-c,s. uN.^r.
no.
r^.^
/S/S-'ii
'^/?At^^^
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
VARIABLE DIMENSION COMPLEXES, PART I:
BASIC THEORY
by
Robert M. Freund
Sloan W.P.. No. 1515-8A
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
July 1983
VARIABLE DIMENSION COMPLEXES, PART I:
BASIC THEORY
by
Robert M. Freund
Sloan W.P., No. 1515-84
July 1983
Research supported by the Department of Energy Contract DE-AC03-76-SF00326,
PA No. DE-AT-03-76ER72018; the National Science Foundation Grants MCS79-031A5
and SOC78-16811; U.S. Army Research Office Contract DAAG-29-78-G-0026.
Abstract
Over the past several yenrs, much attention in the field
of siiiiplicial pivoting algorithms has been focused on the new
class of variable-dimension algorithms, wherin the sequence
of simplices generated varies in dimension.
This study
is aimed at the combinatorial nature of variable-dimension
algorithms.
sliort for
In Part I,
we introduce the notion of
variable-dimension complex.
labelling function is introduced on
characterization of paths on
Part II of
t :is
a
is
I
a
I
to orovide construc-
variety of combinatorial lemmas
demonstrate two new dual lemmas on the n-dimen-
sionsl cube, and use
generalizati
V-c nplex, certain path-
study uses the path-following theory of
tive algorithmic proofs of
VJe
:n of the
Lemma on tne sinnlex.
a
Generalized Sperner Lemma to prove
a
Knaster-Kuratowski-Kazurkiewicz Covering
'Ve
also show that Tucker's LeTima can
be derived directly from the Borsuk-Ulam Theorem.
We report
the interrelationships between these results, Brouv/er
'
s
Point Theorem, /^the existence of stationary points on the
simplex^
a
labelled V-complex.
labelled V-complexes d:ve]oned in Part
in topology.
V-complex,
show that when
The main result of Part
following properties arise.
a
a
-7e
a
Fixed
Key Words
complementary pivot, fixed points, simnlex, pseudomanif old,
orientation, stationary point, combinatorial topology,
V-complex, antipodal point.
Introduction
Over the past several years, much attention in the field of sim-
plicial pivoting algorithms has been focused on the new class of variable-
dimension algorithms, wherein the sequence of simplices generated varies in
dimension, see van der Laan and Talman [12, 13, 14, 15], Reiser [20],
Luthl [19]
,
and Ko j ima and Yamamoto [9].
Like other simplicial algorithms,
these algorithms usually can be executed with integer or vector labels.
When vector labels are used, it is appropriate to interpret these
algorithms as piecewise-linear (PWL) path tracing as in Eaves [2].
and Yamamoto in
[
9]
Kojima
present a theory of dual pairs of subdivided manifolds
that is a basis for interpreting these algorithms via PWL path tracing.
Although the integer-labelled variable dimension algorithms can
also be interpreted as PWL path tracing, it seems more appropriate to use
a combinatorial framework in
which to study these algorithms.
The integer-
labelled algorithms usually are associated with some combinatorial lemma in
topology, e.g. Sperner
'
s
Lemma
,
and the algorithms result in constructive
proofs.
This study is aimed at the combinatorial nature of variable-dimension
algorithms.
In Part I, we introduce the notion of a V-complex, short for
variable-dimension complex.
We show that when a labelling function is
introduced on a V-complex, certain path-following properties arise.
main result of Part
I
The
is a characterization of paths on a labelled V-complex.
Whereas most of the variable-dimension integer-labelled algorithms can be
interpreted as path-following on a V-complex, see Freund [6], our interest
here lies in the study of some combinatorial lemmas in topology.
In Part II of this study, we give constructive,
algorithmic proofs to a
number of combinatorial lemmas in topology, namely Sperner's Lemma [26],
Kuhn's Cubical Lemma [10], Scarf's Dual Sperner Lemma [22], a Generalized
Sperner Lemma [4], Tucker's Lemma on the cube [29], Gale's Hex Theorem [7],
and two new dual lemmas on cube [6].
We also show the variety of relationships
between these lemmas, the Brouwer fixed point theorem, and the Borsuk-Ulam
Antipodal Point Theorem, and a new set covering theorem.
This study is based on the author's doctoral disseration [6].
Notation
Define e to be the vector of I's, namely e = (1,
represented by
<^
,
I
I
X
I
...,1).
€
IR^^I
x>0}.
The empty set is
For two sets S and T, define the symmetic difference operator
.
SAT = {x|x€S'JT, x^SAT} and define
For xeIR
= {x
be real n-dimensional space, and let IR
Let IR
L
='Vx,+
.
.
.+x*",
S\T
I
x
I
= {x| x<S,
I
|
x^T}.
= max x
I
.
|
,
the Euclidean and maximum norms, respectively.
Let e
be the
i
unit vector in IR
column of A, and A,
1.
be the
i
.
For a matrix A, let A
row of A.
.
be the
i'^'^
- 4 -
Preliminaries - Complexes, Pseudomanlfolds, Orientation, Trlangulatlons
An abstract complex consists of a set of vertices K
of finite nonempty subsets of K
1)
?£
X
denoted K, such that
implies {v} f K
V e K
ii)
,
C
y 6 K implies
x€
The elements of K are called simplices
where
|
.
denotes cardinality.
|
and a set
or simply an n-simplex
.
K.
.
Suppose x f K and
|xl
= n+1,
Then x is called an n-dimensional simplex ,
Condition (i) above means that all members of K
are 0-simplices, and condition (ii) means that K is closed under subsets.
Technically, an abstract complex is defined by the pair (K,K
is implied by K,
since the set K
).
However,
it is convenient to simply denote the
complex by K alone.
An abstract complex K is said to be finite if the set K
finite.
V
6
K
,
is
An abstract complex K is said to be locally finite if for each
the set of simplices containing v is a finite set.
K is locally finite if and only if for each v
{x 6
k|v
fi
6
K
More formally,
,
x} is a finite set.
A subset L of K is said to be a subcomplex of K if L itself is a complex.
A particular class of complexes, called pseudomanlfolds, is central
to the theory to be developed.
An n-dimensional pseudomanifold (where n^l)
or more simply an n-pseudomanifold, is a complex K such that
i)
X
e
K implies there exists y e K with
|y|
= n+1,
and X C y.
ii)
If X e
K and
|x|
that contain x.
= n,
then there are at most two n-simplices
.
.
- 5 -
Let K be an n-pseudomanifold.
The boundary of K, denoted 3K, is
defined to be the set of simplicies x € K such that x is contained in an
(n-1) -simplex y e K, and y is a subset of exactly one n-simplex of K.
An n-pseudomanifold K is said to be homogeneous if for any pair of
n-simplices x, y 6 K, there is a finite sequence x of n-simplices in K such that 7i^f\ *i+i
i'l,
x,
Xj* ^2*
,
* * * »
'Sn
" ^
^s an (n-l)-simplex in K, for
•••f m— 1
Let R be an n-pseudomanifold and let x be an n-simplex in K.
On
Let X = {v„, ..., V
w€ K
In
Let y = {v., .... v
}.
such that {w, v
exchanging v
. . . ,
,
v
}
}.
If y ^ 3K, there is unique
Is an n-simplex in K.
The process of
for w to obtain a new n-simplex is called a pivot
.
In general,
if X and z are n-simplices and z can be obtained from x by a pivot, x and z
are said to be a neighboring pair , or simply neighbors .
For the purposes of this study a 0-dimensional pseudomanifold K
is defined to be a set of one of the following two forms:
i)
ii)
K -
{0,ia}],
where K° - {a}, or
K = {0,{4,ib}}, where K° = {a,b}.
Note that K contains 0, the empty set, as a member, and so is not
properly a complex, by the usual definition.
If K is of type (i)
,
then 3K = {0}.
Here, however,
If K is of type (ii)
,
is a -1-simplex.
then 3K = 0, i.e.
K has no boundary, and {a} and {b} are neighbors.
Let R be a homogeneous n-pseudomanifold, and let x be an n-simplex
in R.
Let (v.,
..., v
n
)
be some fixed ordering of the vertices of x.
For
an arbitrary ordering (v. , ..., v. ) of x, this ordering is said to have a
JO
Jn
(+)
orientation if and only if the permutation
%,
....
is even; otherwise the orientation is (-)
V
- 6 -
Now let us extend this notion to all of
all n-slmpllces of K.
K.
Fix an ordering of
Let x be an n-slmplex and let y be an n-slmplex
by w.
of x and replacing v.
obtained by pivoting on an element v.
