acyclic valued correspondence, then F has a fixed point. Unfortunately it would take many pages to define acyclicity, so we will simply say that
acyclicity is a property that is invariant under homeomorphism, and is weaker than
contractibility. The known examples of spaces that are acyclic but not contractible
are not objects one would expect to encounter “in nature,” so it seems farfetched
that the additional strength of the Eilenberg-Montgomery theorem, beyond that of
the result below, will ever figure in economic analysis.
Theorem 14.1.5. If X is a nonempty compact absolute retract and F : X → X
is an upper semicontinuous contractible valued correspondence, then F has a fixed
point.
14.2. THE HOPF THEOREM
197
Proof. Recall (Proposition 7.5.3) that an absolute retract is an ANR that is contractible. Theorem 9.1.1 implies that F can be approximated in the sense of
Continuity by a continuous function, so ΛX (F ) = ΛX (f ) for some continuous
f : X → X. Let c : X × [0, 1] → X be a contraction. Then (x, t) 7→ c(f (x), t) (or
(x, t) → f (c(x, t))) is a homotopy between f and a constant function, so Homotopy
[fix this] and Normalization imply that ΛX (f ) = 1. Now the claim follows from the
last result.
14.2
The Hopf Theorem
Two functions that are homotopic may differ in their quantitative features, but
from the perspective of topology these differences are uninteresting. Two functions
that are not homotopic differ in some qualitative way that one may hope to characterize in terms of discrete objects. A homotopy!invariant may be thought of
as a function whose domain is the set of homotopy classes; equivalently, it may be
thought of as a mapping from a space of functions that is constant on each homotopy class. A fundamental method of topology is to define and study homotopy
invariants.
The degree is an example: for compact manifolds M and N of the same dimension it assigns an integer to each continuous f : M → N, and if f and f ′ are
homotopic, then they have the same degree. There are a great many other homotopy invariants, whose systematic study is far beyond our scope. In the study of
such invariants, one is naturally interested in settings in which some invariant (or
collection of invariants) gives a complete classification, in the sense that if two functions are not homotopic, then the invariant assigns different values to them. The
prototypical result of this sort, due to Hopf, asserts that the degree is a complete
invariant when N is the m-sphere.
Theorem 14.2.1 (Hopf). If M is an m-dimensional compact connected smooth
manifold, then two maps f, f ′ : M → S m are homotopic if and only if deg(f ) =
deg(f ′ ).
We provide a rather informal sketch of the proof. Since the ideas in the argument
are geometric, and easily visualized, this should be completely convincing, and little
would be gained by adding more formal details of particular constructions.
We already know that two homotopic functions have the same degree, so our
goal is to show that two functions of the same degree are homotopic. Consider a
particular f : M → S m . The results of Section 10.7 imply that CS (M, S m ) is locally
path connected, and that C ∞ (M, S m ) is dense in this space, so f is homotopic to a
smooth function. Suppose that f is smooth, and that q is a regular value of f . (The
existence of such a q follows from Sard’s theorem.) The inverse function theorem
implies that if D is a sufficiently small disk in S m centered at q, then f −1 (D) is a
collection of pairwise disjoint disks, each containing one element of f −1 (q).
Let q− be the antipode of q in S m . (This is −q when S m is the unit sphere
centered at the origin in Rm+1 .) Let j : S m × [0, 1] → S m be a homotopy with
j0 = IdS m that stretches D until it covers S m , so that j1 maps the boundary of D
and everything outside D to q− . Then f = j0 ◦ f is homotopic to j1 ◦ f .
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CHAPTER 14. TOPOLOGICAL CONSEQUENCES
We have shown that the f we started with is homotopic to a function with
the following description: there are finitely many pairwise disjoint disks in M,
everything outside the interiors of these disks is mapped to q− , and each disk is
mapped bijectively (except that all points in the boundary are mapped to q− ) to
S m . We shall leave the peculiarities of the case m = 1 to the reader: when m ≥ 2, it
is visually obvious that homotopies can be used to move these disks around freely,
so that two maps satisfying this description are homotopic if they have the same
number of disks mapped onto S m in an orientation preserving manner and the same
number of disks in which the mapping is orientation reversing.
The final step in the argument is to show that a disk in which the orientation
is positive and a disk in which the orientation is negative can be “cancelled,” so
that the map is homotopic to a map satisfying the dsecription above, but with
one fewer disk of each type. Repeating this cancellation, we eventually arrive at a
map in which the mapping is either orientation preserving in all disks or orientation
reversing in all disks. Thus any map is homotopic to a map of this form, and any two
such maps with the same number of disks of the same orientation are homotopic.
Since the number of disks is the absolute value of the degree, and the maps are
orientation preserving or orientation reversing according to whether the degree is
positive or negative, we conclude that maps of the same degree are homotopic.
For the cancellation step it is best to adopt a concrete model of the domain and
range. We will think of S m as the unit disk D m = { x ∈ Rm : kxk ≤ 1 } with the
boundary points identified with a single point, which will continue to be denoted
by q− . We will think of Rm as representing an open subset of M containing two
disks that are mapped with opposite orientation. Let e1 = (1, 0, . . . , 0) ∈ Rm . After
sliding the disks around, expanding or contracting them, and revising the maps on
their interiors, we can achieve the following specific f : Rm → S m :


kx − e1 k < 1,
x − e1 ,
f (x) = x − (−e1 ) − 2(hx, e1 i − h−e1 , e1 i)e1 , kx − (−e1 )k < 1,


q− ,
otherwise.
Visually, f maps the unit disk centered at e1 to S m preserving orientation, it maps
the unit disk centered at −e1 reversing orientation, and everything else goes to q− .
We now have the following homotopy:


kx − (1 − 2t)e1 k < 1 and x1 ≥ 0,
x − (1 − 2t)e1 ,
ht (x) = x − (1 − 2t)e1 − 2hx, e1 ie1 , kx − (1 − 2t)(−e1 )k < 1 and x1 ≤ 0,


q− ,
otherwise.
Of course the first two expressions agree when x1 = 0, so this is well defined and
continuous, and h1 (x) = q− for all x.
In preparation for an application of the Hopf theorem, we introduce an important
concept from topology. If X is a topological space and A ⊂ X, the pair (X, A)
has the homotopy!extension property if, for any topological space Y and any
function g : (X × {0}) ∪ (A × [0, 1]) → Y , there is a homotopy h : X × [0, 1] → Y
such that is an extension of g: h(x, 0) = g(x, 0) for all x ∈ X and h(x, t) = g(x, t)
for all (x, t) ∈ A × [0, 1].
14.2. THE HOPF THEOREM
199
Lemma 14.2.2. The pair (X, A) has the homotopy extension property if and only
if (X × {0}) ∪ (A × [0, 1]) is a retract of X × [0, 1].
Proof. If (X, A) has the homotopy extension property, then the inclusion map from
(X × {0}) ∪ (A × [0, 1]) to X × [0, 1] has a continuous extension to all of X × [0, 1],
which is to say that there is a retraction. On the other hand, if r is a retraction,
then for any g there is continuous extension h = g ◦ r.
We will only be concerned with the example given by the next result, but it
is worth noting that this concept takes on greater power when one realizes that
(X, A) has the homotopy extension property whenever X is a simplicial complex
and A is a subcomplex. It is easy to prove this if there is only one simplex σ in X
that is not in A; either the boundary of σ is contained in A, in which case there
is an argument like the proof of the following, or it isn’t, and another very simple
construction works. The general case follows from induction because if (X, A) and
(A, B) have the homotopy extension property, then so does (X, B). To show this
suppose that g : (X × {0}) ∪ (B × [0, 1]) → Y is given. There is a continuous
extension h : A × [0, 1] → Y of the restriction of g to (A × {0}) ∪ (B × [0, 1]). The
extension of h to all of (X × {0}) ∪ (A × [0, 1]) defined by setting h|X×{0} = g|X×{0}
is continuous because it is continuous on X × {0} and A × [0, 1], both of which
are closed subsets of X × [0, 1] (here the requirement that A is closed finally shows
up) and since (X, A) has the homotopy extension property this h can be further
extended to all of X × [0, 1].
Lemma 14.2.3. The pair (D m , S m−1 ) has the homotopy extension property.
Proof. There is an obvious retraction
r : D m × [0, 1] → (D m × {0}) ∪ (S m−1 × [0, 1])
defined by projecting radially from (0, 2) ∈ Rm × R.
We now relate the degree of a map from D m to Rm with what may be thought
of as the “winding number” of the restriction of the map to S m−1 .
Theorem 14.2.4. If f : D m → Rm is continuous, 0 ∈
/ f (S m−1 ), and f˜ : S m−1 →
m−1
S
is the function x 7→ f (x)/kf (x)k, then deg0 (f ) = deg(f˜).
Proof. For k ∈ Z let fk : D m → Rm be the map
(r cos θ, r sin θ, x3 , . . . , xm ) 7→ (r cos kθ, r sin kθ, x3 , . . . , xm ).
It is easy to see that deg0 (fk ) = k = deg(f |S m−1 ).
Now let k = deg(f˜). The Hopf theorem implies that there is a homotopy
h̃ : S m−1 × [0, 1] → S m−1 with h̃0 = f˜ and h̃1 = fk |S m−1 . Let h : S m−1 × [0, 1] → Rm
be the homotopy with h0 = f |S m−1 and h1 = fk |S m−1 given by
h(x, t) = (1 − t)kf (x)k + t h̃(x, t),
and extend this to g : (D m × {0}) ∪ (S m−1 × [0, 1]) → Rm by setting g(x, 0) = f (x).
The last result implies that g extends to a homotopy j : D m × [0, 1] → Rm . There
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CHAPTER 14. TOPOLOGICAL CONSEQUENCES
is an additional homotopy ℓ : D m × [0, 1] → Rm with ℓ0 = j1 and ℓ1 = fk given by
setting
ℓ(x, t) = (1 − t)j1 (x) + tfk (x).
Note that ℓt |S m−1 = fk |S m−1 for all t. The invariance of degree under degree admissible homotopy now implies that
deg(f˜) = k = deg0 (fk ) = deg0 (j1 ) = deg0 (j0 ) = deg0 (f ).
14.3
More on Maps Between Spheres
Insofar as spheres are the simplest “nontrivial” (where, in effect, this means
noncontractible) topological spaces, it is entirely natural that mathematicians would
quickly investigate the application of degree and index theory to these spaces, and
to maps between them. There are many results coming out of this research, some
of which are quite famous.
Our discussion combines some purely topological reasoning with analysis based
on concrete examples, and for the latter it is best to agree that
S m := { x ∈ Rm+1 : kxk = 1 }.
Some of our arguments involve induction on m, and for this purpose we will regard
S m−1 as a subset of S m by setting
S m−1 = { x ∈ S m : xm+1 = 0 }.
Let am : S m → S m be the function
am (x) = −x.
Two points x, y ∈ S m are said to be antipodal if y = am (x). Regarded topologically, am is a fixed point free local diffeomorphism whose composition with itself
is IdS m , and one should expect that all the topological results below involving am
and antipodal points should depend only on these properties, but we will not try to
demonstrate this (the subject is huge, and our coverage is cursory) instead treating
am as an entirely concrete object.
Let
Em = { (x, y) ∈ S m × S m : y 6= am (x) }.
There is a continuous function rm : Em × [0, 1] → S m given by
rm (x, y, t) :=
tx + (1 − t)y
.
ktx + (1 − t)yk
Proposition 14.3.1. Suppose f, f ′ : S m → S n are continuous. If they do not map
any point to a pair of antipodal points—that is, f ′ (p) 6= an (f (p)) for all p ∈ S m —
then f and f ′ are homotopic.
14.3. MORE ON MAPS BETWEEN SPHERES
201
Proof. Specifically, there is the homotopy h(x, t) = rn (f (x), f ′ (x), t).
Consider a continuous function f : S m → S n . If m < n, then f is homotopic
to a constant map, and thus rather uninteresting. To see this, first note that the
smooth functions are dense in C(S m , S n ), and a sufficiently nearby function does
not map any point to the antipode of its image under f , so f is homotopic to a
smooth function. So, suppose that f is smooth. By Sard’s theorem, the regular
values of f are dense, and since n > m, a regular value is a y ∈ S n with f −1 (y) = ∅.
We now have the homotopy h(x, t) = rn (f (x), an (y), t).
When m > n, on the other hand, the analysis of the homotopy classes of maps
from S m to S n is a very difficult topic that has been worked out for many specific
values of m and n, but not in general. We will only discuss the case of m = n, for
which the most basic question is the relation between the index and the degree.
Theorem 14.3.2. If f : S m → S m is continuous, then
Λ(f ) = 1 + (−1)m deg(f ).
Proof. Hopf’s theorem (Theorem 14.2.1) implies that two maps from S m to itself are
homotopic if they have the same degree, and the index is a homotopy invariant, so
if suffices to determine the relationship between the degree and index for a specific
instance of a map of each possible degree.
We begin with m = 1. For d ∈ Z let f1,d : S 1 → S 1 be the function
f1,d (cos θ, sin θ) := (cos dθ, sin dθ).
