c.
3i
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
VARIABLE DIMENSION COMPLEXES, PART II:
A UNIFIED APPROACH TO SOME COMBINATORIAL
LEMMAS IN TOPOLOGY
by
Robert M. Freund
Sloan W.P., No. 1516-84
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
July 1983
VARIABLE DIMENSION COMPLEXES, PART II:
A UNIFIED APPROACH TO SOME COMBINATORIAL
LEMMAS IN TOPOLOGY
by
Robert M. Freund
Sloan W.P., No. 1516-84
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.
Key Words
complementary pivot, fixed points, simplex, pseudomanif old,
orientation, stationary point, combinatorial tooology,
V-complex, antipodal point.
Abstract
Part II of this study uses the path-following theory of
labelled V-complexes developed in Part
I
to provide construc-
tive algorithmic proofs of a variety of combinatorial lenmas
in topology.
\Je
demonstrate two new dual lemmas on the n-dimen-
sional cube, and use a Generc.lized
S ^erner
Lemma to prove
a
gener lization of the Knaster-Kuratowski-y.azurkiewicz Covering
Lemma on the simplex.
We also show that Tucker's lemma can be
derived directly from the Borsuk-Ulam Theorem.
We report
the interrelationships between these results, Brouwer's Fixed
Point Theorem, -^the existence of stationary points on the
simplex^
Introduction
Part II of this study uses the path-following theory of labelled
V-complexes developed in Part
I to
provide constructive algorithmic proofs of
a variety of combinatorial lemmas in topology.
We demonstrate two new dual
lemmas on the n-dimensional cube, and use a Generalized Sperner Lemma to prove
a generalization of the Knaster-Kuratowski-Mazurkiewicz Covering Lemma on the
simplex.
We also show that Tucker's Lemma can be derived directly from the
Borsak-Ulam Theorem.
We report the interrelationships between these results,
the Brouwer Fixed Point Theorem, the Borsak-Ulam Theorem, the existence of
stationary points on the simplex, and two new theorems on the simplex relating
to the Generalized Covering Theorem.
In Section I, we give constructive proofs of the Sperner Lemma,
Scarf's Dual Sperner Lemma, and a Generalized Sperner Lemma,
We show that the
Generalized Sperner Lemma leads to simple proofs of the Sperner Lemma and the
Dual Sperner Lemma.
We show how each of these results is related to Brouwer 's
Fixed Point Theorem.
Section II introduces a Generalized Covering Theorem that generalizes
the Knaster-Kuratowski-Mazurkiewicz Lemma [12].
The Generalized Covering
Lemma is used to provide a proof of the existence of stationary points on the
simplex, as well as two new results on the simplex.
Section III deals with combinatorial lemmas on the cube that relate
to Brouwer 's Theorem.
Two new dual lemmas on the cube are introduced and
given constructive proofs.
We also give constructive proofs of Gale's Hex
Theorem and Kuhn's Strong Cubical Lemma.
and one of the new lemmas are equivalent.
It is shown that the Hex Theorem
Finally, the interrelationship
between these results and Brouwer 's Theorem are exposited.
- 2 -
In Section IV, we give a constructive proof of Tucker's Combinatorial
Lemma on the cube, and show that the Borsak-Ulam Theorem leads to a direct
proof of this lemma.
Notation
In addition the notation used in Part I, let S
the standard (n-1) -simplex.
<1
V
,
. .
.
,
Let c"* (x 6 IR
|
m\
V } denote the convex hull of
= (x
O^xie},
v
1
,
.
.
.
,
K
and
{x={v, ...,v}|^v,
=
K
= {v|v is a vertex of some
0-dimensional, then
I.
...,v
K
includes
C
/
D,
IR^je^x
the unit n-cube.
v
m
.
triangulation of H.
Let H be a set in Ir" and let C be a locally finite
By the pseudomanifold corresponding to
e
we mean the pseudomanifold
is a face of some
simplex^ of
simplex^ of C}, where if
H
C}
is
the empty set, as an element.
0,
Combinatorial Lemmas on the Simplex
In this section, we give constructive proofs of three combinatorial
lemmas on the simplex, Sperner's Lemma [17], Scarf's Dual Sperner Lemma [15],
and a Generalized Sperner Lemma.