We say
that the pair (x,y) is coherently-oriented if the orderings (v^ , ..., v. )
JO
Jn
and (v.
JO
,
..-, V.
J
,
1-1
w, v.
J
. . • ,
,
v.
are differently oriented, i.e.
)
Jn
1+1
K is said to be orientable if it is
one la (+) and the other is (-).
possible to specify orientations on all n-simplices of K in a way that all
neighboring n-slmplices x, y are coherently-oriented.
Finally, we define induced orientation on the boundary of K.
K be
Let
a homogeneous orientable n-pseudomanlfold such that 3K is not empty.
Then there is a unique n-simplex x
Let y be an (n-l)-slmplex in 3K.
such that y
C
X.
y " x
\
{v.
}
for some 1 uniquely determined.
^1
the orientation of the ordering (v^ ,
JQ
K
Let (v^ , .... Vj ) be an ordering
JO
Jn
Orient K coherently.
of the vertices of x.
e
.
.
.
,
v^
Jn
)
by Or(v^ ,
JO
.
. .
,
v^
Jn
).
Then we define the induced orientation on y as
Or(vj^. ..., vj^_^, vj^_^^. ..., vj^) - (-1)
Or(vj^, ..., v^^),
Denote
- 7 -
Proposition
PROOF
Induced orientation is well-defined.
1
Let y be an (n-1) -simplex in 3K, and let x be the unique n-simplex
.
in K that contains y.
of y, and let (j^,
Let (i
•••»
of x, from which (i
j
i
(-1)''
i^ .) =
...,
Or(j
n-1
be an ordering of the vertices
_•, )
•••
^
be orderings of the vertices
)
is derived,
_•, )
= v„
for some unique r, s.
V = V.
s
^r
If r = s, then (j^, ..., j^) = (l^,
Or(i-,
O
i
and (^^,
)
...,
,
...,
,
U
,
...,
...,
£^)
j
n
y = x
\ v for some unique vf
and
,
= (-1)^ Or(£
)
x.
...,
U
£
n
)
trivially.
So suppose s
•••»
(Jq.
=
(-1)''
to
j^)
It takes s-r transpositions to change
r.
>
...,
(£q,
(-1)^"'' Or(£Q,
Hence
£^).
....
(-1)^ Or(£Q,
=
£^)
OrCj^,
(-1)
...,
...,
£^)
j^)
.
®
An n-dimensional pseudomanifold is an abstraction of a triangulation
of an n-dimensional set in IR
The m-simplices of pseudomanifolds
.
correspond to geometric objects, which are also called m-simplices.
Let^ V
»
,
.
.
.
,
V
ro
m
^
,
.
be vectors
T„n
IR
.
v
,
.
.
.
,
v
m
1
are saidJ ^to be af^rrfinely
.
•
i_
independent if the matrix
V
has rank m+1.
If v
hull, denoted \v
,
.
.
Let {v
\v
/is
,
..., V
Any k"face of
\v
,
m-simplex is called
or more
,
All m-simplices are closed and bounded polyhedral
,
..., v
called
.
m
v / is said to be an m- dimensional simplex
.
simply an m-simplex.
convex sets.
v
are af finely independent then their convex
..., v
,
...
.
.
,
a
k
}
be a subset of {v
,
.... v
m
k-dimensional face of k- face of ^v
v / is a k-simplex itself.
a facet of the
m-simplex.
Then
}.
,
.
.
.
,
v
An (m-l)-face of an
>.
.
.
- 8 -
Let H be an m-dlmensional convex set In IR
of m-simplices a together vlth all of their faces.
.
Let C be a collection
C is a triangulation
of H if
1)
g
H -
a
atC
ii)
iii)
a,T
€
C imply o A t e C
If a is an (m-l)'^8±mplex of C, a is a face of at most
two m-simplices of C.
The connection between triangulations and pseudomanlfolds should be clear.
Corresponding to each simplex o in C is its set of vertices {v
.
,
•
.
v
,
k
}
Let K be the collection of these sets of vertices together with their
nonempty subsets.
Then K is an m-dimensional pseudomanlf old
C is said to be locally finite if for each vertex v£H, the set of
simplicies
fl-^C
that contain v is a finite set.
Pertinent references for complexes and pseudonanifolds are
Some of the material on orientation was
Lefschetz C16D or Spanier C25D.
taken from Lemke and Grotzinger
C18I1.
The notion of orienting pseudo-
manifolds can be extended to triangulations by the use of determinants.
The interested reader can refer to Eaves E2D and Eaves and Scarf C33
for an exposition on orienting triangulations.
V-Complexes
This section defines
a
particular combinatorial framework called
a
V-complex that is used in the study of combinatorial lemmas in topology, as
'
well as in the study of variable-dimension simplicial pivoting algorithms for
obtaining solutions of equations.
in IR
,
When applied to a triangulation of
a set S
a V-complex constitutes a division of S into a number of regions of
varying dimension.
As an example, consider the set S = {x<IR^|
-
e< x
^ e) triangulated in such a manner that each coordinate axis is also
triangulated.
Consider the regional division defined as follows:
R (I) - {0)
R ({!)) = { xtS\ Xj iO, x^ =
R ({-!)) =
^
R ({2}) =
xeS| x^ SO, x^ =
{
)
x€S| x^ = 0, X2 ^0
{
R ({-2}) =
)
x«S| Xj = 0, Xj SO
{
R ({1,2)) =
)
)
x€S| x^ ^ 0, x^ ^ 0)
{
R ({1.-2)) =
{
x<S| Xj a 0, x^ S 0}
R ({-1.2)) =
{
x«S| Xj S 0. x^ ^ 0)
R ({-1,2)) =
{
x(S| Xj S 0. x^ S 0)
Let us define J to be the domain over which R(.) is defined, namely" =
{0,
{1),{-1). {2). (-2). {1.2
{1,-2),
),
T<5
{-1,2). {-1,-2,)), and note that for S,
Also note that each R(T) is a
R(S)r\R(T).
given T<7, suppose there is a
j f
.
|T|
S(M*J
and R(SnT) =
.
-dimensional manifold.
T such that Tu{j)f J
.
For a
Then, because C
I
triangulates each region R(.)» each |T|-simplex x =^v**,...,
v'
T
I
/
in R(T) will have a imique vertex v in R(Ti'{j)) such that
<v°,...,v''^',
v>
is a (|T|
+
1)
-
simplex in R(TU{j)).
Because the above-mentioned properties do not depend on the particular
triangulation of
S and
are more combinatoric than geometric, it is convenient
to restate them in the more abstract framework of complexes and pseudomanifolds.
Let K be the pseudomanifold corresponding to C, and K° its set of
vertices.
For each 7^7, define A(T) to be the pseudomanifold corresponding to
the restriction of C to R(T).
The above properties of R(.) carry over to
A(.), namely A(Sr\T) = A(S)f\A(T) for S,
pseudomanifold.
Te'J,
and A(T) is a |T|- dimensional
ItI
Furthermore, if x = {v",..., v
)
is a
|T|-
dimensional simplex in A(T) and j^T but Tu{j)etJ, then there is a unique
vertex v of K° such that
{
v*,..., v'^'
.
v) €
A (TU{j)).
In a typical application of a V-complex, we have a function L(.) that
In the above example
assigns a label to each vertex v of the trlangulation.
consider a fimction L(.):
of A(.),^
,
K**
- N, where N={ 1,2, -1,-2)
.
The properties
and N, which are formalized in a more general setting below, will
be central to the development of algorithms which will find a simplex each of
whose labels have certain desirable properties.
We now proceed to define a
V-cooplex.
Let K be a locally finite slmpliclal complex with vertices K
Let N be a fixed finite nonempty set, which we call the label set
.
.
Let
'J
subsets
denote a collection of subsets of N, which we call the admissible
of N.
T
Let A(.) be a set-to-set map. A: J"*2
collection of subsets of a set
met:
\{0}, where
S
2
denotes the
K Is said to be a V-complex with
S.
operater A(') and admlssable sets
K
J
,
if the following eight conditions are
- 9 -
1)
11)
111)
Iv)
v)
vl)
Vll)
K Is a locally finite complex with vertices K
![
C 2^
6
7
For any x € K, there is a Tf
7
T
€
J
,
S 6
7
Implies S
T
rv
A(.):'J->2^\ {0}
For any S, T
For T
€ J
,
6
J
,
A(S
r\
such that x
T) - A(S)
O
«
A(T)
A(T)
A(T) is a subcomplex of K and is a pseudomanlfold
of dimension |t|, where
denotes the cardinality
|»|
of the set.
vill)
T«7.TU{j}€^ ,j^
T implies A(T)C aA(T
Let us examine these properties.