−1
If d > 0, then f1,d
(1, 0) consists of d points at which f1,d is orientation preserving,
when d = 0 there are points in S 1 that are not in the image of f1,0 , and if d > 0,
−1
then f1,d
(1, 0) consists of d points at which f1,d is orientation reversing. Therefore
deg(f1,d ) = d.
Now observe that f1,1 is homotopic to a map without fixed points, while for
d 6= 1 the fixed points of f1,d are the points
2πk
2πk
, sin d−1
(k = 0, . . . , d − 2).
cos d−1
If d > 1, then motion in the domain is translated by f1,d into more rapid motion
in the range, so the index of each fixed point is −1. When d < 1, f1,d translates
motion in the domain into motion in the opposite direction in the range, so the
index of each fixed point is 1. Combining these facts, we conclude that
Λ(f1,d ) = 1 − d,
which establishes the result when m = 1.
Let em+1 = (0, . . . , 0, 1) ∈ Rm+1 . Then
S m = { αx + βem+1 : x ∈ S m−1 , α ≥ 0, α2 + β 2 = 1 }.
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CHAPTER 14. TOPOLOGICAL CONSEQUENCES
We define fm,d inductively by the formula
fm,d αx + βem+1 = αfm−1,−d (x) − βem+1 .
If fm−1,−d is orientation preserving (reversing) at x ∈ S m−1 , then fm,d is clearly
orientation reversing (preserving) at x, so deg(fm,d ) = − deg(fm−1,−d ). Therefore,
by induction, deg(fm,d ) = d.
The fixed points of fm,d are evidently the fixed points of fm−1,−d . Fix such an
x. Computing in a local coordinate system, one may easily show that the index of
x, as a fixed point of fm,d , is the same as the index of x as a fixed point of fm−1,−d ,
so Λ(fm,d ) = Λ(fm−1,−d ). By induction,
Λ(fm,d ) = Λ(fm−1,−d ) = 1 + (−1)m−1 deg(fm−1,−d ) = 1 + (−1)m deg(fm,d ).
Corollary 14.3.3. If a map f : S m → S m has no fixed points, then deg(f ) =
(−1)m+1 . If f does not map any point to its antipode, which is to say that am ◦ f
has no fixed points, then deg(f ) = 1. Consequently, if f does not map any point
either to itself or its antipode, then m is odd.
Proof. The first claim follows from Λ(f ) = 0 and the result above. In particular,
am has no fixed points, so deg(am ) = (−1)m+1 . The second result now follows from
the multiplicative property of the degree of a composition (Corollary 12.4.2):
(−1)m+1 = deg(am ◦ f ) = deg(am ) · deg(f ) = (−1)m+1 deg(f ).
Proposition 14.3.4. If the map f : S m → S m never maps antipodal points to
antipodal points—that is, am (f (p)) 6= f (am (p)) for all p ∈ S m —then deg(f ) is
even. If m is even, then deg(f ) = 0.
Proof. The homotopy h : S m × [0, 1] → S m given by
h(p, t) := rm (f (p), f (am (p)), t)
shows that f and f ◦ am are homotopic, whence deg(f ) = deg(f ◦ am ). Corollary
12.4.2 and Corollary 14.3.3 give
deg(f ) = deg(f ◦ am ) = deg(f ) deg(am ) = (−1)m+1 deg(f ),
and when m is even it follows that deg(f ) = 0.
Since f is homotopic to a nearby smooth function, we may assume that it is
smooth, in which case each ht is also smooth. Sard’s theorem implies that each ht
has regular values, and since h1/2 = h1/2 ◦ am , any regular value of h1/2 has an even
number of preimages. The sum of an even number of elements of {1, −1} is even,
so it follows that deg(f ) = deg(h1/2 ) is even.
Combining this result with the first assertion of Corollary 14.3.3 gives a result
that was actually applied to the theory of general economic equilibrium by Hart
and Kuhn (1975):
14.3. MORE ON MAPS BETWEEN SPHERES
203
Corollary 14.3.5. Any map f : S m → S m either has a fixed point or a point p
such that f (am (p)) = am (f (p)).
Of course am extends to the map x 7→ −x from Rm+1 to itself, and in appropriate
contexts we will understand it in this sense. If D ⊂ Rm+1 satisfies am (D) = D, a
map f : D → Rn+1 is said to be antipodal if
f ◦ am |D = an ◦ f.
An antipodal map f : S m → S m induces a map from m-dimensional projective
space to itself. If you think about it for a bit, you should be able to see that a map
from m-dimensional projective space to itself is induced by such an f if and only if
it maps orientation reversing loops to orientation reversing loops.
The next result seems to be naturally paired with Proposition 14.3.4, but it is
actually much deeper.
Theorem 14.3.6. If a map f : S m → S m is antipodal, then its degree is odd.
Proof. There are smooth maps arbitrarily close to f . For such an f ′ the map
p 7→ rm (f ′ (p), −f ′ (−p), 21 )
is well defined, smooth, antipodal, and close to f , so it is homotopic to f and has
the same degree. Evidently it suffices to prove the claim with f replaced by this
map, so we may assume that f is smooth. Sard’s theorem implies that there is a
regular value of f , say q.
After rotating S m we may assume that q = (0, . . . , 0, 1) and −q = (0, . . . , 0, −1)
are the North and South poles of S m . We would like to assume that
(f −1 (q) ∪ f −1 (−q)) ∩ S m−1 = ∅,
and we can bring this about by replacing f with f ◦ h where h : S m → S m is an
antipodal diffeomorphism than perturbs neighborhoods of the points in f −1 (q) ∪
f −1 (−q) while leaving points far away from these points fixed. (Such an h can easily
be constructed using the methods of Section 10.2.)
Since a sum of numbers drawn from {−1, 1} is even or odd according to whether
the number of summands is even or odd, our goal reduces to showing that f −1 (q)
has an odd number of elements. When m = 0 this is established by considering the
two antipode preserving maps from S 0 to itself. Proceeding inductively, suppose
the result has been established when m is replaced by m − 1.
For p ∈ S m , p ∈ f −1 (q) if and only if −p ∈ f −1 (−q), because f is antipodal,
so the number of elements of f −1 (q) ∪ f −1 (−q) is twice the number of elements of
f −1 (q). Let
S+m := { p ∈ S m : pm+1 ≥ 0 } and S−m := { p ∈ S m : pm+1 ≤ 0 }
be the Northern and Southern hemispheres of S m . Then p ∈ S+m if and only if
−p ∈ S−m , so S+m contains half the elements of f −1 (q) ∪ f −1 (−q). Thus it suffices to
show that (f −1 (q) ∪ f −1 (−q)) ∩ S+m has an odd number of elements.
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CHAPTER 14. TOPOLOGICAL CONSEQUENCES
For ε > 0 consider the small open and closed disks
Dε := { p ∈ S m : pm+1 > 1 − ε } and D ε := { p ∈ S m : pm+1 ≥ 1 − ε }
centered at the North pole. Since f is antipode preserving, −q is also a regular value
of f . In view of the inverse function theorem, f −1 (Dε ∪ −D ε ) is a disjoint union
of diffeomorphic images of D ε , and none of these intersect S m−1 if ε is sufficiently
small. Concretely, for each p ∈ f −1 (q) ∪ f −1 (−q) the component C p of f −1 (D ε ∪
−D ε ) containing p is mapped diffeomorphically by f to either Dε or −D ε , and the
various C p are disjoint from each other and S m−1 . Therefore we wish to show that
f −1 (D ε ∪ −D ε ) ∩ S+m has an odd number of components.
Let M = S+m \ f −1 (Dε ∪ −Dε ). Clearly M is a compact m-dimensional smooth
∂-manifold. Each point in S m \ {q, −q} has a unique representation of the form
αy + βq where y ∈ S m−1 , 0 < α ≤ 1, and α2 + β 2 = 1. Let j : S m \ {q, −q} → S m−1
be the function j αy + βq := y, and let
g := j ◦ f |M : M → S m−1 .
Sard’s theorem implies that some q ∗ ∈ S m−1 is a regular value of both g and g|∂M .
Theorem 12.2.1 implies that degq∗ (g|∂M ) = 0, so (g|∂M )−1 (q ∗ ) has an even number of
elements. Evidently g maps the boundary of each Cp diffeomorphically onto S m−1 ,
so each such boundary contains exactly one element of (g|∂M )−1 (q ∗ ). In addition,
j maps antipodal points of S m \ {q, −q} to antipodal ponts of S m−1 , so g|S m−1 is
antipodal, and our induction hypothesis implies that (g|∂M )−1 (q ∗ ) ∩ S m−1 has an
odd number of elements. Therefore the number of components of f −1 (D ε ∪ −D ε )
contained in S+m is odd, as desired.
The hypotheses can be weakened:
Corollary 14.3.7. If the map f : S m → S m satisfies f (−p) 6= f (p) for all p, then
the degree of f is odd.
Proof. This will follow from the last result once we have shown that f is homotopic to an antipodal map. Let h : S m × [0, 1] → S m be the homotopy h(p, t) =
rm (f (p), −f (−p), 2t). The hypothesis implies that this is well defined, and h1 is
antipodal.
This result has a wealth of geometric consequences.
Theorem 14.3.8 (Borsuk-Ulam Theorem). The following are true:
(a) If f : S m → Rm is continuous, then there is a p ∈ S m such that f (p) =
f (am (p)).
(b) If f : S m → Rm is continuous and antipodal, then there is a p ∈ S m such that
f (p) = 0.
(c) There is no continuous antipodal f : S m → S m−1 .
(d) There is no continuous g : D m = { (y1, . . . , ym , 0) ∈ Rm+1 : kyk ≤ 1 } → S m−1
such that g|S m−1 is antipodal.
14.3. MORE ON MAPS BETWEEN SPHERES
205
(e) Any cover F1 , . . . , Fm+1 of S m by m + 1 closed sets has a least one set that
contains a pair of antipodal points.
(f ) Any cover U1 , . . . , Um+1 of S m by m + 1 open sets has a least one set that
contains a pair of antipodal points.
Proof. We think of Rm as S m with a point removed, so a continuous f : S m → Rm
amounts to a function from S m to itself that is not surjective, and whose degree is
consequently zero. Now (a) follows from the last result.
Suppose that f : S m → Rm is continuous and f (p) = f (−p). If f is also
antipodal, then f (−p) = −f (p) so f (p) = 0. Thus (a) implies (b).
Obviously (b) implies (c).
Let π : p 7→ (p1 , . . . , pm , 0) be the standard projection from Rm+1 to Rm . As
in the proof of Theorem 14.3.6 let S+m and S−m be the Northern and Southern
hemispheres of S m . If g : D m → S m−1 was continuous and antipodal, we could
define a continuous and antipodal f : S m → S m−1 by setting
(
g(π(p)),
p ∈ S+m ,
f (p) =
g(π(am (p))), p ∈ S−m .
Thus (c) implies (d).
Suppose that F1 , . . . , Fm+1 is a cover of S m by closed sets. Define f : S m → Rm
by setting
f (p) = d(x, F1 ), . . . , d(x, Fm )
where d(x, x′ ) = kx − x′ k is the usual metric for Rm+1 . Suppose that f (p) =
f (−p) = y. If yi = 0, then p, −p ∈ Fi , and if all the components of y are nonzero,
then p, −p ∈ Fm+1 . Thus (a) implies (e).
Suppose U1 , . . . , Um+1 is a cover of S m by open sets and ε > 0. For i = 1, . . . , m+
1 set Fi := { p ∈ S m : d(p, S m \ Ui ) ≥ ε }. Then each Fi is a closed subset of Ui ,
and these sets cover S m if ε is sufficiently small. Thus (e) implies (f).
In the argument above we showed that (a) ⇒ (b) ⇒ (c) ⇒ (d) and (a) ⇒ (e) ⇒
(f). There are also easy arguments for the implications (d) ⇒ (c) ⇒ (b) ⇒ (a) and
(f) ⇒ (e) ⇒ (c), so (a)-(f) are equivalent in the sense of each being an elementary
consequence of each other. The proofs that (d) ⇒ (c) and (c) ⇒ (b) are obvious
and can be safely left to the reader. To show that (b) ⇒ (a), for a given continuous
f : S m → Rm we apply (b) to f − f ◦ am . To show that (f) ⇒ (e) observe that if
F1 , . . . , Fm+1 are closed and cover S m , then for each n the sets U1/n (Fi ) are open
and cover S m , so there is a pn with pn , −pn ∈ U1/n (Fi ) for some i. Any limit point
of the sequence {pn } has the desired property.
The proof that (e) ⇒ (c) is more interesting. Consider an m-simplex that is
embedded in D m with the origin in its interior. Let F1 , . . . , Fm+1 be the radial
projections of the facets of the simplex onto S m−1 . These sets are closed and cover
S m−1 , and since each facet is separated from the origin by a hyperplane, each Fi
does not contain an antipodal pair of points. If f : S m → S m−1 is continuous,
then f −1 (F1 ), . . . , f −1 (Fm+1 ) are a cover of S m by closed sets, and (e) implies the
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CHAPTER 14. TOPOLOGICAL CONSEQUENCES
existence of p, −p ∈ f −1 (Fi ) for some i. If f was also antipodal, then f (p), f (−p) =
−f (p) ∈ Fi , which is impossible.