The Generalized Sperner Lemma was independently
and the author [4], and inadvertantly by Luthi [lA].
developed by Ky Fan [3]
We show the interrelationships between these lemmas and the Brouwer Fixed
Point Theorem [1]
Our method of proof will be consistent throughout.
s"~
and a triangulation
pseudomanifold A(T)
of the vertices of
the V-complex.
C
of s"
,
we will define a V-complex, where each
corresponds to a face of
C,
Given a simplex
we will examine the sets
The elements of BUG\{0}
S
.
B
Given a labelling
and
G
L(')
associated with
will be shown to have the desired
- 3 -
properties of the conclusions of the lemmas.
BUG ^
By lemma 13 of Part 1,
must have an odd number of elements.
Let C be a locally finite triangulation of S
(an (n-l)-pseudomanlfold) corresponding to
of vertices
of
{l|v^>0 for some
.... n
{l,
v € x}.
containing x.
S
Is the collection
C.
completely labelled if L(x) =
face of
K
C, where
L('):K - {1, ..., n} be a labelling of K
Let
set
and let K be the complex
For x-fiK, x Is called
.
The carrier of x, denoted F(x), is the
}.
F(x) is the "Index set" of the smallest
L(x) = F(x)
If
x
,
is called completely
labelled in its face .
Sperner's Lemma [17]
of
K
v € K
such that If
PROOF
{1,2,3,
..., n-1}}.
T = {l,
..., m}
A('). and
,
n)
and
,
7
= iH)},
A(0) = {0, e^}
Define
Let
T
such that
But then
whence
Suppose
= {1,
?<
..., m}.
v^ =
L(x) ^ T
B = {0}.
C
x
x f
^
€
B,
since
Then
B.
3'A(T
for all v €
,
{1,2},
For
*
T
{v € s"~ |v
to
constitute a V-complex.
We first examine B.
0-slmplex.
.
(D.
{1,2,3},
< "5
....
,
m < n-1, and define A(T) to be the pseudomanifold
for some
N
L(v) ^ 1 (such a labelling
Then there exists an odd number of completely
.
corresponding to the restriction of
K,7
be a labelling
n)
. . . ,
K.
€
N = {l
Let
;
x
"Kl,
v^ = 0, then
and
Is called a proper labelling)
labelled simplices
L('): K
Let
.
x.
)
Let
=
be a proper labelling.
L(')
A(0) = {0,{e }}
L(x) = T
X
and
for l>m+l}.
contains only one
x < 3'A(T
X
).
implies that there Is an 1, 1< i< m,
Thus
1^
L(x), since
contradicting the definition of
B.
L(-) is proper.
Therefore
x ^ B,
- 4 -
Now let us examine
T
- {1,
T <
.... m}
J
Because
.
L(x) ^ J
ahJ
Because
=1,
|B|
L(x)
G,
«
is proper,
L(')
L(x) - {1,
,
x
If
G.
...
must be odd.
|g|
L(x)
T
X
C
,
L(x) 4
{1,
1
•
Again let
Since
..., ntfl}.
Thus x is completely labelled.
n}.
,
D
Therefore there are an odd
®
number of completely labelled simplices.
Kuhn's algorithm for Spemer's Lemma [11], independently developed
by Shapley [16], consists of tracing the path of adjacent simplices from
0*B
to an element
xtf G.
Sperner's Lemma requires that the labelling
the boundary of s""^.
L(.)
be restricted on
If we omit this restriction, we obtain the following
generalization of Sperner's Lemma:
Generalized Sperner Lemma (3, A, 14].
Then there exists an odd number of simplices
Let
L('):
x € K
{1,
K^-*-
such that
..., n}.
x
is
completely labelled in its face.
As in the proof of Sperner's Lemma, we first construct a V-complex.
PROOF:
Let
for
N = {1, ..., n}
?*
T
^
J
define
,
restriction of
A(),7
,
A(T
)
1
=
X € 8'A(T
X
S""^| v^ - 0,
F(x) = T
.
x
V
=0
n
Thus
1^
T
\J
{n}}.
constitute a V-complex.
and N
Suppose
),
Define A(0) = {0, {e"}}, and
to be the pseudomanlfold corresponding to the
We first examine the set
O-slmplex.
{TCN|n^ T}.
to
C
{V «
K,
and let
?*
Then
x € B.
for all
v
B.
6
x.