(11) and
(i),
what has been said in the preceding paragraph.
on 7
,
map A(
covers all simplices of K.
morphism with respect to intersections.
(iv) reiterate
(v)
states that A('
(vl)
{j}).
(ill) imposes some structure
namely that it is closed under Intersections.
.)
(J
states that the
)
is a homo-
(vll) states that each A(T)
is an appropriately-dimensioned pseudomanlfold.
Condition (vill) stipulates
how the pseudomanifolds A(T) are arranged relative to each other, namely
that A(T) is part of the boundary of A(T
U
{j}).
The nomenclature "V-complex" is short for variable-dimension complex, and
derives from property (vii) above, where the dimension of the pseudomanifolds
A(T) varies over the range of T in 7.
As an example of a V-complex, consider a triangulation C of
IR
that also triangulates the coordinate axes.
Let N = {±1,±2}.
Let
7
be the collection of sets {1}, {-1}, {2}, {-2}, {1,2}, {1,-2}, {-1,2}, {-1,-2}
and 0.
Let K be the complex (actually a pseudomanlfold itself) corresponding
to the triangulation C.
A(T) - {x«
Then for each
k|v € X implies i-vi
.i
T*7.
>0 for each
we define
1 6 T,
and v
if 1 f T and -1 ^ T},
=
.
- 10 -
Figure 1(a)
Note that for matters of convenience,
illustrates this V-complex.
the set brackets
have been deleted.
}
{
In the fiaure, A(0) is the origin,
A(i) corresponds to one of the four axes emanating from the origin, and
A(i,j) corresponds to one of the four quadrants.
Figure 1(a) is by no means the only V-complex associated with
IR
2
IR
Figures 1(b) and 1(c) demonstrate other V-complexes associated with
.
2
,
with the triangulations omitted.
is a V-complex.
K
Suppose
T
O
=
Tt^
X
Let
x
We define
K.
e
T
xtA(T)
T
X
if
then is the smallest set
=
|x|
simplex in
|t
I
A(T
such that
T
x 6 A(T)
.
We say
is full
x
+1.
X is a full simplex if it is a maximum-dimension
TfJ.
we also define
)
For each
8'A(T)
=
as
3'A(T)
{x €
3A(T)1t = T}.
We illustrate the above definitions in the V-complex in Figure 2.
In the fiflure,
the left-most vertex of the 2-simplex is
line segment is
A(l)
,
the simplex itself is
For X
For
=
X
{e,h,f}
8'A(1,2)
b to c.
{k,Jl}
=
the left-sided line segment is
T
T
are all full.
=
the "bottom"
A(2), and
{1,2}, for
{1,2}.
=
x
The simplices
9'A(1)
=
{c},
{f,g},
{a},
8'A(2)
T
=
{1}.
{f,g}, and
=
{b}, and
pseudomanifold corresponding to the line segment from
is the
9'A(1,2), whereas
For T
=
We have
Thus, while both
(.
,
A(l,2).
{d,e},
{e,f,h},
A(0)
=
0,
{k,£,}
{f,g}
and
^
{f,g}
are elements of
3A(1,2),
3'A(1,2).
A(T) contains only one vertex, the origin, and the
empty set 0, and therefore
3'A(0)
=
{0}.
Figure 1(a)
Af2)
A0.2)
^(2.3)
-ACD-
"MW
AC3)
A(1,3)
Figure
1
(b^
A(4)
A(6)
Figure
1(c)
- 11 -
Let
K
be a V-complex with label set
that assigns to each
v<
is a labelling function .
^^^^
^^
"
L(v).
L(x)
K
an element
For a simplex
=
be a function
L(.)
Such a function
i e N.
x
Let
N.
{v
,
. . . ,
L(.)
€ K, we define
v""}
is the set of labels spanned by the elements
of X.
We define two distinct simplices
X
"v
«
X and y are full
L (xny) = T
ii)
X
U T
Note that adjacency is symmetric:
y
.
x
'V'
y
if and only if
if X ~ y, L(x)D Lixi\y) =
Figure
3
T^^T^ D
y
'v
x.
To see this, observe that
Also note that if x - y for some y, L(x)DT^.
•
K to be adjacent (written
y) if
1)
and
x,y
T^.
represents a V-complex whose vertices K
have labels L(.)-
In the figure, we have the following adjacent simplices:
{a}'v{a,b}'\.{b,c}'v{c,d}'v{c,d,u}'\-{u,d,w}'\,{u,w,v}
{u,s,vK{s,u,t}'v{t,s,p}'\,{p,t,q}'\,{p,q}'v,{p,n}'v{n,m}'v{n,m,r}'\,{m,r,k}'v,{k,r,s}'\,{s,k,v}
{h,g,w}'v.{g,w,e}'\,{w,e,d}'v,{d,e}'\,{e,f}.
Observe that, in the figure, any full simplex is adjacent to at most
two other simplices.
We shall see later on that this is true in general.
Observe also that the adjacency relationship results in the formation of three
distinct "paths" of simplices, each path being a string of simplices
adjacent to one another.
The purpose of the remainder of Part
characterization of these paths.
I
of this study is to give a
However, we must develop the theory of
V-complexes further before a complete characterization is possible.
,3A(2)
- 12 -
H-Complexes
K
We wish to "lift"
-
n
q
,
CL.
Into a pseudomanlfold of dimension
Without loss of generality, asstnne
In],
be the set of vertices of
Let K
..., q
-
and admissible sets 7
N
Let R be a V-complex with label set
Let
.
Q|1 € N
{q, €
-
Q
N,
{q
,
N
-
n
.
where
.... n}.
{1,
We define artlflcal vertices
K.
Define
..., q^}.
t}. The H-complex K associated with the V-complex K is defined
-
K
{x U Q|x
U
Q
i
X6K, QCQ
0,
},
^x
where K* is the set of 0-simpllces of K.
The nomenclature "H-complex" is
short for homogeneous -dimension complex and derives from Theorem 2, below,
which states that K is an n-dimensional pseudomanlfold, i.e., its dimension
does not vary.
K
Is an n-dlmenslonal pseudomanlfold.
Theorem
2
PROOF
Clearly
.
.
Let
X U Q € K.
Let
P
"
Q
is closed under nonempty subsets, and so Is a complex.
K
Then there exists
Then we have
.
y € A(T
)
that Is full and
xUQCy^P€
K.
lyl+n-|T^|
-
y
D
-a.
Furthermore,
^x
|yUp|
-
lyl
+
|P|
Therefore every simplex in
-
I^
is a subset of an
|y|+n-(|y|-l)
n-simplex in
=
n+1.
It only
K.
remains to show that each (n-l)-simplex of K is contained in at most two
n-simplices.
Let
X
(n-1) -simplex In
«
X
O
Q^
be an n-simplex in
Suppose
K.
y
C
z i x,
and
R, and let
z
Is an n-slmplex,
z
is uniquely determined by
x is full and
0,.
"
Q-j
'
C
x
be an
Is an n-simplex In
_
We aim to show that
9*
x
-
"v
and
y.
K.
Since
^® ^^^^ three cases:
x
.
- 13 -
Case 1
= Q
then Q
z
z
D
= T
T
z
w, and hence
Case
2
x\{v}
U
z = y
have
= T
X
•
U
x
Therefore
Ifz
z
Therefore
for some w.
{w}
zUQ.
Let z =
= x,
But since
x.
?*
N
= Q
Q
{q.},
By property (viii) of V-complexes, the choice of
'--'
\{v}.
y = x
where
Q
w
for some
{w}
v 6 x, and
for some
The choice of
.
Q
is unique.
y = y
can write
z
z =
V^ {i}.
X
z,
y =
.
q. €
and so z = x, a contradiction.
,
X
for some
^q.)
X, we must have
and so
T
\
y = x
.
K
fe
w
,
^ y,
x\
y is not full, we must
Since
and hence
is uniquely determined,
w
We
is not full.
{v}
Q^ = Q^, whence
since
A(T X ) is a
pseudomanif old
Case
3
y = x \{v}
.
y = y
write
for some
i €
UQ X
T
X
,
,
where
K
y = x
v
€ x,
\
(v).
and
x
Since
\
{v}
is full.
z
= y
U
containing
{w} for any
'^
is
=
z
w € K
.
Again we
= T
is full, T
y
and by property (viii) of V-complexes,
Hence we cannot have
n-simplex of
for some
y
X
y €
9A(T
\
X
{i}
).
Therefore the unique
yUQU {q.}. ®
We illustrate this result in Figure
4.