As a consequence of the Borsuk-Ulam theorem, the following “obvious” fact is
actually highly nontrivial.
Theorem 14.3.9. Spheres of different dimensions are not homeomorphic.
Proof. If k < m then, since S k can be embedded in Rm , part (a) of the Borsuk-Ulam
theorem implies that a continuous function from S m to S k cannot be injective.
14.4
Invariance of Domain
The main result of this section, invariance of domain, is a famous result with
numerous applications. It can be thought of as a purely topological version of the
inverse function theorem. However, before that we give an important consequences
of the Borsuk-Ulam theorem for Euclidean spaces.
Theorem 14.4.1. Euclidean spaces of different dimensions are not homeomorphic.
Proof. If k 6= m and f : Rk → Rm was a homeomorphism, for any sequence {xj } in
Rk with {xj } → ∞ the sequence {f (xj )} could not have a convergent subsequence,
so kf (xj )k → ∞. Identifying Rk and Rm with S k \ {ptk } and S m \ {ptm }, the
extension of f to S k given by setting f (ptk ) = ptm would be continuous, with a
continuous inverse, contrary to the last result.
The next two lemmas develop the proof of this section’s main result.
Lemma 14.4.2. Suppose S+m is the Northern hemisphere of S m , f : S+m → S m
is a map such that f |S m−1 is antipodal, and p ∈ S+m \ S m−1 is a point such that
−p ∈
/ f (S+m ) and p ∈
/ f (S m−1 ). Then degp (f ) is odd.
Proof. Let f˜ : S m → S m be the extension of f given by setting f˜(p) = −f (−p)
when pm+1 < 0. Clearly f˜ is continuous and antipodal, so its degree is odd. The
hypotheses imply that f˜−1 (p) ⊂ S+m \ S m−1 , and that f is degree admissible over p,
so Additivity implies that degp (f ) = degp (f˜).
Lemma 14.4.3. If f : D m → Rm is injective, then degf (0) (f ) is odd, and f (D m )
includes a neighborhood of f (0).
Proof. Replacing f with x 7→ f (x) − f (0), we may assume that f (0) = 0. Let
h : D m × [0, 1] → Rm be the homotopy
x
h(x, t) := f ( 1+t
) − f ( −tx
).
1+t
Of course h0 = f and h1 is antipodal. If ht (x) = 0 then, because f is injective,
x = −tx, so that x = 0. Therefore h is a degree admissible homotopy over zero, so
deg0 (h0 ) = deg0 (h1 ), and the last result implies that deg0 (h1 ) is odd, so deg0 (h0 ) =
deg0 (f ) is odd. The Continuity property of the degree implies that degy (f ) is odd
for all y in some neighborhood of f (0). Since, by Additivity, degy (f ) = 0 whenever
y∈
/ f (D m ), we conclude that f (D m ) contains a neighborhood of 0.
14.5. ESSENTIAL SETS REVISITED
207
The next result is quite famous, being commonly regarded as one of the major
accomplishments of algebraic topology. As the elementary nature of the assertion
suggests, it is applied quite frequently.
Theorem 14.4.4 (Invariance of Domain). If U ⊂ Rm is open and f : U → Rm
is continuous and injective, then f (U) is open and f is a homeomorphism onto its
image.
Proof. The last result can be applied to a closed disk surrounding any point in the
domain, so for any open V ⊂ U, f (V ) is open. Thus f −1 is continuous.
14.5
Essential Sets Revisited
Let X be a compact ANR, let C ⊂ X be compact, and let f : C → X be an
index admissible function. If Λ(f ) 6= 0, then the set of fixed points is essential.
What about a converse? More specifically, if Λ(f ) = 0, then of course f may have
fixed points, but is f necessarily index admissible homotopic to a function without
fixed points? When C = X, so that Λ(f ) = L(f ), this question amounts to a
request for conditions under which a converse of the Lefschetz fixed point theorem
holds. We can also ask whether a somewhat more demanding condition holds: does
every neighborhood of f in C(C, X) contains a function without any fixed points?
If C is not connected, the answers to these questions are obtained by combining
the answers obtained when this question is applied to the restrictions of f to the
various connected components of C, so we should assume that C is connected. If
C1 , . . . , Cr are pairwise disjointP
subsets of C with F P(f ) contained in the interior
of C1 ∪ . . . ∪ Cr , then Λ(f ) = i Λ(f |Ci ), and of course when r > 1 it can easily
happen that Λ(f |Ci ) 6= 0 for some i even though the sum is zero. Therefore we
should assume that F P(f ) is also connected.
Our goal is to develop conditions under which a connected set of fixed points
with index zero can be “perturbed away,” in the sense that there is a nearby function
or correspondence with no fixed points near that set. Without additional assumptions, there is little hope of achieving positive answers. For the general situation in
which the space is an ANR, the techniques we develop below would lead eventually
to composing a perturbation with a retraction, and it is difficult to prevent the
retraction from introducing undesired fixed points. An approach to this issue for
simplicial complexes is developed in Ch. VIII of Brown (1971).
Our attention is restricted to the following settings: a) X is a “well behaved”
subset of a smooth manifold; b) X is a compact convex subset of a Euclidean space.
The gist of the argument used to prove these results is to first approximate with a
smooth function that has only regular fixed point, which are necessarily finite and
can be organized in pairs of opposite index, then perturb to eliminate each pair.
Proposition 14.5.1. If g : D m → Rm is continuous, 0 ∈
/ g(S m−1 ), and deg0 (g) =
0, then there is a continuous ĝ : D m → Rm \ {0} with ĝ|S m−1 = g|S m−1 .
Proof. Let g̃ : S m−1 → S m−1 be the function g̃(x) = g(x)/kg(x)k. Theorem 14.2.4
implies that deg(g̃) = 0, so the Hopf theorem implies that there is a homotopy
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CHAPTER 14. TOPOLOGICAL CONSEQUENCES
h : S m−1 × [0, 1] → S m−1 with h0 = g̃ and h1 a constant function. For (x, t) ∈
S m−1 × [0, 1] we set
ĝ(tx) = tkg(x)k + (1 − t) h1−t (x).
If (xr , tr ) is a sequence with tr → 0, then g(tr , xr ) converges to the constant value
of h1 , so this is well defined and continuous. For x ∈ S m−1 we have g(x) 6= 0, so
the origin is not in the image of ĝ, and ĝ(x) = kg(x)kh0 (x) = g(x).
The first of this section’s principal results is as follows.
Theorem 14.5.2. Let M be a smooth C r manifold, where 2 ≤ r ≤ ∞, let X ⊂ M
be a compact ANR for which there is a homotopy h : X × [0, 1] → X such that
h0 = IdX and, for each t > 0, ht : X → ht (X) is a homeomorphism whose image
ht (X) is contained in the interior of X. Let C be a compact subset of X, and let
f : C → X be an index admissible function. If F P(f ) is connected and Λ(f ) = 0,
then F P(f ) is an inessential set of fixed points.
Proof. It suffices to show that for a given open W ⊂ C × X containing the graph
of f there is a continuous f ′ : C → X with F P(f ′) = ∅. We have Gr(ht ◦ f ) ⊂ W
for small t > 0, so it suffices to prove the result with f replaced by ht ◦ f , which
means that we may assume that the image of f is contained in the interior of X.
Recall that Proposition 10.7.8 gives a continuous function λ : M → (0, ∞) and
a C r−1 function κ : Vλ → M, where Vλ = { (p, v) ∈ RM : kvk < λ(p) }, such that
κ(p, 0) = p for all p ∈ M and
κ̃ = π × κ : Vλ → M × M
is a C r−1 embedding, where π : T M → M is the projection. Let Ṽλ = κ̃(Vλ ).
Let Y0 = { p ∈ C : (p, f (p)) ∈ Ṽλ }; of course this is an open set containing
F P(f ). Let Y1 and Y2 be open sets such that F P(f ) ⊂ Y2 , Y 2 ⊂ Y1 , Y 1 ⊂ Y0 ,
and Y2 is path connected. (Such a Y2 can be constructed by taking a finite union
of images of D m under C r parameterizations.) We can define a vector field ζ on a
neighborhood of Y0 by setting
ζ(p) = κ̃−1 (p, f (p)).
Proposition 11.5.2 and Corollary 10.2.5 combine to imply that there is a vector field
ζ̃ on Y0 with image contained in κ̃−1 (W ) that agrees with ζ on Y0 \ Y1 , is C r−1 on
Y2 , and has only regular equilibria, all of which are in Y2 . The number of equilibria
is necessarily finite, and we may assume that, among all the vector fields on Y0 that
agree with ζ on Y0 \ Y1 , are C r−1 on Y2 , and have only regular equilibria in Y2 , ζ̃
minimizes this number. If ζ̃ has no equilibria, then we may define a continuous
function f˜ : C → X without any fixed points whose graph is contained in W by
setting f˜(p) = κ(ζ̃(p)) if p ∈ Y0 and setting f˜(p) = f (p) otherwise.
Aiming at a contradiction, suppose that ζ̃ has equilibria. Since the index is
zero, there must be two equilibria of opposite index, say p0 and p1 , and it suffices
to show that we can further perturb ζ̃ in a way that eliminates both of them.
There is a C r embedding γ : (−ε, 1 + ε) → Upp with γ(0) = p0 and γ(1) = p1 .
14.5. ESSENTIAL SETS REVISITED
209
(This is obvious, but painful to prove formally, and in addition the case m = 1
requires special treatment. A formal verification would do little to improve the
reader’s understanding, so we omit the details.) Applying the tubular neighborhood
theorem, this path can be used to construct a C r parameterization ϕ : Z → U where
Z ⊂ Rm is a a neighborhood of D m .
Let g : Z → Rm be defined by setting g(x) = Dϕ(x)−1 ζϕ(x) . Proposition 14.5.1
gives a continuous function ĝ : Z → Rm \ {0} that agrees with g on the closure of
Z \ D m . We extend ĝ to all of Z by setting ĝ(x) = g(x) if x ∈
/ D m . Define a new
vector field ζ̂ on ϕ(Z) by setting
ζ̂(p) = Dϕ(ϕ−1 (p))ĝ(ϕ−1 (p)).
There are two final technical points. In order to insure that ζ(p) ∈ κ̃−1 (W ) for
all p we can first multiplying ĝ by a C r function β : D m → (0, 1] that is identically
1 on Z \ D m and close to zero in the interior of D m outside of some neighborhood of
S m−1 . We can also use Proposition 11.5.2 and Corollary 10.2.5 to further perturb ζ̂
to make is C r−1 without introducing any additional equilibria. This completes the
construction, thereby arriving at a contradiction that completes the proof.
Economic applications call for a version of the result for correspondences. Ideally
one would like to encompass contractible valued correspondences in the setting of
a manifold, but the methods used here are not suitable. Instead we are restricted
to convex valued correspondences, and thus to settings where convexity is defined.
Theorem 14.5.3. If X ⊂ Rm is compact and convex, C ⊂ X is compact, F :
C → X is an index admissible upper semicontinuous convex valued correspondence,
Λ(F ) = 0, and F P(F ) is connected, then F is inessential.
Caution: The analogous result does not hold for essential sets of Nash equilibria,
which are defined by Jiang (1963) in terms of perturbations of the game’s payoffs.
Hauk and Hurkens (2002) give an example of a game with a component of the set
of Nash equilibria that has index zero but is robust with respect to perturbations
of payoffs.
Proof. Let W ⊂ C ×X be an open set containing the graph of F . We will show that
there is a continuous f : C → X with Gr(f ) ⊂ W and F P(f ) = ∅. Let x0 be a point
in the interior of X, let h : X ×[0, 1] → X be the contraction h(x, t) = (1−t)x+tx0 ,
and for t ∈ [0, 1] let ht ◦F be the correspondence x 7→ ht (F (x)). This correspondence
is obviously upper semicontinuous and convex valued, and Gr(ht ◦F ) ⊂ W for small
t > 0, so it suffices to prove the result with F replaced by ht ◦F for such t. Therefore
we may assume that the image of F is contained in the interior of X.
For each x ∈ F P(F ) we choose convex neighborhoods Yx ⊂ C of x and Zx ⊂ X
of F (x) such that Yx ⊂ Zx and Yx ×Zx ⊂ W . Choose x1 , . . . , xk such that F P(F ) ⊂
Yx1 ∪ . . . ∪ Yxk , and let
Y 0 = Y x1 ∪ . . . ∪ Y xk
and Z0 = (Yx1 × Zx1 ) ∪ . . . ∪ (Yxk × Zxk ).
Note that for all (x, y) ∈ Z0 , Z0 contains the line segment { (x, (1 − t)y + tx) }. Let
Y1 and Y2 be open subsets of C with F P(F ) ⊂ Y2 , Y 2 ⊂ Y1 , Y 1 ⊂ Y0 , and Y2 is
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CHAPTER 14. TOPOLOGICAL CONSEQUENCES
path connected. Let α : C → [0, 1] be a C ∞ function that is identically one on Y 2
and identically zero on C \ Y1 .