€ B
since
L(x) = T
And since
A(0) contains only one
and
x
x €
8'A(T ).
contains
It
'
F(x) = L(x) and
x
I
Since
elements,
x'
is completely labelled in its face.
- 5 -
Next we examine
L(x)^ J
and
Then
.
F(x) - T
elements,
G.
1/
{n}.
Therefore
F(x), X 6 B.
U
L(x) - T
Conversely, suppose
n^
Let
x
G.
€
Then
Is full,
x
contains
x
{n}, and since
L(x)
|T
D
T
,
+1
]
L(x) = F(x).
Thus
F(x) = L(x).
bO G\0
G;
if
is the set of simplices each of which
Since
is completely labelled In its face.
n € F(x), x C
Then if
B
are disjoint and have the
and G
same parity by Lemma 13 of Part I, there are an odd number of simplices
x€
K
is completely labelled in its face. S)
x
such that
GUB\0
An algorithm for computing an element of
consists of
following the path of adjacent simplices where the endpolnt is 0.
This is
essentially Luthi's algorithm in [14].
As a corollary to the Generalized Sperner Lemma, we have:
Sperner Lemma
Ihxal
(See Scarf [15], e.g.).
.
be a labelling such that
no simplex
of
o
C
v € 9S ~
v.>
,
L('): K
Let
->-{l,
L(v) ^ 1, and suppose
implies
has a nonempty intersection with every face of
Then there are an odd number of completely labelled simplices
PROOF
X
€
€
such that
K
If
x
And since
X.
X
is completely labelled in its face.
L(x) ^ {1,
whence for each
Thus
x
S
€ K.
By the Generalized Sperner Lemma, there are an odd number of simplices
.
simplex.
V
..., n}
1 €
..., n},
F(x) = L(x), each
L(x), there is a
meets every face of
S
1^
then for all
,
v € x
v € x
Let
L(x), v
x
=
be one such
for all
meets the boundary of
with
a contradiction.
v
= 0, by design of
Therefore
S
,
L(0.
L(x) = {1,
An algorithm for finding the completely labelled simplex of this
lemma is the same as that for the Generalized Sperner Lemma.
G
U B\0
will be completely labelled.
Any element of
...,
- 6 -
The Generalized Spemer Lennna also Implies the Sperner Lenma, where
by "Implies" we mean that one leads to simple straightforward proof of the
other.
We have
The Generalized Sperner Lemma Implies the Sperner Lenmia
PROOF
;
Our proof Is by Induction on
N = {1, .... n}, and for
C
T
j*
N,
1
^ T}.
proper labelling of
Implies
L(v) € T.
Note that
N
S
the two lemmas are
k<n.
N
For
T
i*
S"~
(number of slmpllces In
" S
S
x>0,
and note that If
,
It Is also a proper labelling of
,
odd number of slmpllces In
.
1,
S^ - {x € IR"I
let
n— 1
- O for
n
Suppose the Implication is true for all
trivially Identical.
Xj^
For
n.
z
(//
S
T
e* x = 1,
TO!
L(')
S
Let
,
Is a
v
I.e.
6
K
N, we Inductively have that there are an
?^
whose set of labels Is
T.
We have:
that are completely labelled)
slmpllces In
S
that have label set
T)
0^T?'N
TCN
= an odd number
by the Generalized Sperner Lemma.
by induction, and there are
2^-2
Each term In the summation
terms, an even number for
the total number of slmpllces In the summation
number of slmpllces In
S
Z
Z
Is odd
n>l.
Thus
Is even, and so the
which are completely labelled Is odd.
®
Each of the above three combinatorial lemmas bears an Interesting
relationship to the Brouwer Fixed Point Theorem [1], stated here for the
simplex.
r\ S
-
Brouwer's Theorem on the Simplex
function.
-
7
Let
.
f:
s"~
v € s"~
Then there exists a point
s"~
•*
be a continuous
such that
f (v)
" v.
(X>
That Scarf's Dual Sperner Lemma Implies and Is Implied by the Brouwer
Fixed Point Theorem has been shown In [15], for example.
It has also been
shown, see [11] for example, that Sperner's Lemma Implies Brouwer's Theorem.
We also have the following implications.