In Figure 4(c), A(0) consists of the north and south "poles" of the circle
and
A(l)
,
A(2) are the right and left arcs, respectively.
Our next task is a characterization of the boundary of
Lemma
3
.
3K
= S, (J S2, where
Si = iy
^
Qy
(^
K|y € 3'A(Ty)}, and
S2 = {y
^Qy
^
KlNXdlq^e
Q„}< J
}.
K.
'2K f1.2})
3F= (*, (i;.
Figure 4(a)
9== (0,
A(2)
(1),
A(1)
A(«)
Figure 4(b)
^=
(4)
,
(1
},
Figure
(2)]
4
(c)
[2})
.
- 14 -
We first prove that
PROOF.
element of
contains
S^
is of the form
y
,
1
C
y £ 3'A(T
Then
.
1
S
^
be a maximal
Q
y
U
{v}
The only n-simplex that
.
Q = Q
'•y
y
and
)
y
U
y
y = y
Let
9^.
Q
is uniquely determined
where v
,
y
y€
since
Therefore
9'A(Ty).
S„C9K.
Next we prove that
of
Then
S„.
L
U
X = y
Let
T
{i}\
IJ
and so
= Q„
iy
a t K
,
j
?*
{5!, whereby
j
= i,
C &K.
S2
y
Therefore S^
U
\
*-*
be a maximal element
Q
1
\)
is uniquely
a
{i}^
U {i}\{j}^tJ
T
J
.
.
Hence
and in fact
Therefore
a contradiction.
C
.
y
since the set T
Then
U {i}^^
for some i, where T
{q.}
We need to show
K.
i.
S2
y = y
Let
Suppose
for some j.
= T
Q
8K.
{a} be an n-simplex in
We cannot have
determined.
a = q
is full and
y
C
S^
a = q^,
8K.
Let
Now let us prove the converse.
x = x
U
be an n-simplex
Q_
x
in
and let
K
C
y
be an element of
x
8K,
y = y
where
U
Q.
We have
two cases:
Case 1
X = y
.
Case
{v} for some v <
U
y = y
full, then
Hence
U
Q
.
Q
Tx
= Q L/ {q^
y ^ 9K.
U
T
X
there is a unique
U
{v}
{i|q^
6
for some q. ^
J
)
J
This means that
N\
Clearly we must have
Therefore
y
y €
8A(T^)
If y were
.
is not full.
y 6 S-
2
y = y
and so
,
y.
W
v
Q t R,
Q}^ a
Therefore
,
(j)
<f
K°
^l
.
s.t.
and hence
and
Q.
Suppose
N
\{ilq^
€
Q} ^ J
.
But then, by property (viii) of V-complexes,
y
U
{v} € A(T
X
\J
(j)).
y ^ 3K, a contradiction.
y € S^.
9K C S^ 17 S2, so
9K = S^
U S2. ®
Hence
Therefore
- 15 -
Labelling Vertices and Adjacency on H-Complexes
R
Let
K
L(.):
»•
L(q,) " 1
a labelling function on
K
M.
Its associated H-complex.
be a labelling function on
N
by the simple rule that
L(i) -
K
be a V-complex and
for each
x
Let
.
1)
and
11)
Q,
be a simplex In
We define two distinct n-slmpllces
x
q. £
L(.)
K°
to
thereby obtaining
K.
We define
L(v).
v«x
(written
We extend
K°.
Let
'^
L(x
to be adjacent
If
y)
X
K
x, y €
and
y
A
are n-slmpllces
y) - N
The above definition of adjacency Is quite standard for labelling functions
on pseudomanlfolds (see Gould and Tolle [8] or Lemke and Grotzinger [18]).
Note that If
x
y, x
'v-
and
must be neighbors.
y
Characterization of Paths on H-Complexes
Let
K
K
be a V-complex,
be a labelling function on
L(.)
its associated H-complex, and let
K, extended to
K.
The following theorem,
whose proof we omit, follows from the standard "ghost story" argument
of complementary pivot theory (see Lemke [17], Gould and Tolle [8],
Kuhn [11], Eaves [l], or Scarf [21].
Theorem
4
.
Let
be an n-slmplex of
x
most two other n-slmpllces of
of
K.
If
Then
K.
x
y e 3K.
is adjacent to at
is adjacent to only one n-simplex
K, then there is a unique (n-1) -simplex
and
x
y
C
^
G
-=
such that
L(y) « N
Q
We define
The notations
B - {x € S jKx) - N}
B
and
{x 6 S |L(x)
and G are short for "bad" and "good".
•=
N}.
In most applications,
a path- following scheme will terminate with an element of B or G.
G
typically
contains those simplices with pre-specif ied desirable properties, whereas B does
not.
«
.
- 16 -
Proposition
PROOF
5
G =
(9 .
Then
x € G.
Suppose
.
A
B
.
U
We can write
x = x
otherwise
is not full.
x
where
Q,
Therefore
Let
K.
i
But then
4, we can construct and
andx
L(x^
)
of the sequence, define
i+1
= N
and
x
If
dK.
€
x
to be the unique subset of
x
k-1
= N
)
and
x,
added, is called a path on
I
1
.
X
x.-v x.^
ii)
Type II
.
Note that endpoints are elements of G
K.
iii)
iv)
as one of six types.
K
for all
-°°<
for any i
^^
i
<-H=°
j
i
X.
0,
1
ii)
such that
x
where the sequence has no endpoints, and
\
i
i)
4 X,
-x.
{x
is a lef t-endpoint
where the sequence has no endpoints, and
r
i)
k
K.
We can characterize paths on
Type
x,
The new sequence, with possible endpoints
€ 9K.
,
k-1
If
to be the unique subset
x
iC~ J.
L(x
since
characterize
for any i.
^ x
i-1
i+1
i
such that
x
,
x ^ B. (S
is a right-endpoint of the sequence, define
of
x ^ S.
X
.
x...
1+1
for all
There is an m>2
X
1
^ X
1
I
-<»
for all -«
X
?*
K
.
be a maximal sequence of n-simplices of K
i^.f
)=N, x-vx
L(x
such that
is a maximal element of S
x
is full.
x
With the help of Theorem
"paths" on
= n, so
jx]
<i<+»
< i <
such that
for any
0<k<m.
+
x_,
1
= x
.
,
i+m
for
all-»<i<+"'
L/ B.
.
- 17 -
Type III
ix
,
f
where the sequence consists of only three elements.
.
Xq, x^, X2, and
say
Xq, X2 t
i)
iii)
Type IV
ii)
iii)
Type V
i)
.
/ X
,
C
x„
x^
.
and
B,
x„ / x
and
for any
.
U
< i <
i^'j,
m
m-1
0<i,j<m.
has only a left endpoint, say
Xq € G
X
X.
^ X.
x^, and
Ub
^
i>0
for all
for all
i,
j
2 0,
i
^
j
has only a right endpoint, say
x.f.
G W
m e
X
ii)
X.
,
'V
iii)
X.
i"
X.
I
1
1+1
iii)
i)
A type
X
X
"v
^
,
for all
'\'
1
X.
.
m
X
ii)
Type VI
x
U
^x.{.
.
and
X.
x^, X € G
u
m
.
x^.,
has more than three elements, and has two endpoints,
f
say
i)
Xq / X2
B,
is an n-simplex and
x^
^x
.
U
= N
L(x^)
ii)
G
1-1
x
,
and
B
X.
1
for all
for all
1
i,
<m
j
<m,
i
?^
j
path stretches infinitely in both directions.
is a loop.
A type II path
A type III path is a "degenerate" path consisting of one
n-simplex and two of its (n-1) subsimplices.
with two endpoints.
A type IV path is a path
A Type V or Type VI path consists of one endpoint
and stretches infinitely in one direction.
In the applications of V-complexes and H-complexes, it is the
endpoints of paths that are of interest.
We have the following lemma
.
.
- 18 -
Lemma
6
X € G
<J
PROOF
Let
.
x
If X is an endpolnt of a path, by definition
x^ GU
Conversely, let
a 6 K
for some
7
i
G
(7
z
= x
B.
^
{a}
We can construct a path starting at
K
and
is finite, B
K
If
.
If
.
There is a unique n-simplex
B.
L(z) = N.
and
,
x
= z, etc. (S
X.
,
Corollary
PROOF
is an endpolnt of a path if and only if
x
B.
.
X = x
Then
K.
fe
have the same parity.
G
the total number of endpoints of paths is finite
is finite,
Each endpolnt is in exactly one of the two sets above; hence,
and even.
they have the same parity.