Let
W0 = Z0 ∪ (W ∩ ((C \ Y1 ) × X)) \ { (x, x) : x ∈ C \ Y2 }.
This is an open set that contains the graph of F , so Proposition 10.2.7 implies that
there is a C ∞ function f : C → X with Gr(f ) ⊂ W0 that has only regular fixed
points. We assume that among all functions with these properties, f is minimal for
the number of fixed points.
There is some ε > 0 such that {x} × Uε (x) ⊂ Z0 for all x ∈ Y 2 . For any
δ ∈ (0, 1] the function
f ′ : x 7→ (1 − α(x))f (x) + α(x + δ(f (x) − x))
is C ∞ , its graph is contained in W0 , and it has only regular fixed points. If δ > 0
is sufficiently small, then f ′ (x) ∈ Uε (x) for all x ∈ Y 2 . Therefore we may assume
that f (x) ∈ Uε (x) for all x ∈ Y 2 .
Define a function ζ : Y2 → Rm by setting ζ(x) = f (x) − x. Aiming at a
contradiction, suppose that ζ has zeros. Since the Λ(f ) = 0, there must be two zeros
of opposite index, say x0 and x1 . As in the last proof, there is a C r embedding γ :
(−ε, 1 + ε) → Y2 with γ(0) = x0 and γ(1) = x1 . Applying the tubular neighborhood
theorem, this path can be used to construct a C ∞ parameterization ϕ : T → Y2
where T ⊂ Rm is a neighborhood of D m .
Let g : T → Rm be defined by setting g(x) = Dϕ(x)−1 ζϕ(x) . Proposition 14.5.1
gives a continuous function ĝ : T → Rm \ {0} that agrees with g on the closure of
T \ D m . We extend ĝ to all of T by setting ĝ(x) = g(x) if x ∈
/ D m . Define a new
vector field ζ̂ on ϕ(T ) by setting
ζ̂(p) = Dϕ(ϕ−1 (p))ĝ(ϕ−1 (p)).
There are two final technical points. In order to insure that kζ̂(x)k < ε for all
p we can first multiply ĝ by a C ∞ function β : D m → (0, 1] that is identically 1
on T \ D m and close to zero in the interior of D m outside of some neighborhood of
S m−1 . We can also use Proposition 11.5.2 and Corollary 10.2.5 to further perturb
ζ̂ to make it C ∞ without introducing any additional zeros. We can now define
a function f ′ : C → X by setting f (x) = x + ζ̂(x) if x ∈ T and f ′ (x) = f (x)
otherwise. Since f ′ has all the properties of f , and two fewer fixed points, this is a
contradiction, and the proof is complete.
Chapter 15
Vector Fields and their Equilibria
Under mild technical conditions, explained in Sections 15.1 and 15.2, a vector
field ζ on a manifold M determines a dynamical system. That is, there is a function
Φ : W → M, where W ⊂ M × R is a neighborhood of W × {0}, such that the
derivative of Φ at (p, t) ∈ W , with respect to time, is ζΦ(p,t) . In this final chapter
we develop the relationship between the fixed point index and the stability of rest
points, and sets of rest points, of such a dynamical system.
In addition to the degree and the fixed point index, there is a third expression of
the underlying mathematical principle for vector fields. In Section 15.3 we present
an axiomatic description of the vector field index, paralleling our axiom systems
for the degree and fixed point index. Existence and uniqueness are established by
showing that the vector field index of ζ|C , for suitable compact C ⊂ M, agrees
with the fixed point index of Φ(·, t)|C for small negative t. Since we are primarily
interested in forward stability, it is more to the point to say that the fixed point
index of Φ(·, t)|C for small positive t agrees with the vector field index of −ζ|C .
The notion of stability we focus on, asymptotic stability, has a rather complicated definition, but the intuition is simple: a compact set A is asymptotically
stable if the trajectory of each point in some neighborhood of A is eventually drawn
into, and remains inside, arbitrarily small neighborhoods of A. In order to use the
fixed point index to study stability, we need to find some neighborhood of such an A
that is mapped into itself by Φ(·, t) for small positive t. The tool we use to achieve
this is the converse Lyapunov theorem, which asserts that if A is asymptotically
stable, then there is a Lyapunov function for ζ that is defined on a neighborhood of
A. Unlike the better known Lyapunov theorem, which asserts that the existence of
a Lyapunov function implies asymptotic stability, the converse Lyapunov theorem
is a more recent and difficult result. We prove a version of it that is sufficient for
our needs in Section 15.5.
Once all this background material is in place, it will not take long to prove the
culminating result, that if A is a asymptotically stable, and an ANR, then the vector
field index of −ζ is the Euler characteristic of A. This was proved in the context of
a game theoretic model by Demichelis and Ritzberger (2003). The special case of A
being a singleton is a prominent result in the theory of dynamical systems, due to
Krasnosel’ski and Zabreiko (1984): if an isolated rest point is asymptotically stable
for ζ, then the vector field index of that point for −ζ is 1.
211
212
CHAPTER 15. VECTOR FIELDS AND THEIR EQUILIBRIA
Paul Samuelson advocated a “correspondence principle” in two papers Samuelson (1941, 1942) and his famous book Foundations of Economic Analysis Samuelson
(1947). The idea is that the stability of an economic equilibrium, with respect to
natural dynamics of adjustment to equilibrium, implies certain qualitative properties of the equilibrium’s comparative statics. There are 1-dimensional settings
in which this idea is regarded as natural and compelling, but Samuelson’s writings discuss many examples without formulating it as a general theorem, and its
nature and status in higher dimensions has not been well understood; Echenique
(2008) provides a concise summary of the state of knowledge and related literature.
The book concludes with an explanation of how the Krasnosel’ski-Zabreiko theorem
allows the correspondence principle to be formulated in a precise and general way.
15.1
Euclidean Dynamical Systems
We begin with a review of the theory of ordinary differential equations in Euclidean space. Let U ⊂ Rm be open, and let z : U → Rm be a function, thought
of as a vector field. A trajectory of z is a C 1 function γ : (a, b) → U such that
γ ′ (s) = zγ(s) for all s. Without additional assumptions the dynamics associated
with z need not be deterministic: there can be more than one trajectory for the
vector field satisfying an initial condition that specifies the position of the trajectory
at a particular moment. For example, suppose that m = 1, U = R, and
(
0,
t ≤ 0,
√
z(t) =
2 t, t > 0.
Then for any s0 there is a trajectory γs0 : R → M given by
(
0,
s ≤ s0 ,
γs0 (s) =
2
(s − s0 ) , s > s0 .
For most purposes this sort of indeterminacy is unsatisfactory, so we need to find
a condition that implies that for any initial condition there is a unique trajectory.
Let (X, d) and (X ′ , d′ ) be metric spaces. A function f : X → X ′ is Lipshitz if
there is a constant L > 0 such that
d′ (f (x), f (y)) ≤ Ld(x, y)
for all x, y ∈ X. We say that f is locally Lipschitz if each x ∈ X has a neighborhood U such that f |U is Lipschitz. The basic existence-uniqueness result for
ordinary differential equations is:
Theorem 15.1.1 (Picard-Lindelöf Theorem). Suppose that U ⊂ Rm is open, z :
U → Rm is locally Lipschitz, and C ⊂ U is compact. Then for sufficiently small
ε > 0 there is a unique function F : C × (−ε, ε) → U such that for each x ∈ C,
F (x, 0) = x and F (x, ·) is a trajectory of z. In addition F is continuous, and if z
is C s (1 ≤ s ≤ ∞) then so is F .
213
15.2. DYNAMICS ON A MANIFOLD
Due to its fundamental character, a detailed proof would be out of place here,
but we will briefly describe the central ideas of two methods. First, for any ∆ > 0
one can define a piecewise linear approximate solution going forward in time by
setting F∆ (x, 0) = x and inductively applying the equation
F∆ (x, t) = F∆ (x, k∆) + (t − k∆) · z(F∆ (x, k∆)) for k∆ < t ≤ (k + 1)∆.
Concrete calculations show that this collection of functions has a limit as ∆ → 0,
that this limit is continuous and satisfies the differential equation (∗), and also
that any solution of (∗) is a limit of this collection. These calculations give precise information concerning the accuracy of the numerical scheme for computing
approximate solutions described by this approach.
The second proof scheme uses a fixed point theorem. It considers the mapping
F 7→ F̃ given by the equation
F̃ (x, t) = x +
Z
t
z(F (x, s)) ds.
0
This defines a function from C(C × [−ε, ε], U) to C(C × [−ε, ε], Rm ). As usual,
the range is endowed with the supremum norm. A calculation shows that if ε is
sufficiently small, then the restriction of this function to a certain neighborhood
of the function (x, t) 7→ x is actually a contraction. Since C(C × [−ε, ε], Rm ) is a
complete metric space, the contraction mapping theorem gives a unique fixed point.
Additional details can be found in Chapter 5 of Spivak (1979) and Chapter 8 of
Hirsch and Smale (1974).
15.2
Dynamics on a Manifold
Throughout this chapter we will work with a fixed order of differentiability
2 ≤ r ≤ ∞ and an m-dimensional C r manifold M ⊂ Rk . Recall that if S is a
subset of M, a vector field on S is a continuous function ζ : S → T M such
that π ◦ ζ = IdS , where π : T M → M is the projection (p, v) 7→ p. We write
ζ(p) = (p, ζp ), so that ζ is thought of as “attaching” a tangent vector ζp to each
p ∈ S, in a continuous manner. A trajectory of ζ is a C 1 function γ : (a, b) → S
such that γ ′ (s) = ζγ(s) for all s.
We wish to transport the Picard-Lindelöf theorem to M. To this end, we study
how vector fields and their associated dynamical systems are transformed by changes
of coordinates. In addition to the vector field ζ on M, suppose that N ⊂ Rℓ is a
second C r manifold and h : M → N is a C r diffeomorphism. Let η be the vector
field on N defined by
ηq = Dh(h−1 (q))ζh−1(q) .
(∗)
This formula preserves the dynamics:
Lemma 15.2.1. The curve γ : (a, b) → M is a trajectory of ζ if and only if h ◦ γ
is a trajectory of η.
214
CHAPTER 15. VECTOR FIELDS AND THEIR EQUILIBRIA
Proof. For each s the chain rule gives
(h ◦ γ)′ (s) = Dh(γ(s))γ ′ (s)
and Dh(γ(s)) is a linear isomorphism because h is a diffeomorphism.
In our main application of this result N will be an open subset of Rm , to which
Theorem 15.1.1 can be applied. The given ζ will be locally Lipschitz, and it should
follow that η is also locally Lipschitz. Insofar as M is a subset of Rk , T M inherits a
metric, which gives meaning to the assumption that ζ is locally Lipschitz, but this
is a technical artifice, and it would be troubling if our concepts depended on this
metric in an important way. One consequence of the results below is that different
embeddings of M in a Euclidean space give rise to the same class of locally Lipschitz
vector fields.
Lemma 15.2.2. If U ⊂ Rm is open and f : U → Rn is C 1 , then f is locally
Lipschitz.
Proof. Consider a point x ∈ U. There is an ε > 0 such that the closed ball B of
radius ε centered at x is contained in U. Let L := maxy∈B kDf (y)k. Since B is
convex, for any y, z, ∈ B we have
Z 1
kf (z) − f (y)k = k
Df (y + t(z − y))(z − y) dtk
0
≤
Z
0
1
kDf (y + t(z − y))k · kz − yk dt ≤ Lkz − yk.
Lemma 15.2.3. A composition of two Lipschitz functions is Lipschitz, and a composition of two locally Lipschitz functions is locally Lipschitz.
Proof. Suppose that f : X → X ′ is Lipschitz, with Lipschitz constant L, that
(X ′′ , d′′ ) is a third metric space, and that g : X ′ → X ′′ is Lipschitz with Lipschitz
constant M. Then
d′′ (g(f (x)), g(f (y))) ≤ Md′ (f (x), f (y)) ≤ LMd(x, y)
for all x, y ∈ X, so g ◦ f is Lipschitz with Lipschitz constant LM.
Now suppose that f and g are only locally Lipschitz. For any x ∈ X there is a
neighborhood U of x such that f |U is Lipschitz and a neighborhood V of f (x) such
that g|V is Lipschitz. Then f |U ∩f −1 (V ) is Lipschitz, and, by continuity, U ∩ f −1 (V )
is a neighborhood of x. Thus g ◦ f is locally Lipschitz.
In preparation for the next result we note the following immediate consequences
of equation (∗):
Dh(p)ζp = ηh(p)
and Dh−1 (q)ηq = ζh−1 (q)
for all p ∈ M and q ∈ h(M). We also note that for everything we have done up to
this point it is enough that r ≥ 1, but the following result depends on r being at
least 2.
15.2. DYNAMICS ON A MANIFOLD
215
Lemma 15.2.4. ζ is locally Lipschitz if and only if η is locally Lipschitz.