The Generalized Sperner Lemma implies Brotjwers Theorem
PROOF
:
f:S""^-^s""^
Let
L(.): K ^ {1,
Let
S""-^.
be continuous. Let
be defined as
.... n}
L(v) = some
X
Let
fj^(v)
so
v
And for
€ x.
in its face, we have a point
Since
4 F(x)
1
=
v^^
,
Then for
for any
v
such that
x
with
fj^(v)
x
:
Extend
Let
f
iv^
e f(v) = e v = 1, we must have
L(-): K°->{1,
for all
a
S
into
(PWL)
S
~
set of vertices of
<r
by the continuity
.
f(v) - e^
fashion on all of
for
s"~
.
v
€ iP
f
is
C
containing
v, and let
v
x
.
of
L(x) = F(x)
,
whereby
x
f.
be the
.
Then we must have
®
1,
Thus there is a fixed point
.
be the smallest real simplex In
in its face.
and
®
f(v) = v.
.... n}, and define
in a piecewise linear
continuous and maps
Let
v € x
Is completely labelled
Brouwer's Theorem implies the Generalized Sperner Lemma
PROOF
= L(x),
1 € F(x)
Taking a sequence of finer triangulations and choosing a
limiting subsequence of simplices
of f.
fAv)iv^.
such that
1
be completely labelled in its face.
iv^ for some
f£(v)>v^.
triangulation of
be a locally finite
C
is completely labelled
- 8 -
Finally we have:
Brouwer's Theorem Implies the Sperner Lennna
L('): K - tl,
Let
PROOF:
-
L(v)
v€
K
.
and extend
f
to all of s"
n}
L(x)
If
0.
such that
opposite the vertex e
if
i
>
2
n
if
i
=
1
L(.) is proper, then
k
^
[^
But since
the argument, we obtain
..., n)
and
,
x
Figure
f
f
be a fixed point of
maps
o
let
f,
be the set of
x
i
is
€
{1,
...,
into the face of
S
where
,
i-1
r
=
k
v
..., n}, there is an
Therefore
L(x).
i
i
{1,
s«
in a PWL fashion,
containing v, and let
C
e-'
1
Let
.
f(v) =
2
L(v) =
be the smallest simplex of
vertices of
i
.
into s"
continuous and maps s""
Define
be a proper labelling.
n}
,
L(v)
if
,
.
if
1
n
for all
.
L(x)
=
is not an element of
0,
L(x)
Thus
a contradiction.
.
L(x)
Continuing
=
{1,
completely labelled.®
is
summarizes the relationships demonstrated herein er elsewhere,
1
where the implication is in the direction of the arrow.
II.
A Generalized Covering Lemma on the Simplex, and Extensions
In this section we prove a Generalized Covering Lemma on the simplex,
"generalized" in that it generalizes the Knaster-Kuratowski-Mazurkiewicz
Lemma [12].
We use this lemma to demonstrate the existence of stationary
points (see Eaves
[2]
,
for example) and two other results on the simplex.
Generalized Covering Lemma
such that
such that
n
V>
.
Let
C^,
i = 1,
.... n
be closed sets in IR
..
C
D
{ilx*>0}
s"~
C
.
Then there is at least one point
{i|x* € C^}.
x*
in
S
SCARFS DUAL SPERNER LEMN4A
Figure
1
- 9 -
PROOF
;
Let
C
be a locally finite triangulation of S°
let L(.) be a labelling of K°, where for each
K
v €
with vertex set K
,
L(v) € {i|v € C^}.
By the Generalized Sperner Lemma, there is a simplex
completely labelled in its face.
of
that is
C
Take a sequence of triangulations whose
diameter goes to zero in the limit.
a,
a
Then there is a sequence of simplices
completely labelled in their faces, that have a limit point, say
Since each
the theorem.
is a closed set, we have
C
{i|x*>0}C{i|x*€C
},
x*.
proving
®
This theorem is illustrated in Figure
of "degeneracy" occurs at
In Figure 2(b) a tjrpe
2.
x*, showing that strict inclusion
of
"c"
the theorem can indeed occur.
Our next lemma demonstrates the existence of stationary points
(see Eaves [2], and Liithi [lA]).
x>0, w>0}, where
s" = {(x,w) €
it is understood that
d" = {x € IR'^je x<l, x>0}.
the x-coordinates.