®
Characterization of Paths on V-Complexes
The characterization of paths on V-complexes is achieved by
certain equivalence relationships between V-complexes and
establishing
The first equivalence is given in the following lemma.
H-complexes.
Lemma
8
y = y
U
PROOF
.
Q
and
x
Let
.
Then
.
Suppose
N = L(x
x
A
x
'^
-v
y.
y)
y
be n-simplices of
y
if and only if
This means
= L(x
L(Q^
A
(N\T
U
Q
,
x ^ y
L(x
A y)U
x = x
Let
K.
r\
A
y)
We have
= N.
Q^) = L(x
f\y)U ((N\(T^) U (N\Ty))
Therefore
L(x
r\
y)
Thus we see that
=
N\((N-ST
L(x
A
in reverse shows that if
Define
X
)
= T
y)
x
G = {x f k|x
-x-
X
y,
(J
T
,
y
N^(N\(T X U
=
))
and so
x
'v^
y.
T )) = T U T
X
y
y
The same argument
y
then
is full,
x
'^
y
L(x)
.
D
^
T^, and
L(x) ^
T
}•
.
- 19 -
G can be thought of as the goal set , for in most applications of V-complexes,
the algorithm searches for an element of
G.
B = {x € KJx 6 a'A(T ), and
X
Define
L(x) = T
}.
We have the
x € B
if and only
X
following lemmas:
Lemma
if
9
Let
.
U
x = x
x € K, and let
Q
Then
.
X 6 B.
PROOF
Let
.
L(x) = T
X € B.
x€
Furthermore,
9'A(T
X
Conversely let
Lemma 10
PROOF.
Let
T^^ iiX'J
N\{i|q
L(x) =
.
e Q
X
Li
=
\(x)
\X/
=
}
L(x)U L(Q
X
= L(x)
)
^
Let
X € K
x €
Q
<J
x £ K,
From Lemma 8
PROOF.
X = X
For any
.
X
U
'
(J
and y = y
Q
X
so is X.
(^
,
we have
W
Q
.
y
for some
{1}
^°
{j})) = T
X
^
{j) ^
= N.
"^
x
T
^
~
i
X
N\L(Q
)
,
€
G.
where
Furthermore,
.
Therefore
=N\{i|q
K
N.
^^^°'
^ ^ ^-
Then L(x)
deg(x)
L(x) =
Then x £ G if and only if x
H.X;
to be the number of distinct simplices of
Lemma 11
=
)
^
®
€
Q
i
We define the degree of
be full.
\ T
B.
and
)
(N
V^
*
^
U (N\(L(x))
G.
= T
)
xe
Therefore
x € 9'A(T
^''j^^^T
N\(N\(T U
Hx^
x € G.
"^
^T
Conversely, let
Hence
x € B.
L(x) = T
Then
X € G.
.
1
Then
Let x e K and let x = x
.
L(Q
^x
xfe S
x € B.
= N\(N\T3j) = T^, whence
U
L(x) = L(x)
.
so
,
)
X
x e G.
s) i
L (x^
1
.
written
x,
deg(x),
adjacent to x.
< 2.
'^
y
Since x
if and only if
x
^'
y, where
is adjacent to at most two simplices,
.
- 20 -
With the help of Lemma 11, we can construct paths on
K
be a maximal sequence of full simplices in
^i-1 *
Let
such that
be a left endpoint of the sequence.
of the associated sequence in
define
such that
x^
vise, if
Define
K.
= x^_
x^_
U
Q
\.
,
x^
Lemma 12
Let
.
x
x € G
PROOF
X
Let
.
K.
fe
and only if
U
Lemma 13
.
B
.
Then
^c
,,
C
Q)
x€GUB.®
If K
if finite,
K.
K
B
is finite,
G have the same parity.
K.
K
Then
x = x
G
and
U
i)
X
.
ii)
X
.
'^^
?^
'^\+-[
X
.
Q
(for
have the same parity.
so is
K.
B
(G)
There is
and elements of B
Thus, by Corollary 7, B and
K
as one of six types.
where the sequence has no endpoints, and
,
if
®
Hence we can classify paths on
\
Like-
The new
analogously.
We thus see a complete equivalence between paths on
^x
C Q.
G, by definition, have no simplices in common.
and
Also, if
.
Q
is an endpoint of a path on K.
a one-to-one correspondence between elements of
I
>
as in the last section and
is an endpoint of a path on
x
be an endpoint of a path on
So
x€GUB.
Type
^j,-,
B.
appropriate choice of Q
.
'^
is a left endpoint
for appropriate
is a right endpoint, define
jL
Then
sequence, with possible endpoints added, is a path on
(G)
x
V.
\+l*
x^
PROOF
^x
Let
K.
^°^ ^^^ ^
for all
i
?^
j
K
and on
K.
- 21 -
Type II
'(x
.
x^
1)
X
'\'
x^_
ii)
iv)
x^ = x
1
i+m
such that
> 2
for all i
0<k<m.
for all i, all
Xq, x^, x^, and
^0'
"^2
L(xJ
ii)
iii)
x
iii)
Type V.
Xu
and x
m
,
x
,
C x
.
and
U
x^
for all
'V'
x^_j_j^
x^ / X.
and
x^
?i
< i <
i
=^
j,
x
m
m-1
0<i, j<m.
has only a left endpoint, say
f
G
ii)
x^
'\-
x^^^
iii)
x^
7^
x.
for all
for any i,
i >
j
>0,
i ^ j
has only a right endpoint, say
i)
x^e
G KJB
ii)
x^_-|^
'V
x^ / X.
Xq, and
UB
Xq
^x^*^^
B
for any
i)
iii)
x
x^, x € G
U
m
^'^±(±
.
x
has more than three elements, and has two endpoints,
say
ii)
T
^
is full and
{x^}^
i)
^^
^
2>
1
Type VI
m
\x^^^, where the sequence consists of only three elements,
.
i)
.
for all i
^ X
x^ / x^^^
say
Type IV
for all i
There is an
ill)
Type III
where the sequence has no endpoints, and
,
r
x^
for all
for all i,
i<m
j
<m,
i
/
j
x^, and
.
- 22 -
There are two ways to develop an algorithm based on a V-complex
L(.). depending on the nature of the set
and a labelling function
If
set
consists of a single 0-almplex, say
A(0)
0, then
L(0) -
and
6 3'A(0)
since
« B,
If
have
V
-v-
consists of two 0-slmpllces, say
A(0)
L({v}
w, since
{w}) = L(0) -
(\
=
U
and the empty
{w}
Thus our
= T^.
endpolnt is
algorithm consists of following a path whose
-
0.
and
{v}
T^v}^
{w}
we
,
"^^^
"^{w}'
{v}
the algorithm consists of following the path containing
A(0)
and
{w}
in one or both directions.
The purpose of the preceding exposition has been to show how to construct
We used the construction of an H-complex to
and follow paths on a V-complex.
expedite the development of the theory.
It should be noted that the
characterization of paths on a V-complex can be demonstrated without resorting
to the H-complex superstructure.
However, the introduction of the H-complex
renders the proofs less cumbersome, and shows the equivalence of
path-following on a V-complex and path- following on n-dimensional
pseudomani folds.
The latter is
ein
"ordinary" phenomenon familiar to most
researchers in complementary pivot theory.
When viewed properly, path
following on a V-complex is equivalent to path- following on n-dimensional
pseudomanifolds
and can be viewed as the "projection" onto
,
of
K
path- following on K.
In most algorithms based on V-complexes, we search for an element
of
G.
We have seen that the set
properties of J
A(T)
,
G
is derived from the structural
and hence the way our complex
K
is divided up into the
is intimately connected to what we can expect to look for In an
algorittin on K.
Conversely, suppose we wish to find elements
L(x) 6 G, where
with certain labels
the space Into
A(T)
,
to our stated purpose.
T ^
J
,
such that
is some set.
G
G
arises from
x
of
K
If we can divide
*]
,
we are close
- 23 -
Orientation
This section develops an orientation theory on V- and H-complexes.
At
issue are conditions which guarantee that paths of siinplices have certain
orientation properties.
This is accomplished in theorems 27 and 28, which
give necessary and sufficient conditions on a V-complex for its associated
H-complex to be orientable.
The orientation theory is developed in the
context of H-complexes, which are pseudomanifolds
.
This material could be
developed for V-complexes without explicit reference to the associated
H-complex.
However, the development would be much more cumbersome, and would
not show the implicit equivalence with orientation on pseudomanifolds.
As this material is not central to this study, and the subsequent
development is not very elegant, the reader can omit this section without
detracting from the exposition.