Proof. Suppose that ζ is locally Lipschitz. For any p, p′ ∈ M we have
kηh(p) − ηh(p′ ) k = kDh(p)(ζp − ζp′ ) + (Dh(p) − Dh(p′ ))ζp′ k
≤ kDh(p)k · kζp − ζp′ k + kDh(p) − Dh(p′ ))k · kζp′ k.
Any p0 ∈ M has a neighborhood M0 ⊂ M such that ζ|M0 is Lipschitz, say with
Lipschitz constant L1 , and there are constants C1 , C2 > 0 such that kDh(p)k ≤ C1
and kζp k ≤ C2 for all p ∈ M0 . Since h is C 2 , Dh is C 1 and consequently locally
Lipschitz, so we can choose M0 such that Dh|M0 is Lipschitz, say with Lipschitz
constant L2 . Then η ◦ h|M0 is Lipschitz with Lipschitz constant C1 L1 + C2 L2 .
Now suppose that η is locally Lipschitz. By the definition of a C r function,
there is an open W ⊂ Rk containing h(M) and a C r function Ψ : W → Rm whose
restriction to h(M) is h−1 . Replacing W with Ψ−1 (M), we may assume that the
image of Ψ is contained in M. We extend η to W be setting η = η ◦ h ◦ Ψ. This is
locally Lipschitz because η is locally Lipschitz and h ◦ Ψ is C 1 . The remainder of
the proof follows the pattern of the first part, with ζ in place of η, η in place of ζ,
and Ψ in place of h.
With the preparations complete, we can now place the Picard-Lindelöf theorem
in a general setting.
Theorem 15.2.5. Suppose that ζ is locally Lipschitz and C ⊂ M is compact. Then
for sufficiently small ε > 0 there is a unique function Φ : C × (−ε, ε) → U such
that for all p ∈ C, Φ(p, 0) = p and Φ(p, t) is a trajectory for ζ. In addition Φ is
continuous, and if ζ is C s (1 ≤ s ≤ r) then so is Φ.
Proof. We can cover C with the interiors of a finite collection K1 , . . . , Kr of compact
subsets, each of which is contained in the image of some C r parameterization ϕi :
Ui → M. For each i let zi be the vector field on Ui derived from ζ and ϕ−1
i , as
−1
described above, and let Fi : ϕ (Ki ) × (−εi , εi ) → Ui be the function given by
Theorem 15.1.1. Then the function Φi : Ki × (−εi , εi ) → M given by
Φi (p, t) = ϕi (Fi (ϕ−1
i (p), t))
inherits the continuity and smoothness properties of Fi , and for each p ∈ Ki ,
Φi (p, 0) = p and Φi (p, ·) is a trajectory for ζ. If ε ≤ min{ε1, . . . , εr }, then we
must have Φ(p, t) = Φi (p, t) whenever p ∈ Ki , so Φ is unique if it exists. In fact
Φ unambiguously defined by this condition: if p ∈ Ki ∩ Kj , then ϕ−1
i ◦ Φj (p, ·) is
−1
trajectory for zi , so it agrees with Fi (ϕi (p), ·), and thus Φi (p, ·) and Φj (p, ·) agree
on (−ε, ε).
Taking a union of the interiors of the sets C × (−ε, ε) gives an open W ′ ∈ M × R
such that:
(a) for each p, { t : (p, t) ∈ W ′ } is an interval containing 0;
(b) there is a unique function Φ′ : C × (−ε, ε) → U such that for all p ∈ C,
Φ′ (p, 0) = p and Φ′ (p, ·) is a trajectory for ζ.
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CHAPTER 15. VECTOR FIELDS AND THEIR EQUILIBRIA
If W ′′ and Φ′′ is a second pair with these properties, then W ′ ∪ W ′′ satisfies (a), and
uniqueness implies that Φ′ and Φ′′ agree on W ′ ∩ W ′′ , so the function on W ′ ∪ W ′′
that agrees with Φ′ on W ′ and with Φ′′ on W ′′ satisfies (b). In fact his logic extends
to any, possibly infinite, collection of pairs. Applying it to the collection of all such
pairs shows that there is a maximal W satisfying (a), called the flow domain of
ζ, such that there is a unique Φ : W → M satisfying (b), which is called the flow
of ζ. Since the flow agrees, in a neighborhood of any point, with a function derived
(by change of time) from one of those given by Theorem 15.2.5, it is continuous,
and it is C s (1 ≤ s ≤ r) if ζ is C s .
The vector field ζ is said to be complete if W = M × R. When this is the case
each Φ(·, t) : M → M is a homeomorphism (or C s diffeomorphism is ζ is C s ) with
inverse Φ(·, −t), and t 7→ Φ(·, t) is a homomorphism from R (thought of as a group)
to the space of homeomorphisms (or C s diffeomorphisms) between M and itself.
It is important to understand that when ζ is not complete, it is because there
are trajectories that “go to ∞ in finite time.” One way of making this rigorous is
to define the notion of “going to ∞” as a matter of eventually being outside any
compact set. Suppose that Ip = (a, b), where b < ∞, and C ⊂ M is compact. If we
had Φ(p, tn ) ∈ C for all n, where {tn } is a sequence in (a, b) converging to b, then
after passing to a subsequence we would have Φ(p, tn ) → q for some q ∈ C, and we
could used the method of the last proof to show that (p, b) ∈ W .
15.3
The Vector Field Index
If S ⊂ M and ζ is a vector field on S, an equilibrium of ζ is a point p ∈ S
such that ζ(p) = 0 ∈ Tp M. Intuitively, an equilibrium is a rest point of the
dynamical system defined by ζ in the sense that the constant function with value p
is a trajectory.
The axiomatic description of the vector field index resembles the corresponding
descriptions of the degree and the fixed point index. If C ⊂ M is compact, a
continuous vector field ζ on C is index admissible if it has no equilibria in ∂C.
(As before, int C is the topological interior of C, and ∂C = C \int C is its topological
boundary.) Let V(M) be the set of index admissible vector fields ζ : C → T M where
C is compact.
Definition 15.3.1. A vector field index for M is a function ind : V(M) → Z,
ζ 7→ ind(ζ), satisfying:
(V1) ind(ζ) = 1 for all ζ ∈ V(M) with domain C such that there is a C r parameterization ϕ : V → M with C ⊂ ϕ(V ), ϕ−1 (C) = D m , and Dϕ(x)−1 ζϕ(x) = x
for all x ∈ D m = { x ∈ Rm : kxk ≤ 1 }.
Ps
(V2) ind(ζ) =
i=1 ind(ζ|Ci ) whenever ζ ∈ V(M), C is the domain of ζ, and
C1 , . . . , Cs are pairwise disjoint compact subsets of C such that ζ has no equilibria in C \ (int C1 ∪ . . . ∪ int Cs ).
(V3) For each ζ ∈ V(M) with domain C there is a neighborhood A ⊂ T M of Gr(ζ)
such that ind(ζ ′ ) = ind(ζ) for all vector fields ζ ′ on C with Gr(ζ ′) ⊂ A.
15.3. THE VECTOR FIELD INDEX
217
A vector field homotopy on S is a continuous function η : S × [0, 1] → T M
such that π(η(p, t)) = p for all (p, t), which is to say that each ηt = η(·, t) : S → T M
is a vector field on S. A vector field homotopy η on C is index admissible if each
ηt is index admissible. If ind(·) is a vector field index, then ind(ηt ) is locally constant
as a function of t, hence constant because [0, 1] is connected, so ind(η0 ) = ind(η1 ).
Our analysis of the vector field index relates it to the fixed point index.
Theorem 15.3.2. There is a unique index for M. If ζ ∈ V(M) has an extension
to a neighborhood of its domain that is locally Lipschitz and Φ is the flow of this
extension, then
ind(ζ) = Λ(Φ(·, −t)|C ) = (−1)m Λ(Φ(·, t)|C )
for all sufficiently small positive t. Equivalently, ind(−ζ) = Λ(Φ(·, t)|C ) for small
positive t.
Remark: In the theory of dynamical systems we are more interested in the future
than the past. In particular, forward stability is of much greater interest than
backward stability, even though the symmetry t 7→ −t makes the study of one
equivalent to the study of the other. From this point of view it seems that it would
have been preferable to define the vector field index with (V1) replaced by the
normalization requiring that the vector field x 7→ −x ∈ Tx Rm has index 1.
The remainder of this section is devoted to the proof of Theorem 15.3.2. Fix
ζ ∈ V(M) with domain C. The first order of business is to show that ζ can be approximated by a well enough behaved vector field that is defined on a neighborhood
of C.
Since C is compact, it is covered by the interiors of a finite collection K1 , . . . , Kk
of compact sets, with each Ki contained in an open Vi that is the image of a C r
parameterization ϕi . Each ϕi induces an isomorphism between T Vi and Vi × Rm ,
so that the Tietze extension theorem implies that there is a vector field on Vi that
agrees with ζ on C∩Vi . There is a partition of unity {λi } for K1 ∪. . .∪K
S k subordinate
to the cover V1 , . . . , Vk , and we may define an extension of ζ to V = i Vi by setting
X
ζ(p) =
λi (p)ζi (p).
p∈Vi
Suppose 2 ≤ r ≤ ∞. We will need to show that ζ can be approximated by a
vector field that is locally Lipschitz, but in fact we can approximate with a C r−1
vector field. In the setting of the last paragraph, we may assume that the partition
of unity {λi } is C r . Proposition 10.2.7 allows us to approximate each vector field
x 7→ Dϕi (x)−1 (ζi,ϕi(x) )
on ϕi (Vi ) with a C r vector field ξi , and
p 7→ Dϕi (ϕ−1
i (p))ξi,ϕ−1
i (p)
is then a C r−1 vector field ζ̃i on Vi that approximates ζi . The vector field ζ̃ on V
given by
X
ζ̃(p) =
λi (p)ζ̃i (p)
p∈Vi
218
CHAPTER 15. VECTOR FIELDS AND THEIR EQUILIBRIA
is a C r−1 vector field that approximates ζ.
Actually, we wish to approximate ζ with a C r−1 vector field satisfying an additional regularity condition. Recall that T(p,0) (T M) = Tp M × Tp M, and let
π2 : Tp M × Tp M → Tp M,
π2 (v, w) = w,
be the projection onto the second component. We say that p is a regular equilibrium of ζ̃ if p is an equilibrium of ζ̃ and π2 ◦ D ζ̃(p) is nonsingular. (Intuitively, the
derivative at p of the map q 7→ ζ̃q has rank m.) We need the following local result.
Lemma 15.3.3. Suppose that K ⊂ V ⊂ V ⊂ U ⊂ Rm with U and V open and
K and V compact, and λ : U → [0, 1] is a C r−1 (2 ≤ r ≤ ∞) function with
λ(x) = 1 whenever x ∈ K and λ(x) = 0 whenever x ∈
/ V . Let D be a closed subset
of U, and let f : U → Rm be a C r−1 function whose zeros in D are all regular.
Then any neighborhood of the origin in Rm contains a y such that all the zeros of
fy : x 7→ f (x) + λ(x)y in D ∪ K are regular.
Proof. The equidimensional case of Sard’s theorem implies that the set of regular
values of f |V is dense, and if −y is a regular value of f |V , then all the zeros of fy |K
are regular. If the claim is false, there must be a sequence yn → 0 such that for
each n there is a xn ∈ V ∩ D such that xn is a singular zero of fyn . But V ∩ D is
compact, so the sequence {xn } must have a limit point, which is a singular zero of
f |D by continuity, contrary to assumption.
PnUsing the last result, we first choose a perturbation ζ̂1 of ζ̃1 such that λ1 ζ̂1 +
i=2 λi ζ̃i has no zeros in K1 . Working inductively, we then choose perturbations
ζ̂2 , . . . , ζ̂n of ζ̃2 , . . . , ζ̃n one at a time, in such a way that for each i,
i
X
h=1
λi ζ̂i +
n
X
λi ζ̃i
h=i+1
has only
P regular equilibria in Di−1 ∪ Ki = (K1 ∪ . . . ∪ Ki−1 ) ∪ Ki . At the end of this
ζ̂ = i λi ζ̂i is an approximation of ζ̃ that has only regular equilibria in C.
We can now explain the proof that the vector field index is unique. In view
of (V3), if it exists, the vector field index is determined by its values on those
ζ ∈ V(M) that are C r−1 and have only regular equilibria. Applying (V2), we find
that the vector field index is in fact fully determined by its values on the ζ that are
C r−1 and have a single regular equilibria.
For such ζ the main ideas here are essentially the ones that were developed
in connection with our analysis of orientation, so we only sketch them briefly and
informally. By the logic of that analysis, there is a C r parameterization ϕ : V → M
whose image contains the unique equilibrium, such that either x 7→ Dϕ(x)−1 ζϕ(x)
is admissibly homotopic to either the vector field x 7→ x or the vector field x 7→
(−x1 , x2 , . . . , xm ). In the first of these two situations, the index is determined by
(V1). In addition, one can easily define an admissible homotopy transforming this
situation into one in which there are three regular equilibria, two of which are of
the first type and one of which is of the second type. Combining (V1) and (V2), we
219
15.3. THE VECTOR FIELD INDEX
find that the index of the equilibrium of the second type is −1, so the vector field
index is indeed uniquely determined by the axioms.