Let
Let
Clearly
f('):D
->
IR
d"
x f IR
i)
ii)
iii)
iv)
z €
y>0, z>0
f(x) = y - ee
x.y =
z(l - e*x) =
IR
and
e*x + w = 1,
Let
.
is the projection of
be continuous.
is said to be a stationary point of the pair
and only if there exists
Ir"'*"-'-|
y € IR
(f ,D
)
A point
onto
s"
x
in
D
(see Eaves [3]) if
such that
,
and
Figure 2(a)
Figue 2Cb)
- 10 -
We have the following:
Lemma (Hartman and Stampacchia [7], and Karamardian [8a], [8b], [9]).
There exists a stationary point
PROOF
1
n,
(f,D^).
Define
c""*"^
s"|f(x);fO
is closed, and that
ii)
in
w*>0
implies
(x*, w*)
implies
f.(x*)<0, but
f(x*)>0, therefore
z (1 - e
T
and so
f (x*) > 0,
-x*) =
.... n+1)
1 = 1,
since
f
(f,D ).
z = 0,
y>0.
(x*) = 0,
z = 0.
..., n, and
c""*"-"-.
In this case, let
(x*,w*) € C
Finally,
e
,
is a stationary point of
.
C^ (1 = 1,
.... n}
such that
S
(x*, w*) e C
(x*,w*) £ c"
= 1,
j
Thus by the Generalized Covering Lemma,
S'^.
implies
x*
for any
(x)
Note that each
X* >
We now show that
.
D
U._. C
(x*, w*)
i)
f^(x)<f.
and
s"|f(x)>0}.
- {(x,w) 6
there exists
For
define
= {(x,w) e
C
I
of
Our proof is based on the Generalized Covering Lemma.
:
1
Case
x*
We have two cases:
y = f(x*).
and let
Also,
x*
implies
>
Thus x*y = x*f(x*) = 0,
Therefore;
is a stationary
x*
point.
n+1
,
Case II
z>0.
(x*,w*) f C
.
Let
y = f(x*) + ze.
y = f(x*) + ze> 0.
j,
and since
Finally, since
implies
z =
Let
.
x? >
f(x*);<0, f.(x*) = -z.
Ti+1
(x*,w*) ^ C
z(l - e x*) = 0.
-z< f^(x*)
We have
Furthermore,
,
Thus
implies
Therefore
we must have
x*
Note iK*t
-min{fi(x*), .... f^Cx*)},
for each
Hence
i.
f ^ (x*) < f ^ (x*)
y.
for any
= 0, and so x*'y = 0.
w* = 0, so
T
e x* = 1, which
is a stationary point of
(f.o").
®
- 12 -
Combinatorial Lemmas on the Cube Related to Brouwer's Theorem
III.
In this section, we present two new dual combinatorial lemmas on
We present and give constructive algorithmic proofs to Kuhn's
the cube.
Cubical Lemma and Gale's Hex Theorem.
We show the Interrelationship between
these results and the Brouwer Fixed Point Theorem.
Let C be a locally finite triangulation of C
and let K be the pseudomanifold corresponding to
= {x e IR |0^x^e)
C, with vertex set
K
.
Our
first two results are the following:
Lemma
(Freund,
1
Let
[4]).
v € K
labelling such that if
L(')5K -^{1. •••» n, -1, ..., -n}
,
V.
=
V
= 1 implies
implies
Then there exists a pair of vertices
L(v) ^ -1, and
L(v) ^ 1.
in some simplex of
v', v"
K such
L(v') = -L(v").
that
PROOF
;
L(0
Let
and let
^
T^7
=
,
B
and
{SCNJi^S implies
define
restriction of
C
We proceed by defining a V-complex
be as described above.
and examining the sets
/
be a
A(T)
G.
N = {1, ..., n, -1, ..., -n}
Define
i>0}.
Define A(0) = {0, {0}},
and for
to be the pseudomanifold corresponding to the
to
{y € C"|y^ =
for
1
^
T},
- 13 -
K. A(*)»J
N
and
t
constitute a V-complex.
Examining B, observe that
z € 3'A(T
there exists an
X ),
XX
Then
^ x £ B.
Suppose
O-slmplex.
1 € T
x € 3'A(Tj^)
for which
Then by the definition of the labelling rule
L(x) - T
fact that
X
B -
Thus
.
Now let us examine
|b
number of elements, by Lemma 13 of Part
L(x)^J
and
1 ^ T
Then
.