- 24 -
The use of orientation In complementary pivot schemes was first
developed by Shapley [24] for the linear complementarity problem, advanced
[3] and Eaves
by Eaves and Scarf
[2]
for subdivided manifolds, and extended to
An extensive treatment of orientation
pseudomanlfolds by Lemke and Grotzlnger [18],
P^vot algorithms Is presented in [28).
in the context of complementary
Our set-up
the interested
than that of Lemke and Grotzlnger; however,
i. .lightly different
r«ader can easily establish the similarity.
Pivots and C-Plvots on Pseudomanlfolds
be an orlentable H-complex of dimension n, oriented
K
Let
K
by Or(.). with vertex set
N = (l,
Let
.
.
. . ,
L(-):K
and let
n)
•*
N.
We define the set
D={x€K||x|
= n+1, L(x) = N}(J {x £ 3K| |x|
K
slmpllcles on the boundary of
whose labels exhaust
K
D consists then of n-slmpllces of
whose labels exhaust
that the two sets above whose union is
by
D,
X.
,
X € D
If
,
Denote these sets
There is a very natural way to order the elements
X € D.
,
we can write
x = {v
1
(v,
We remark
N.
are disjoint.
D
N, and
Dj, respectively.
and
Let
of
= n, L(x) = N}.
V
^0
)
of
On
,
.
.
. ,
v
}
.
The ordering
Is called a C -orderlng if and only if:
X
^n
L(v^
)
= j.
j
Note there are always twofbrderings of
For
of
X
have the label
of
X
must be the unique vertex
by
v'
and
v"
r.
j <
v
those two vertices in
two C-orderings of
x
are:
r €
N\{r}, the
C x
x
..., n.
The reason for this is that
x.
x, there is some unique
among the labels of
= 1,
N
j—
such that two vertices
component of a C-orderlng
for which L(v
)
whose labels are
" j.
Denote
r.
Then the
.
.
- 25 -
(v", V
..., V
,
V,
,
V
r-1
1
,
.... V
^r+1
)
n
and
(v', V
.... V
,
v", V
,
r-1
1
,
V
,..,
)
n
"-r+l
Also note that these two orderings have opposite orientations, i.e. one
is
and the other is (-)
(+)
X 6 D
If
in
we can write
,
x = {v.,
2
(v
.
,
.
.
V
,
^1
..., v }.
The ordering
is called a C -ordering if and only if
)
^n
L(v.
_
The C-ordering for
= j,
)
j
= 1,
..., n.
^j
is unique.
x € D2
With the notion of a pivot in mind, we now define a C- pivot on
elements of
For
D.
x € D
A C-pivot is performed on
Case
1
{v.
.
.
,
.
.
,
V
.
,
y = {v
.
.
v^
,
..., v.
)
be a C-ordering of
In this case, simply drop
from
v.
x,
^0
..., v
,
x.
The derived ordering on
}.
is
y
n
)
n
1
Case
,
as follows:
€ 8K.
1
(v^
(v^
n
1
and let
x
}
.
let
.
2
{v.
il
.
such that
..,, v
,
{v^
,
..., v^
,
v}
n
,
.. .,
r+1
If
,
Let
n-simplex of
L(v)
v} ^ K.
V.
)
^n
X ^ D
r^
for some
and form the new ordering
,
of y.
let
V
(v
In
.
,
.
.
,
V
,
v
Set
N.
,
v.
L(v) = r
A
x.
:
for some
r 6
K
such that
N.
Set
y =
U
x
{v,
X4
,
is an
{v}
.
.
.
,
1
and from the new ordering
,
^r-1
be the C-ordering of
)
v
...,
,
^1
X
X^
be the unique element of
K.
(v
^r
is performed as follow^
X
= r
_
^
C -pivot on
v 6 K
_
_
n
1
V, V.
In this case, there is a unique
_
..., v^
,
1
y = {v^
d 9K.
}
in
(v
,
r
v
,
^1
. . .
,
v
,
^r-1
v, v
,
^r+1
v
,
,
X
..., v
)
^n
v}
n
of
y.
?*
v.
^0
- 26 -
We have the following results on C-pivots:
Proposition 14
ordering of
Let
.
x 6 D
be derived from a C-pivot on
y
is a C-ordering and the orderings on
y
Then the
.
and
x
as specified
y
in the C-pivot have the same orientation.
The first conclusion of the proposition follows immediately from
PROOF.
the ordering defined on
The second conclusion follows from a case
y.
analysis.
Case 1.
Then Or( v^
y € 9K.
..., v^
,
Case
2
Or(v
.
,
..., V
)
= -Or(v, v
,
.... v
= 0r(v. , V. ,
^1
^r
ordering of
Let
.
y
..., V
= Or(v
)
n
,
^1
)
Ti
-^1
^n
Proposition 15
,
^1
n
Then
y i 3K.
^0
(-1)° Or(v
=
)
^0
..., v.
V, v.
,
..., v.
,
x 6 D
be derived from a CSpivot on
y
).
®
n
^r+1
^r-1
is a C-ordering and the orderings on
x
and
.
Then the
y, as specified
in the C-pivot, have opposite orientation.
PROOF.
y.
The first conclusion follows directly from the ordering fixed on
For the second conclusion, note that
Or(v, , V
,
^1
^r
,..,
v
V, V
,
r-1
= -Or(v, v^
,
In
,
.
..., V
"-r+l
.
.
,
v^
)
= -Or(vj^
)
n
In
,
.
.
.
,
v^^
)
®
.
Orientations on Paths Generated by C-Pivots
Let
K
-0
Or(-), K
be an orientable H-complex oriented by
vertex set, and assume, without loss of generality, that
Let
L(-):K°-^N
be a labelling function.
Let
possibly without left and/or right endpoints.
(
x
(
N = {1,
be a path on
its
..., n}.
K,
.... v
)
n
- 27 -
Choose
Note that
an element of the path.
X
L(x) = N.
Is an endpoint of the path (say a left endpoint, and we can assume
without loss of generality), there is a unique C-ordering of x
be derived from
by a C-pivot on
x
x
Lemma 16
)
Let
.
= Or(x
\X
is obtained from
f
Let
.
Or(x
and
)
,
y
= -Or(x
We have just proved the following
by Proposition 14.
)
be a path with left endpoint
.
by a C-pivot,
x
Or(x
)
x
= -Or(x
x
If
.
i>0.8>
for all
)
In particular, we have
Corollary 17
x„
and
X
m
,
Let
.
\x
be a path with left- and right-endpoints
)
generated by a series of C-pivots starting at
i)
x
.
Then
Or(x^) = -Or(x ),
m
U
and
11)
Or(x
)
= Or(x.)
0<i,
for all
j<m.
^
Corollary 17 is analogous to other path orientation theorems
presented elsewhere, see, for example, Shapley [23], Eaves and Scarf [3],
Eaves [2], and Lemke and Grotzinger [18].
All of these theorems assert
that the orientation along a path is constant except at the endpoints,
whose orientations are opposite in sign.
Now suppose that
is an element of the path
x.
1
X
is not an endpoint.
C-orderings of
x,
x
Then
,
Nx.V
i
x
6 D
.
and
i
Since L(x) = N, we can choose two
each one opposite in sign.
orderings will yield
)
For each of these pivots, we
the right endpoint of the path, if it exists.
Or(x
,
x = x
We can keep performing C-pivots on the x., until we reach
from Proposition 15.
have
y = x
Then
.
x
If
C-pivoting on one of these
and the C-ordering of
x
will have the same
- 28 -
orientation as the C-ordering of
x
we will generate the path elements
\x V has
only if
for all
j > i.
a
Continuing the C-pivot process,
.
x
,
x
,
x
,
...,
terminating if and
By Proposition 14,
right endpoint.
Or(x
)
= Or(x
)
A parallel arginnent for the other C-ordering completes the
proof of the following.
LeTTima
18
Let
.
\x
be a path on
)
and let
be an element of this
x
Let the entire path be generated from
path that is not an endpoint.
We have
by its two C-orderings.
K
Or(x.) = -Or(x^)
for all
j < i <
k.
x
®
In particular, we have
Corollary 19
X
and X
m
,
.
Let
\x
)
respectively.
with left and right endpoints
If this path is generated from
x
,
then
Or(x^) = -Or(x ),
m
and
ii)
x
,
i
by the two C-orderings of
i)
be a path on K
Or(x
)
=
-Or(x
)
for all
j<i<k.
(X>
0<l<m,
- 29 -
By way of concluding thus far, we remark that the usual path
orientation results for manifolds carry over to H-complexes.
they do more than this
— they
Actually,
carry over to orientable n-pseudomanifolds.