We still need to construct the index. One way to proceed would be to define
the vector field index to be the index of nearby smooth approximations with regular equilibria. This is possible, but the key step, namely showing that different
approximations give the same result, would duplicate work done in earlier chapters.
Instead we will define the vector field index using the characterization in terms of
the fixed point index given in the statement of Theorem 15.3.2, after which the
axioms for the fixed point index will imply (V1)-(V3).
We need the following technical fact.
Lemma 15.3.4. If C ⊂ M is compact, ζ is a locally Lipschitz vector field defined
on a neighborhood U of C, Φ is the flow of ζ, and ζ(p) 6= 0 for all p ∈ C, then there
is ε > 0 such that Φ(p, t) 6= p for all (p, t) ∈ C × ((−ε, 0) ∪ (0, ε)).
Proof. We have C = K1 ∪ . . . ∪ Kk where each Ki is compact and contained in
the domain Wi of a C r parameterization ϕi . It suffices to prove the claim with
C replaced with Ki , so we may assume that C is contained in the image of a C r
parameterization. We can use Lemma 15.2.1 to move the problem to the domain of
the parameterization, so we may assume that U is an open subset of Rm and that
ζ is Lipschitz, say with Lipschitz constant L.
Let V be a neighborhood of C such that V is a compact subset of U, and let
M := max kζ(p)k and m := min kζ(p)k.
p∈V
p∈V
Let ε > 0 be small enough that: a) V × (−ε, ε) is contained in the flow domain W ζ
of ζ; b) Φ(C × (−ε, ε)) ⊂ V ; c) LMε < m.
We claim that
kΦ(p, t) − pk < M|t|
(∗)
for all (p, t) ∈ C × (−ε, ε). For any (p, s) ∈ C × (−ε, ε) and v ∈ Rm we have
d
hΦ(p, s) − p, vi = hζΦ(p,s), vi ≤ kζΦ(p,s) k · kvk ≤ Mkvk.
ds
Therefore the intermediate value theorem implies that |hΦ(p, t) − p, vi| ≤ M|t| · kvk.
Since v may be any unit vector, (∗) follows this.
Now suppose that Φ(p, t) = p for some (p, t) ∈ C × ((−ε, 0) ∪ (0, ε)). Rolle’s
theorem implies that there is some s between 0 and t such that
0=
d
hΦ(p, s)
ds
− p, ζpi = hζΦ(p,s) , ζp i,
but among the vectors that are orthogonal to ζp , the origin is the closest, and
kζΦ(p,s) − ζp k ≤ LkΦ(p, s) − pk ≤ LM|s| < LMε < m < kζp k,
so this is impossible.
220
CHAPTER 15. VECTOR FIELDS AND THEIR EQUILIBRIA
We now define the vector field index of the pair (U, ζ) to be Λ(Φζ̃ (·, t)|C ), where
ζ̃ is a nearby C r−1 vector field, for all sufficiently small negative t. Since such a ζ̃
is vector field admissible, the last result (applied to ∂C) implies that Φζ̃ (·, t)|C is
index admissible for all small negative t, and it also (by Homotopy) implies that
the choice of t does not affect the definition.
We must also show that the choice of ζ̃ does not matter. Certainly there is a
neighborhood such that for ζ̃0 and ζ̃1 in this neighborhood and all s ∈ [0, 1], ζ̃s =
(1 − s)ζ̃0 + sζ̃1 is index admissible. In addition, that Φζ̃s (p, t) is jointly continuous
as a function of (p, t, s) follows from Theorem 15.2.5 applied to the vector field
(p, s) 7→ (ζ̃s (p), 0) ∈ T(p,s) (M × (−δ, 1 + δ)) on M × (−δ, 1 + δ), where δ is a suitable
small positive number. Therefore Continuity for the fixed point index implies that
Λ(Φζ̃0 (·, t)|C ) = Λ(Φζ̃1 (·, t)|C ).
We now have to verify that our definition satisfies (V1)-(V3). But the result
established in the last paragraph immediately implies (V3). Of course (V2) follows
directly from the Additivity property of the fixed point index. Finally, the flow of
the vector field ζ(x) = x on Rm is Φ(x, t) = et x, so for small negative t there is
an index admissible homotopy between Φ(·, t)|Dm and the constant map x 7→ 0, so
(V1) follows from Continuity and Normalization for the fixed point index.
All that remains of the proof of Theorem 15.3.2 is to show that ζ is locally
Lipschitz and defined in a neighborhood of C, then
ind(ζ) = (−1)m Λ(Φζ (·, t)|C )
for sufficiently small positive t. Since we can approximate ζ with a vector field that
is C r−1 and has only regular equilibria, by (V2) it suffices to prove this when C is
a single regular equilibrium. If ζ is one of the two vector fields x 7→ x ∈ Tx Rm or
x 7→ (−x1 , x2 , . . . , xm ) ∈ Tx Rm on Rm , then Φζ (x, −t) = −Φζ (x, t) for all x and t,
so the result follows from the relationship between the index and the determinant
of the derivative.
15.4
Dynamic Stability
If an equilibrium of a dynamical system is perturbed, does the system necessarily
return to the equilibrium, or can it wander far away from the starting point? Such
questions are of obvious importance for physical systems. In economics the notion
of dynamic adjustment to equilibrium is problematic, because if the dynamics of
adjustment are understood by the agents in the model, they will usually not adjust
their strategies in the predicted way. Nonetheless economists would generally agree
that an equilibrium may or may not be empirically plausible according to whether
there are some “natural” or “reasonable” dynamics for which it is stable.
In this section we study a basic stability notion, and show that a sufficient
condition for it is the existence of a Lyapunov function. As before we work with
a locally Lipschitz vector field ζ on a C r manifold M where r ≥ 2. Let W be the
flow domain of ζ, and let Φ be the flow.
221
15.4. DYNAMIC STABILITY
One of the earliest and most useful tools for understanding stability was introduced by Lyapunov toward the end of the 19th century. A function f : M → R is
ζ-differentiable if the ζ-derivative
ζf (p) =
d
f (Φ(p, t))|t=0
dt
is defined for every p ∈ M. A continuous function L : M → [0, ∞) is a Lyapunov
function for A ⊂ M if:
(a) L−1 (0) = A;
(b) L is ζ-differentiable with ζL(p) < 0 for all p ∈ M \ A;
(c) for every neighborhood U of A there is an ε > 0 such that L−1 ([0, ε]) ⊂ U.
The existence of a Lyapunov function implies quite a bit. A set A ⊂ M is
invariant if A × [0, ∞) ⊂ W and Φ(p, t) ∈ A for all p ∈ A and t ≥ 0. The ω-limit
set of p ∈ M is
\
{ Φ(p, t) : t ≥ t0 }.
t0 ≥0
The domain of attraction of A is
D(A) = { p ∈ M : the ω-limit set of p is nonempty and contained in A }.
A set A ⊂ M is asymptotically stable if:
(a) A is compact;
(b) A is invariant;
(c) D(A) is a neighborhood of A;
(d) for every neighborhood Ũ of A there is a neighborhood U such that Φ(p, t) ∈ Ũ
for all p ∈ U and t ≥ 0.
Asymptotic stability is a local property, in the sense A is asymptotically stable if and
only if it is asymptotically stable for the restriction of ζ to any given neighborhood
of A; this is mostly automatic, but to verify (c) for the restriction one needs to
combine (c) and (d) for the given vector field.
Theorem 15.4.1 (Lyapunov (1992)). If A is compact and L is a Lyapunov function
for A, then A is asymptotically stable.
Proof. If L(Φ(p, t)) > 0 for some p ∈ A and t > 0, the intermediate value theorem
would give a t′ ∈ [0, t] with
0<
d
d
L(Φ(p, t))|t=t′ = L(Φ(Φ(p, t′ ), t))|t=0 ,
dt
dt
contrary to (b). Therefore A = L−1 (0) is invariant.
222
CHAPTER 15. VECTOR FIELDS AND THEIR EQUILIBRIA
Let K be a compact neighborhood of A, choose ε > 0 such that L−1 ([0, ε]) ⊂ K,
and consider a point p ∈ L−1 ([0, ε]). Since L(Φ(p, t)) is weakly decreasing,
{ Φ(p, t) : t ≥ 0 } ⊂ L−1 ([0, ε]) ⊂ K,
so the ω-limit of p is a subset of K. Since K is compact and the ω-limit of is
by definition the intersection of nested nonempty closed subsets, the ω-limit is
nonempty. To show that the ω-limit of p is contained in A, consider some q ∈
/ A,
and fix a t > 0. Since L is continuous there are neighborhoods of V0 of q and Vt of
Φ(q, t) such that L(q ′ ) > L(q ′′ ) for all q ′ ∈ V0 and q ′′ ∈ Vt . Since Φ is continuous,
we can choose V0 small enough that Φ(q ′ , t) ∈ Vt for all q ′ ∈ V0 . The significance
of this is that if the trajectory of p ever entered V0 it would continue to Vt , and it
could not then return to V0 because L(Φ(p, t)) is a decreasing function of t, so q is
not in the ω-limit of p.
We have shown that L−1 ([0, ε)) ⊂ D(A), so D(A) is a neighborhood of A.
Now consider an neighborhood Ũ of A. We want a neighborhood U such that
Φ(U, t) ⊂ Ũ for all t ≥ 0, and it suffices to set U = L−1 ([0, δ)) for some δ > 0 such
that L−1 ([0, δ)) ⊂ Ũ . If there was no such δ there would be a sequence {pn } in
L−1 ([0, ε]) \ Ũ with L(pn ) → 0. Since this sequence would be contained in K, it
would have limit points, which would be in A, by (a), but also in K \ Ũ . Of course
this is impossible.
15.5
The Converse Lyapunov Problem
A converse Lyapunov theorem is a result asserting that if a set is asymptotically stable, then there is a Lyapunov function defined on a neighborhood of the
set. The history of converse Lyapunov theorems is sketched by Nadzieja (1990).
Briefly, after several partial results, the problem was completely solved by Wilson
(1969), who showed that one could require the Lyapunov function to be C ∞ when
the given manifold is C ∞ . Since we do not need such a refined result, we will follow
the simpler treatment given by Nadzieja (1990).
Let M, ζ, W , and Φ be as in the last section. This section’s goal is:
Theorem 15.5.1. If A is asymptotically stable, then (after replacing M with a
suitable neighborhood of A) there is a Lyapunov function for A.
The construction requires that the vector field be complete, and that certain
other conditions hold, so we begin by explaining how the desired situation can be
achieved on some neighborhood of A. Let U ⊂ D(A) be an open neighborhood of
A whose closure (as a subset of Rk ) is contained in M. For any metric on M (e.g.,
the one induced by the inclusion in Rk ) the infimum of the distance from a point
p ∈ U to a point in M \ U is a positive continuous function on U, so Proposition
10.2.7 implies that there is a C r function α : U → (0, ∞) such that for each p ∈ U,
1/α(p) is less than the distance from p to any point in M \ U. Let M̃ be the graph
of α:
M̃ = { (p, α(p)) : p ∈ U } ⊂ U × R ⊂ Rk+1 .
The closed subsets of M̃ are the subsets that are closed in Rk+1 :
15.5. THE CONVERSE LYAPUNOV PROBLEM
223
Lemma 15.5.2. M̃ is a closed subset of Rk+1.
Proof. Suppose a sequence {(pn , hn )} in M̃ converges to (p, h). Then p ∈ M, and
it must be in U because otherwise hn = α(pn ) → ∞. Continuity implies that
h = α(p), so (p, h) ∈ M̃ .
Using the map IdM × α : U → M̃ , we defined a transformed vector field:
ζ̃(p,α(p)) = D(IdM × h)(p)ζp .
Since IdM × α is C r , Lemma 15.2.4 implies that ζ̃ is a locally Lipschitz vector field
on M̃ . Let Φ̃ be the flow of ζ̃. Using the chain rule, it is easy to show that
Φ̃((p, h), t) = Φ(p, t), α(Φ(p, t))
for all (p, t) in the flow domain of ζ. Since asymptotic stability is a local property,
à = { (p, α(p)) : p ∈ A } is asymptotically stable for ζ̃.
We now wish to slow the dynamics, to prevent trajectories from going to ∞ in
finite time. Another application of Proposition 10.2.7 gives a C r function β : M̃ →
(0, ∞) with β(p, h) < 1/kζ̃(p, h)k for all (p, h) ∈ M̃ . Define a vector field ζ̂ on M̃
by setting
ζ̂(p, h) = β(p, h)ζ̃(p, h),
and let Φ̂ be the flow of ζ̂. For (p, t) such that (p, α(p), t) is in the flow domain of
ζ̂ let
Z t
B(p, t) =
β(Φ̂(p, α(p)), s) ds.