For the only properties of H-complexes used in these two sections was that
K
is an orientable n-pseudomanifold and that the label set
n
elements.
N
contains
Conditions for which an H-complex is Orientable
In this section we give conditions on
K.J^N, and
Let
is orientable.
the H-complex associated with
16^
for each
i)
,
define a
A(')
Let
K.
A(T)
K
|n|
= n.
K
that guarantee that
V-complex, and let
K
be
Assume that
is orientable, and hence homogeneous,
and
for all
ii)
S^fe
1,1
= 0,
T£ J
S,
..., m,
,
there is a sequence
Sq = S, S^ = T, and
We will show that K,T,N, and
K
only if
A(')
\s^_-^ AS^|
S
,
..., S
^ 1,
1 =
such that
m
1,
.... m.
satisfy the above two assumptions if and
is orientable.
Towards proving our main result, we make the following:
Definition
.
For
A(T) =
A(T)
q
,
T
€
{xU
tJ
,
define
qIx e A(T), Q
C
Q^,
xU
Q ^ 0}.
can be thought of as a conical construction of
1 ^
T.
Lemma 20.
We have:
A(T)
is an orientable n-pseudomanlfold.
A(T)
with each
- 30 -
PROOF
Clearly
.
Then there is a
y
Then
x
manifold.
is closed under nonempty subsets.
A(T)
U
Q
C
A(T)
-simplex in
^
y
Any n-simplex of
T|
such that
A(T)
in
U
y
]
=
Q^|
x U Q
+ n
1
y
^ A(T)
.
is a pseudo-
A(T)
+
T|
UQ
x
-
T|
|
is of the form
A(T)
Let
.
Q^, and
= T, since
T
Let
= n+1.
|
L'
Q
,
where
be an (n-1) -simplex in
is a
y
A(T)
that is a
,
I
y
U
U
Q_.
T
subset of
U
z
Q
y
?^
z
Q
the choice of
But then
are
x =
sequence
^
l^i
.
.
.
X
L/
"
U
s,
s„,
,
A
\is^^ Qj)
...,
s
i =
U
and since
Uq
y
A(T)
is an n-pseudomanif old.
TI
I
(s^+i^ Q^)|
Then
.
-simplices in
x
there is a
such that
A(T)
'
'
Then
x
U
Q^ =
s^U
is a sequence of n-simplices in
Q,r.
A(T)
is homogeneous,
Since A(T)
.
•••. in-1-
1.
is a pseudomanifold,
A(T)
be n-simplices in
= y of
m
and
= n+1,
i
A(T)
z
l"^!'
= y
Q-T.
and
Q
= Q_,
z
This proves
-simplices in
1
^i+ll
s
,
|t|
Q
is unique.
z
Now let
and y
x^QCzt/Q,|zUQl
Suppose
.
= n,
i = 1,
..., m-1.
s^U
Q^,
and
A(T)
Therefore
Q^,
A(T)
is
homogeneous.
coherent orientation of the
n-simplex of
(v
A(T)
.
Let
..., V ), and let
,
those
V. €
Q
1
If
=
Q^.
-simplices of
= t.
A(T)
.
Let
Order the vertices of
Or(')
x
x
c'
U
be a
Q
Q
be an
as
be the number of transpositions needed to transpose
p
to the last n-t places of the ordering, while preserving
Then we define
is obtained from
(-1)^ (-Or(x)) =
v
€
Or(v^,
1
y € Q
|t|
|t|
the local ordering of those
V. e
Let
A(T) is oricntable.
Finally, we show that
u
x
-Or(xUQ^).
and the local ordering of those
A(T)
..., v
UQ^
Thus
n
)
=
P Or(x)
(-1)^
P
Or(y)
by a pivot, Or(yOQ^) = (-1)
A(T)
is
orientable.®
.
.
- 31 -
Lemma 21
PROOF ;
Q
C
^ =
.
Let
A(T)
'^^
Then we can write
x e K.
Q, and
C
Q
Q^
x€
But then
.
U
x = x
A(T^)
.
Q
where
x € K, and
Conversely, let
^Q€
x
A(T)
X
Then
C
Q
x U Q C K.
and so
Q
®
X
Lemma 22
PROOF
Any n-slmplex of
.
Let
;
x
is an element of exactly one
be an n-simplex in
Q
\J
K
K.
Then
Q =
A(T)
and
Q^,
x
is full.
X
X U Q € A(T^).
Thus
and hence
S
^
T
Also
.
Q„
Thus we see that as
partition
K
xU
Suppose
Q t A(S)
S €
for some
CT
'J
C
Qq
T
ranges over all elements of
which implies
into "disjoint" n-pseudomanifolds.
S
Thus
.
x€
Then
.
"J
= T
S
.
(&
A(T)
the
,
A(S)
We use disjoint loosely
since this partitioning only takes place among the n-simplices of
K.
Next we have
Proposition 23
PROOF
Let
:
K
.
is homogeneous.
and
x
y
be n-simplices in
x = x
We can write
K.
f
Q
,
^x
y = y
U
sequence
'^i ^
for appropriate
Q„
=
T^^
Vl'
...,
T-^,
i = 1»
= ^'
x, y £ K.
= T
T^^
By Assumption (ii), there is a
such that
T^i"^
,
i = 1,
.
.
.
m, and
,
•••' ""-I-
We shall now show how to construct a sequence of neighboring simplices
in
K
If
m =
that have
1,
X, y 6 A(T)
y
as endpoints, using an induction argument on m.
then such a sequence of neighboring simplices exists because
C
simplices <s
and
K,
i
) ~
i
either T^ =
and
x
1=0
i=
T^.^^
U
A(T)
exists
{k}
is homogeneous.
Suppose a sequence of neighboring
whose endpoints are
for some
k <
Tjjj.;^'
°^
x
"^m
and
=
z
€ T
.
m-1
T^-i^^^}
Then
for some k ^
Tj^.^.
.
- 32 -
In the former case,
for some unique
w€
z
K
= z \{q,
k
}U{w}
A(T
Since
.
^t
of neighboring n-simplices
V ^_
m
Is an n-simplex in
is homogeneous,
)
where
,
= z,
t
t.
that is in
K,
A(T ),
m
there is a sequence
= y.
Thus the
sequence
X = s^,
of neighboring simplices has
...,
- %
= z, z = tQ,
s
and
y
argument establishes the result when
T
x
...,
= y
tj
An analogous
as Its endpoints.
= T
m
,\{k}.
m-1
®
K
is
J
implies
The next results will also be used in the proof that
orientable.
Proposition 24
s
:>
.
T* € 1 such that
There is a unique set
S ^
T?
PROOF
Define
.
Clearly
T*=
Q^
Proposition 25
.
Let
S, T £
^
,
contains no
(n-1) -simplices.
PROOF
x U Q € A(S) C\ A(T)
Let
X € A(S), X € A(T), so
X
€
< n -1
Therefore
.
x
U
Q
T,
?^
.
(t+1)
.
contains
S ^
and any
|S|
Let
A(S CXl)
Q C Qg r\ Q^ = Qgy^ < n -
Also
S
t "^
T*.
®
is uniquely determined.
T*
.
T*
Then
S.
t
=It|
=
But
.
.
=
|S|
|S
A
A(S)
|T|
We have
.
T| < t-1.
|xL/Q|<t
Thus
A
Then
cannot be an (n-l)-simplex.
-^
A(T)
Thus
[x] < *.
n-t-1
IS>
We are now ready to describe an inductive procedure for brienting
K.
T* t
Let
'3
m = maxlTl
Let
be the set described in Proposition 24.
Then we partition
- d.
m+1
into
tl
Let
classes,
d =
|
5',,
T*
..., "j+^>
"^
in
where
^ ^
i
'J^ =
>
Ju
^
il e ^ ||t|
J
~ ^'
=
k}
.
Note that
5
=
j^i^
Our procedure for orientating
^d+m'
K
^^^ ^°^
|
^-^-^
is as follows:
.
- 33 -
Step
Orient
.
(i = 1,
.
Or(')
A(T),
denote the orientation on
Or(.)
Let
.
K^ = K^_.,^^(
Let
.... b):
K^
TC
'^^.
Extend the orientation
A(T)).
.j,'^
by using the induced orientation on
to
A(T*)
= A(T*).
K
Set
Step i
A(T*)
to orient
K._,
We now show that each step of this procedure is executable and
the result is a coherent orientation of
Note
K.
= K.
K
Our proof is
as follows:
Clearly Step
Suppose steps 0,
tion of
K
Lemma 26
since
T' €
!J
K^_j^A A(T')
Then
^.
By the induction hypothesis
A(T').
So is
and
9K£_-j^
K^_i
9A(T').
is an orientable n-pseudomanifold.
L = K,_^r\A(T')
Let us denote
is an orientable
for notational convenience.
then is closed under nonempty subsets, and so is a complex.
L
Then
x € A(T)
(ii),
there exists
contain
Note
y
x
U
(x
for some
C
remarks, we know
U Q\{a}
|y<J Q^,|
U
T
=
T'\{k}
L,
of
a ^ Q,
3
A(T'\{k}).
= d
+
for some
/ a.
so
i
Let
k e T'
,
L, and suppose
,.
x
y
Q
fe
be full and
n.
Thus
L.
From the preceding
Q = Q„
and
x
\J
,
.
Q\{a} ^
Let
{6}
is an
We need to show that there is at most one choice
a 6
x.
We have two cases:
L.
By assumption
ye A(T'\{k})
+ n -(d +i) =
(n-1) -simplex of
L.
Let
xVQCy(JQ„,.
Then
be an (n-l)-simplex of
Q
QCQ
x € A(T'),
=T'\{k}.
T
be an (n-2)-simplex of
Clearly,
x 6
is a subset of an
L
X
d+i, and
such that
Furthermore
(n-1) -simplex of
6.
|t| <
such that
,
Q^, € L.
Now let
T,
k 6 T'
y)
every element of
X
is orientable.
The following lemma serves as a basis for our proof:
.
(n-l)-pseudomanifold and is a subset of
PROOF.
A(T*)
are executable and result in a coherent orienta-
i-1
...,
Suppose
.
is executable,
,
.
.
- 3A -
Case
x\{a}
I.
x\{a}
2.
for some
Since
(d+i-1)-
is a
{6}
A(T'\ {k}).
simplex of
Case
must be full,
(6)
^
such that x \ {a}
K
is the unique element of
3
x\{a}U
Then since
is not full.
^ k,
j
x \ {a}
U
T
.
T'\{k}\
=
.
x\{a}
{j }
£ T'.
j
must be a full
{6}
K
element of
In this case
is full.
x\{a} U
such that
l)-simplex,
(d +. i -
is a
{g}
+
(d
is the unique
B
- l)-simplex of
i
A(T'\{j}).
Thus we see that
is a pseudomanif old of dimension
L
Our next task is to show that
UQ„,
y
that each
U
..., s
T
=
z
Q
.
U
.
Q
...,
,
,
z \J
t
U
Q
,
X
Since
L.
VJ
Q„
\
T
{q, }.
= T'
k,
j,
where
a,
A(T'\{j}),
U
is
I
U
{g} =
x
U
\{k}
C
3K._,
.
for some
K
Any n-simpiex of
Let
k
fc
x
f
Q„,
Q„
i - 1.
,
and hence
t
x £ 8K._,
.
s
(J
T
T
.
,
j'
k 6
Q^,
,
T'
...,
,
s^ = z
...,
Q^, =
s
U
{a}
= y a sequence
t
U Q^
,
...,
,
s
l/Q^,,
L is homogeneous,
Thus
L.
be an (n-1) -simplex
is of the foran
K._,
)
....
,
T'
6
T', we can write
Thus the unique n-simplex of
,
such that
is a sequence of neighbors in
L
Q^,
j
x = s^,
_
<d +
X
K
6
Then the sequence
X
|t
k,
?f
j
Let
z
U
Suppose then
L.
Q
of neighbors such
= y
s
U
x
then since A(T
,
Then x U Q^, = s^
k.
A(T'\{k}).
{&) e
Next we show that
of
,
Then there exists
{k}).
A(T'\{k}).
of neighbors in
U
.
T'\{k}, for some
=
be a sequence of neighbors in
t
.
y
A(T'\{j}),
{a} €
=0,
j
,
= T
T
...,
,
is a sequence of neighbors in
,
A(T'\ {j}\
€
z
A(T^)
6
T'\{i}, T
X
Let
s
If
L.
x = s
is homogeneous, there is a sequence
Let
is homogeneous.
L
be distinct (n-l)-simplices in
(n-1)
y
l^
U
x
Q^, =
where
Q^
y
containing
A similar argument shows that
x ^
L
Q_,,
1)
3A(T').
1
,
- 35 -
It only remains to show that
orientable,
on
Or(*)
is orientable.
L
induces an orientation
K._,
Since
on
Or(«)
We need to show that this induced orientation is coherent.
Let us assign labels to elements of
be neighbors.
For
V
R
€
i—
y = {\+x'
i = 2,
U
V ^ X
,
*"' ^n^
"^2'
=1.
L(v)
let
y,
endpoint will be
K
,
K
x
(\
K
and since
,
Or(')
element of
L.
£ K
Define
of A(t')
from
,
y
At least one element of
.
L/Or(-)
.
.
,
,
..., v
L
H
y
L.
®
using the induced
..., v
be a fixed ordered
}
as a "seed".
U
A(T')
T' ^ J
..
.
,
v^} £ A(T')
..., v ), and extend
}
repeat this procedure for all
A(T)
x
the path will have
is coherent on
v^} 6 K^.^^, (3, v^,
= -Or(a, v,,
)
{6, v,
.
K._,
r\
x = irv,,
Now let
.
{a, v^^,
..., v
by using
K.
coherently oriented, and also
A(S)
}
x € ^^i_i» ^ ^ 3A(T'), there exist unique elements
Since
such that
Or(6, v,
n
This will trace
x.
is locally finite,
thus establishing that
orientation
6
v
which if it has a right endpoint, the right
With Lemma 26 established, we can orient
a,
,
.
From the results of the first part of this section,
right endpoint.
,
.
Furthermore, by the nature of our labelling function,
y.
will be an element of
Or(x) = -Or(y)
.
,
L(v^) = L(v^^^) = 1, L(v^) = i,
^^^
all elements of the path will contain
a
x, y ^ L
1
^"*^
a path of simplices of
Let
x = {v
Let us do C-pivots on the C-ordering of
..., n.
L C9Kj^_i.
as follows:
K._,
We can write
is
K.
Or(-) to all
This makes
A(T')
coherently oriented.
since for any S, T ^
A(T) contains no (n-l)-simplices or n-simplices, i.e.
We can
J
A(S)
and
share no common boundary.
Thus step i,
i =
1,
..., m,
results in a coherent orientation of
We have just proved:
of our procedure is executable and
K
,
i
.
Hence
.
K = K is orientable.
m
- 36 -
Theorem 27
Let
.
K
Then
section.
A(
•)
,
N,
.
tJ
K
satisfy assumptions
(i)
and (ii) of this
is orientable. (S
We also have:
Theorem 28
assumption
PROOF
(i)
A(-),'J, N, and K satisfy
and (ii) of this section.
Suppose assumption (i) does not hold.
.
But then
is not orientable.
A(T)
Then
be orientable.
K
Let
.
with the
q
i ^
,
T,
A(T)
,
there is no sequence
S =
i = 0,
..., m-1, and
S.e'J
is orientable,
it is homogeneous.
x = x^
n-simplices
Let
S.
=
! i
we must have
satisfied.
®
,
where
|
Sj
A
x^
,
x.
Sj^-,
|
S^.S,, ....
.
.
i
,
.
K = J^„ A(T) is not
Thus assumption (i) is satisfied.
Then for some
Suppose assumption (ii) does not hold.
K
,
the conical construction of A(T)
is not orientable, whence
orientable, a contradiction.
If
T^J
Then for some
,
= x
^ 1,
x.
*«J
= 0,
= y
.
= T,
S
such that
..., m.
Let
S, T 6 j
|s^ A S.^,
x e A(S)
,
|
,
^ 1
y € A(T).
Hence there is a sequence of
such that
x.
Because
x^^
a contradiction.
and
and
x.
-
x^+]^
are neighbors.
are neighbors,
Thus assumption (ii) is
- 37 -
Conclusion and Remarks
In Part
I
of this study, we have defined a V-complex and characterized
paths induced on simplices of a V-complex by a labelling function
L(«).
We have also demonstrated necessary and sufficient conditions for the H-complex
associated with a V-complex to be orientable.
Part II of this study will
use these results to present constructive algorithmic proofs of a variety
of combinatorial lemmas in topology^ some new, some old.
These lemmas give
^^J<,
to proofs of fixed point, antipodal point, and stationary point theorems,
and to a new set covering theorem on the simplex.
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^l 10
U72
Date Due
-':«•;
Lib-26-67
f>
BAV
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