0
The chain rule computation
i
d h
Φ̃ p, α(p), B(p, t) = β(Φ̂(p, α(p), t))ζ̃Φ̃(p,α(p),B(p,t))
dt
shows that t 7→ Φ̃(p, α(p), B(p, t)) is a trajectory for ζ̂, so
Φ̂(p, α(p), t) = Φ̃ p, α(p), B(p, t) .
This has two important consequences. The first is that the speed of a trajectory
of ζ̂ is never greater than one, so the final component of Φ̂(p, α(p), t) cannot go
to ∞ in finite (forward or backward) time. In view of our remarks at the end of
Section 15.2, ζ̂ is complete. The second point is that since β is bounded below on
any compact set, if { Φ̃(p, α(p), t) : t ≥ 0 } is bounded, then Φ̂(p, ·) traverses the
entire trajectory of ζ̃ beginning at (p, α(p)). It follows that à is asymptotically
stable for ζ̂. Note that if L̂ is a Lyapunov function for ζ̂ and Ã, then it is also a
Lyapunov function for ζ̃ and Ã, and setting L(p) = L̂(p, α(p)) gives a Lyapunov
function for ζ|U and A. Therefore it suffices to establish the claim with M and ζ
replaced by M̃ and ζ̂.
The upshot of the discussion to this point is as follows. We may assume that
ζ is complete, and that the domain of attraction of A is all of M. We may also
assume that M has a metric d that is complete—that is, any Cauchy sequence
converges—so a sequence {pn } that is eventually outside of each compact subset of
M diverges in the sense that d(p, pn ) → ∞ for any p ∈ M.
The next four results are technical preparations for the main argument.
224
CHAPTER 15. VECTOR FIELDS AND THEIR EQUILIBRIA
Lemma 15.5.3. Let K be a compact subset of M. For any neighborhood Ũ of A
there is a neighborhood V of K and a number T such that Φ(p, t) ∈ Ũ whenever
p ∈ V and t ≥ T .
Proof. The asymptotic stability of A implies that A has a neighborhood U such
that Φ(U, t) ⊂ Ũ for all t ≥ 0. The domain of attraction of A is all of M, so for
each p ∈ K there is tp such that Φ(p, tp ) ∈ U, and the continuity of Φ implies that
Φ(p′ , tp ) ∈ U for all p′ in some neighborhood of p. Since K is compact, it has a finite
open cover V1 , . . . , Vk such that for each i there is some ti such that Φ(p, t) ∈ U
whenever p ∈ Vi and t ≥ ti . Set V = V1 ∪ . . . ∪ Vk and T = max{t1 , . . . , tk }.
Lemma 15.5.4. If {(pn , tn )} is a sequence in W = M × R such that the closure
of {pn } does not intersect A, and {Φ(pn , tn )} is bounded, then the sequence {tn } is
bounded below.
Proof. Let Ũ be a neighborhood of A that does not contain any element of {pn }
Since {Φ(pn , tn )} is bounded, it is contained in a compact set, so the last result
gives a T such that Φ(Φ(pn , tn ), t) = Φ(pn , tn + t) ∈ Ũ for all t ≥ T . For all n we
have tn > −T because otherwise pn = Φ(pn , 0) ∈ Ũ.
Lemma 15.5.5. For all p ∈ M \ A, d(Φ(p, t), p) → ∞ as t → −∞.
Proof. Otherwise there is a p and sequence {tn } with tn → −∞ such that {Φ(p, tn )}
is bounded and consequently contained in a compact set. The last result implies
that this is impossible.
Let ℓ : M → [0, ∞) be the function
ℓ(p) = inf d(Φ(p, t), A).
t≤0
If p ∈ A, then ℓ(p) = 0. If p ∈
/ A, then Φ(p, t) ∈
/ A for all t ≤ 0 because t is
invariant, and the last result implies that ℓ(p) > 0.
Lemma 15.5.6. ℓ is continuous.
Proof. Since ℓ(p) ≤ d(p, A), ℓ is continuous at points in A. Suppose that {pn } is a
sequence converging to a point p ∈
/ A. The last result implies that there are t ≤ 0
and tn ≤ 0 for each n such that ℓ(p) = d(Φ(p, t), A) and ℓ(pn ) = d(Φ(pn , tn ), A).
The continuity of Φ and d gives
lim sup ℓ(pn ) ≤ lim sup d(Φ(pn , t), pn ) = ℓ(p).
n
n
On the other hand d(Φ(pn , tn ), A) ≤ d(pn , A), so the sequence Φ(pn , tn ) is bounded,
and Lemma 15.5.4 implies that {tn } is bounded below. Passing to a subsequence,
we may suppose that tn → t′ , so that
ℓ(p) ≤ d(Φ(p, t′ ), A) = lim d(Φ(pn , tn ), A) = lim inf ℓ(pn ).
n
Thus ℓ(pn ) → ℓ(p).
n
225
15.5. THE CONVERSE LYAPUNOV PROBLEM
We are now ready for the main construction. Let L : M → [0, ∞) be defined by
Z ∞
L(p) =
ℓ(Φ(p, s)) exp(−s) ds.
0
The rest of the argument verifies that L is, in fact, a Lyapunov function.
Since A is invariant, L(p) = 0 if p ∈ A. If p ∈
/ A, then L(p) > 0 because
ℓ(p) > 0.
To show that L is continuous at an arbitrary p ∈ M we observe that for any
ε > 0 there is a T such that ℓ(Φ(p, T )) < ε/2. Since Φ is continuous we have
ℓ(Φ(p′ , T )) < ε/2 and |ℓ(Φ(p′ , t)) − ℓ(Φ(p, t))| < ε/2 for all p′ in some neighborhood
of p and all t ∈ [0, T ], so that
Z T
′
ℓ(Φ(p′ , s)) − ℓ(Φ(p, s)) exp(−s) ds −
|L(p ) − L(p)| ≤
0
Z
∞
T
′
′
ℓ(Φ(p , s)) exp(−s) ds −
Z
∞
T
ℓ(Φ(p, s)) exp(−s) ds < ε
for all p in this neighborhood.
To show that L is ζ-differentiable, and to compute its derivative, we observe
that
Z ∞
Z ∞
L(Φ(p, t)) =
Φ(p, t + s) exp(−s) ds = exp(t)
Φ(p, s) exp(−s) ds,
0
t
so that
L(Φ(p, t)) − L(p) = (exp(t) − 1)
Z
∞
t
Φ(p, s) exp(−s) ds −
Dividing by t and taking the limit as t → 0 gives
Z
t
ℓ(Φ(p, t)) exp(−s) ds.
0
ζL(p) = L(p) − ℓ(p).
Note that
L(p) < ℓ(p)
Z
∞
exp(s) ds = ℓ(p)
0
because ℓ(Φ(p, ·)) is weakly decreasing with limt→∞ ℓ(Φ(p, t)) = 0. Therefore ζL(p) <
0 when p ∈
/ A.
We need one more technical result.
Lemma 15.5.7. If {(pn , tn )} is a sequence such that d(pn , A) → ∞ and there is a
number T such that tn < T for all n, then d(Φ(pn , tn ), A) → ∞.
Proof. Suppose not. After passing to a subsequence there is a B > 0 such that
d(Φ(pn , tn ), A) < B for all n, so the sequence {Φ(pn , tn )} is contained in a compact
set K. Since the domain of attraction of A is all of M, Φ is continuous, and K
is compact, for any ε > 0 there is some S such that d(Φ(p, t), A) < ε whenever
p ∈ K and t > S. The function p 7→ d(Φ(p, t), A) is continuous, hence bounded on
the compact set K × [−T, S], so it is bounded on all of K × [−T, ∞). But this is
impossible because −tn > −T and
d(Φ(Φ(pn , tn ), −tn ), A) = d(pn , A) → ∞.
226
CHAPTER 15. VECTOR FIELDS AND THEIR EQUILIBRIA
It remains to show that if U is open and contains A, then there is an ε > 0 such
that L−1 ([0, ε]) ⊂ U. The alternative is that there is some sequence {pn } in M \ U
with L(pn ) → 0. Since L is continuous and positive on M \ U, the sequence must
eventually be outside any compact set. For each n we can choose tn ≤ 1 such that
ℓ(Φ(pn , 1)) = d(Φ(pn , tn ), A), and the last result implies that ℓ(Φ(pn , 1)) → ∞, so
Z 1
Z 1
L(pn ) ≥
ℓ(Φ(pn , t)) exp(−t) dt ≥ ℓ(Φ(pn , t))
exp(−t) dt → ∞.
0
0
This contradiction completes the proof that L is a Lyapunov function, so the proof
of Theorem 15.5.1 is complete.
15.6
A Necessary Condition for Stability
This section establishes the relationship between asymptotic stability and the
vector field index. Let M, ζ, and Φ be as before. If A is a compact set of equilibria
for ζ that has a compact index admissible neighborhood C that contains no other
equilibria of ζ, then ind(ζC ) is the same for all such C; we denote this common
value of the index by indζ (A).
Theorem 15.6.1. If A is an ANR that is asymptotically stable, then
ind−ζ (A) = χ(A).
Proof. From the last section we know that (after restricting to some neighborhood
of A) there is a Lyapunov function L for ζ. For some ε > 0, Aε = L−1 ([0, ε]) is
compact. Using the flow, it is not hard to show that Aε is a retract of Aε′ for some
ε′ > ε, and that Aε′ is a neighborhood of Aε , so Aε is an ANR. For each t > 0,
Φ(·, t)|Aε maps Aε to itself, and is homotopic to the identity, so
χ(Aε ) = Λ(Φ(·, t)|Aε ) = (−1)m ind(ζ|Aε ).
Since A is an ANR, there is a retraction r : C → A, where C is a neighborhood
of A. By taking ε small we may insure that Aε ⊂ C, and we may then replace
C with ε, so we may assume the domain of r is actually Aε . If i : A → C is the
inclusion, then Commutativity gives
χ(A) = Λ(r ◦ i) = Λ(i ◦ r) = Λ(r),
so it suffices to show that if t > 0, then
Λ(Φ(·, t)|C ) = Λ(r).
Let W ⊂ M × M be a neighborhood of the diagonal for which there is “convex
combination” function c : W ×[0, 1] → M as per Proposition 10.7.9. We claim that if
T is sufficiently large, then there is an index admissible homotopy h : Aε ×[0, 1] → Aε
between IdAε and r given by


0 ≤ t ≤ 31 ,
Φ(p, 3tT ),
h(p, t) = c((Φ(p, T ), r(Φ(p, T ))), 3(t − 13 )), 31 ≤ t ≤ 23 ,


2
≤ t ≤ 1.
r(Φ(p, 3(1 − t)T )),
3
15.6. A NECESSARY CONDITION FOR STABILITY
227
This works because there is some neighborhood U of A such that c((p, r(p)), t) is
defined and in the interior of Aε for all p ∈ U and all 0 ≤ t ≤ 1, and Φ(Aε , T ) ⊂ U
if T is sufficiently large.
The following special case is a prominent result in the theory of dynamical
systems.
Corollary 15.6.2 (Krasnosel’ski and Zabreiko (1984)). If {p0 } is asymptotically
stable, then
ind−ζ ({p0 }) = 1.
Physical equilibrium concepts are usually rest points of explicit dynamical systems, for which the notion of stability is easily understood. For economic models,
dynamic adjustment to equilibrium is a concept that goes back to Walras’ notion of
tatonnement, but such adjustment is conceptually problematic. If there is gradual
adjustment of prices, or gradual adjustment of mixed strategies, and the agents understand and expect this, then instead of conforming to such dynamics the agents
will exploit and undermine them. For this reason there are, to a rough approximation, no accepted theoretical foundations for a prediction that an economic or
strategic equilibrium is dynamically stable.
Paul Samuelson (1941, 1942, 1947) advocated a correspondence principle, according to which dynamical stability of an equilibrium has implications for the
qualitative properties of the equilibrium’s comparative statics. Samuelson’s writings consider many particular models, but he never formulated the correspondence
principle as a precise and general theorem, and the economics profession’s understanding of it has languished, being largely restricted to 1-dimensional cases; see
Echenique (2008) for a succinct summary. However, it is possible to pass quickly
from the Krasnosel’ski-Zabreiko theorem to a general formulation of the correspondence principle, as we now explain.
Let U ⊂ Rm be open, let P be a space of parameter values that is an open
subset of Rn , and let z : U × P → Rm be a C 1 function that we understand as
a parameterized vector field. (Working in a Euclidean setting allows us to avoid
discussing differentiation of vector fields on manifolds, which is a very substantial
topic.) For (x, α) ∈ U × P let ∂x z(x, α) and ∂α z(x, α) denote the matrices of partial
derivatives of the components of z with respect to the components of x and α
respectively.
We consider a point (x0 , α0 ) with z(x0 , α0 ) = 0 such that ∂x z(x0 , α0 ) is nonsingular. The implicit function implies that there is a neighborhood V of α0 and C 1
function σ : V → U such that σ(α0 ) = x0 and z(σ(α), α) = 0 for all α ∈ V . The
method of comparative statics if to differentiate this equation with respect to α,
using the chain rule, then rearrange, arriving at
dσ
(α0 ) = −∂x z(x0 , α0 )−1 · ∂α z(x0 , α0 ).
dα
The last result implies that if {x0 } is asymptotically stable for the vector field
z(·, α0 ), then the determinant of −∂x z(x0 , α0 ) is positive, as is the determinant of
dσ
(α0 ) is a positive scalar multiple
its inverse. When m = 1 this says that the vector dα
228
CHAPTER 15. VECTOR FIELDS AND THEIR EQUILIBRIA
of ∂α z(x0 , α0 ). When m > 1 it says that the transformation mapping ∂α z(x0 , α0 )
dσ
(α0 ) is orientation preserving, which is still a qualitiative property of the
to dα
comparative statics, though of course its intuitive and conceptual significance is
less immediate. (It is sometimes argued, e.g., pp. 320-1 of Arrow and Hahn (1971),
that the correspondence principle has no consequences beyond the 1-dimensional
case, but this does not seem quite right. In higher dimensions it still provides a
qualitative restriction on the comparative statics. It is true that the restriction
provides only one bit of information, so by itself it is unlikely to be useful, but one
should still expect the correspondence principle to have some useful consequences
in particular models, in combination with various auxilary hypotheses.)
We conclude with some comments on the status of the correspondence principle
as a foundational element of economic analysis. First of all, the fact that our current
understanding of adjustment to equilibrium gives little reason to expect an equilibrium to be stable is of limited relevance, because in the correspondence principle
stability is an hypothesis, not a conclusion. That is, we observe an equilibrium that
persists over time, and is consequently stable with respect to whatever mechanism
brings about reequilibration after small disturbances. This is given.
In general equilibrium theory and noncooperative game theory, and in a multitude of particular economic models, equilibrium is implicitly defined as a rest
point of some process according to which, in response to a failure of an equilibrium condition, some agent would change her behavior in pursuit of higher utility.
Such a definition brings with it some sense of “natural” dynamics, e.g., the various
prices each adjusting in the direction of excess demand, or each agent adjusting
her mixed strategy in some direction that would be improving if others were not
also adjusting. The Krasnosel’ski-Zabreiko theorem will typically imply that if
ind−z(·,α0 ) ({x0 }) 6= 1, then x0 is not stable for any dynamic process that is natural
in this sense. Logically, this leaves open the possibility that the actual dynamic
process is unnatural, which seemingly requires some sort of coordination on the
part of the various agents, or perhaps that it is much more complicated than we
are imagining. Almost certainly most economists would regard these possibilities
as far fetched. In this sense the correspondence principle is not less reliable or well
founded than other basic principles of our imprecise and uncertain science.
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Index
C r , 126
C r ∂-embedding, 144
C r ∂-immersion, 144
C r atlas, 10
C r function, 127
C r manifold, 10, 131
C r submanifold, 11, 136
Q-robust set, 113
Q-robust set
minimal, 114
minimal connected, 114
T1 -space, 66
ω-limit set, 19, 221
∂-parameterization, 144
ε-domination, 17, 104
ε-homotopy, 17, 104
EXP, 61
FNP, 63
NP, 61
PLS (polynomial local search), 64
PPAD, 64
PPA, 65
PPP (polynomial pigeonhole principle),
64
PSPACE, 61
P, 61
TFNP, 63
Clique, 61
EOTL (end of the line), 64
OEOTL (other end of the line), 65
absolute neighborhood retract, 6, 100
absolute retract, 6, 102
acyclic, 34
affine
combination, 23
dependence, 23
hull, 24
independence, 23
subspace, 24
Alexander horned sphere, 131
algorithm, 60
ambient space, 10, 133
annulus, 144
antipodal function, 203
antipodal points, 200
approximates, 189
Arrow, Kenneth, 2
asymptotic stability, 20, 221
atlas, 10, 131
axiom of choice, 36
balanced set, 116
Banach space, 90
barycenter, 32
base of a topology, 67
bijection, 6
Bing, R. H., 196
Border, Kim, i
Borsuk, Karol, 18
Borsuk-Ulam theorem, 18, 204
bounding hyperplane, 24
Brouwer’s fixed point theorem, 3
Brouwer, Luitzen, 2
Brown, Robert, i
category, 135
Cauchy sequence, 90
Cauchy-Schwartz inequality, 91
certificate, 61
Church-Turing thesis, 60
closed function, 74
codimension, 24, 136
commutativity configuration, 16, 179
compact-open topology, 83
complete invariant, 194
complete metric space, 90
complete vector field, 216
235
236
completely metrizable, 100
component of a graph, 34
computational problem, 60
complete for a class, 62
computable, 60
decision, 61
search, 61
connected
graph, 34
space, 8, 113, 165
continuous, 78
contractible, 5
contraction, 5
converse Lyapunov theorem, 20, 222
convex, 24
combination, 24
cone, 25
hull, 24
coordinate chart, 10, 131
correspondence, 4, 77
closed valued, 77
compact valued, 4
convex valued, 4, 77
graph of, 77
lower semicontinuous, 78
upper semicontinuous, 77
correspondence principle, 19, 212
critical point, 139, 154
critical value, 139, 154
cycle, 34
Debreu, Gerard, 2
degree, 11, 33, 174
degree admissible
function, 12, 14, 171
homotopy, 12, 171
Dehn, Max, 149
Demichelis, Stefano, 19
derivative, 126, 134, 135
derivative along a vector field, 20
Descartes, René, 30
deterministic, 212
diameter, 32
diffeomorphism, 11, 133
diffeomorphism point, 136
differentiable, 126
INDEX
differentiation along a vector field, 221
dimension
of a polyhedron, 26
of a polytopal complex, 30
of an affine subspace, 24
directed graph, 64
discrete set, 131
domain of attraction, 19, 221
domination, 183
dual, 25
Dugundji, James, 93
Dugundji, James, i
edge, 27, 33
Eilenberg, Samuel, 7, 18
Eilenberg-Montgomery theorem, 196
embedding, 6, 131
endpoint, 33, 42
equilibrium, 19, 21
equilibrium of a vector field, 216
regular, 218
essential
fixed point, 7
Nash equilibrium, 8
set of fixed points, 8, 112
set of Nash equilibria, 8
Euclidean neighborhood retract, 6, 99
Euler characteristic, 17, 20, 195
expected payoffs, 37
extension of an index, 183
extraneous solution, 44
extreme point, 29
face, 26
proper, 27
facet, 27
family of sets
locally finite, 85
refinement of, 85
Federer, Herbert, 150
Fermat’s last theorem, 149
fixed point, 3, 4
fixed point property, 3, 6
flow, 19, 216
flow domain, 216
Fort, M. K., 107
four color theorem, 149
237
INDEX
Freedman, Michael, 149
Fubini’s theorem, 150
functor, 135
general position, 41
general linear group, 165
Granas, Andrzej, i
graph, 4, 33
half-space, 24
Hauptvermutung, 196
Hausdorff distance, 70
Hausdorff measre zero, 156
Hausdorff space, 67
have the same orientation, 11, 164
Hawaiian earring, 33, 100
Heegaard, Poul, 149
Hilbert cube, 93
Hilbert space, 91
homology, 2, 3, 177, 196
homotopy, 5, 58–59
class, 5
extension property, 198
invariant, 18, 197
principle, 178
homotopy extension property, 103
Hopf’s theorem, 18, 197–198
Hopf, Heinz, 18, 196
hyperplane, 24
identity component, 165
immersion, 138
immersion point, 136
implicit function theorem, 127
index, 9, 15, 16, 177, 179
index admissible
correspondence, 15, 177
homotopy, 178
vector field, 20
index base, 15, 177
index scope, 15, 178
inessential fixed point, 7
initial point, 27
injection, 6
inner product, 91
inner product space, 91
invariance of domain, 18, 207
invariant, 19
invariant set, 221
inverse function theorem, 127
isometry, 52
Kakutani, Shizuo, 4
Kinoshita, Shin’ichi, 6, 95, 108
labelling, 51
Lefschetz fixed point theorem, 17, 196
Lefschetz number, 17, 196
Lefschetz, Solomon, 17, 196
Lemke-Howson algorithm, 36–49, 62, 64
Lesbesgue measure, 150
lineality space, 26
linear complementarity problem, 45
Lipshitz, 212
local diffeomorphism, 138
locally C r , 130
locally closed set, 98
locally Lipschitz, 212
locally path connected space, 102, 142
lower semicontinuous, 78
Lyapunov function, 221
Lyapunov function for A ⊂ M, 20
Lyapunov theorem, 221
Lyapunov, Aleksandr, 20
manifold, 10
C r , 131
manifold with boundary, 13, 144
Mas-Colell, Andreu, 17, 115
maximal, 34
measure theory, 150
measure zero, 151, 157
mesh, 32
Milnor, John, 150, 196
Minkowski sum, 111
Moise, Edwin E., 196
Montgomery, Deane, 7, 18
Morse-Sard theorem, 157
moving frame, 165
multiplicative, 16, 180
Möbius, August Ferdinand, 149
narrowing of focus, 183
Nash equilibrium
accessible, 43
238
mixed, 37
pure, 37
refinements of, 8
Nash, John, 2
negatively oriented, 164, 168
negatively oriented relative to P , 169
neighborhood retract, 98
neighbors, 33
nerve of an open cover, 105
no retraction theorem, 100
norm, 90
normal bundle, 140
normal space, 67
normal vector, 24
normed space, 90
opposite orientation, 164, 168
oracle, 62
order of differentiability, 126
ordered basis, 164
orientable, 11, 168
orientation, 163–171
orientation preserving, 11, 53, 169
orientation reversing, 11, 53, 169
orientation reversing loop, 167
oriented ∂-manifold, 168
oriented intersection number, 169
oriented manifold, 11
oriented vector space, 11, 164
paracompact space, 85
parameterization, 10, 131
partition of unity, 86
C r , 128
path, 33, 165
path connected space, 164
payoff functions, 37
Perelman, Grigori, 149
Picard-Lindelöf theorem, 212, 215
pivot, 57
pivoting, 48
Poincaré conjecture, 149
Poincaré, Henri, 149
pointed cone, 26
pointed map, 113
pointed space, 113
polyhedral complex, 30
INDEX
polyhedral subdivision, 30
polyhedron, 26
minimal representation of, 27
standard representation of, 27
polytopal complex, 30
polytopal subdivision, 30
polytope, 29
simple, 46
positively oriented, 11, 164, 168
positively oriented relative to P , 169
predictor-corrector method, 58
prime factorization, 63
quadruple
edge, 42
qualified, 42
vertex, 42
quotient topology, 80
Rado, Tibor, 149, 196
recession cone, 25
reduction, 62
regular fixed point, 9
regular point, 11, 139
regular space, 67
regular value, 11, 139
retract, 6, 97
retraction, 97
Ritzberger, Klaus, i, 19
Samuelson, Paul, 19
Sard’s theorem, 182, 218
Scarf algorithm, 56
Scarf, Herbert, 18
separable, 6
separable metric space, 91
separating hyperplane theorem, 24
set valued mapping, 4
simplex, 31
accessible completely labelled, 58
almost completely labelled, 54
completely labelled, 52
simplicial complex, 31
abstract, 32
canonical realization, 32
simplicial subdivision, 31
simply connected, 149
239
INDEX
slack variables, 44
slice of a set, 153
Smale, Stephen, 149
smooth, 11, 156
Sperner labelling, 51
star-shaped, 5
Steinitz, Ernst, 196
step size, 58
Sternberg, Shlomo, 150
strategy
mixed, 37
pure, 37
totally mixed, 39
strategy profile
mixed, 37
pure, 37
strong topology, 83
strong upper topology, 78
subbase of a topology, 67
subcomplex, 30
submanifold, 11
neat, 146
submersion, 138
submersion point, 136
subsumes, 183
support of a mixed strategy, 63
surjection, 6
tableau, 47
tangent bundle, 133
tangent space, 10, 133
tatonnement, 18
Tietze, Heinrich, 196
topological space, 66
topological vector space, 88
locally convex, 89
torus, 10
trajectory, 212, 213
transition function, 10
translation invariant topology, 88
transversal, 139, 146, 158
tree, 34
triangulation, 31
tubular neighborhood theorem, 140
Turing machine, 59
two person game, 37
Ulam, Stanislaw, 18
uniformly locally contractible metric space,
101
upper semicontinuous, 4, 77
Urysohn’s lemma, 87
van Dyke, Walther, 149
vector bundle, 140
vector field, 19, 158, 213
along a curve, 165
index admissible, 216
vector field homotopy, 217
index admissible, 217
vector field index, 216
vertex, 27, 32
vertices, 33
connected, 34
Vietoris topology, 68
Vietoris, Leopold, 66
von Neumann, John, 4
Voronoi diagram, 30
walk, 33
weak topology, 83
weak upper topology, 80
well ordering, 85
well ordering theorem, 85
Whitney embedding theorems, 133
Whitney, Hassler, 10
wild embedding, 131
witness, 61
zero section, 140, 158