=
v
L(v') = -L(v").
Lemma
2
Let
.
V € K
If
V € 3C
for all
v € x.
I.
G
Let
x
must have an odd
6 G.
such that
-1^
and
x
« 1,
L(x)
Then
L(x)
-1 6 L(x).
D
T
Suppose
be definition of
and there exists fv'
,
L(-)
v"]c x
be a labelling such that
n, -1,
...
-n}
Implies
v^^
= 1
implies
v,
= 0.
n
,
L(v) =
1>0
L(v) = -1
<
Then there exists a pair of vertices
v', v" in some simplex of
K
such
L(v') = -L(v").
that
PROOF:
As in the proof of the Lemma 1, we proceed by defining a V-complex
and examining the sets
and let
f*
= 1
Since
®
L(-)5K-*{1» •••»
and
L(x) « T^.
L(x), contradicting the
1 ^
{l, -1}^ L(x)
Thus
This is a contradiction.
such that
v€
for all
|
If N
Thus there Is some
.
v.
and
{0}.
Since
G.
A(0) contains only one
€ B, since
T €
- (ScNJl
tf
J
define
,
restriction of
6
B
S
A(T)
and G.
Implies
.
and
N = {l,
KO}.
Let
..., n, -1,
..., -n}
A(0) = {0, {O}}
to
C
N
and for
to be the pseudomanifold corresponding to the
{y€C"ly
K, A('). J
Let
=
for any
constitute a V-complex.
i>0
and -1 ^ T}.
- 14 -
1>0
there exists an
3'A(T
6.
" 1
v
and
"i ^
contradicting the fact that
" L(x).
T^^
L(x) « T
and
)
L(v) « -1
Thus
Thus
.
v
for all
L(.)» we cannot have
But then from the definition of
V € X,
x
-1 € T
such that
A(0) contains only one
B, since
Then
/ x € B.
Suppose
0-slmplex.
€
B, we see that
Examining
« x.
for any
and
x ^ B
B - {0}.
Therefore
Suppose
Part I.
1>0
such that
Then
-1 ^ T
1€
L(x)
must have an odd number of elements, by Lemma 13 of
G
Then L(x)
x € G.
and so
=
v^
for all
v
L(-).
Thus
by the definition of
L(x)^7
and
T
L(x) = Tj^U{l}.
Thus
1 € L(x).
D
-16 L(x),
{v', v"} In
there exists a pair of vertices
An algorithm for finding
a pair
(v'
{l, -l)
C
and
L(x)
L(v') = -L(v").
such that
x
,
-1 i L(x).
Suppose
But then we cannot have
x.
tf
Then there exists
.
^
of vertices such that
v")
in either of the two lemmas consists of following the path
L(v') = -L(v")
of adjacent slmpllces from
0€
The path must terminate with an element
B.
G, providing the desired pair.
of
It should be noted that Lemma
1
corresponds precisely to the existence of a
j-stopping simplex on the product space of n 1-simplices, as presented in [20].
Before doing so,
We next give a proof of Gale's Hex Theorem [6].
some terminology must be introduced.
labelling of the vertices of
H
is
v €
j
Hj^
= 1,
K.
L(-)jK
Let
Define
Hj^
"*'tlf
..., m, and
j
= 0,
...,
m
m
and
= "'
v.
0, v.
"1 = 1.
'1
{v
If
"path" of vertices from the boundary set
boundary set
{y € C |y. = 1}
,
H
v
is a
}
is
{y € C
be a given
n)
£ K°|L(v) =1}.
= {v
1-connected if there is a sequence of vertices
for all
•••
v
,
.
. . ,
v
We say
such that
1 -simplex of
K,
i-connected, there is a
|
y^^
all of whose labels are
= 0}
1.
to the
We now prove;
- 15 -
Hex Theorem [6]
such that
There exists an i € {1, ..., n}
.
PROOF ;
We proceed by defining a
and let
^
- {T|TtN}.
V-complex on
A(0) - {0, {0}}.
Define
N
Let
K.
H.
For
?*
T €
(l,
..., n)
7
define
to be the pseudomanifold corresponding to the restriction of
{y 6 C"|y^ -
K,
A(.).7. and N constitute
all subsets of
X
all
Let
p
adjacent to
We show that H
.
G =
Note that
such that
1^
T^~
,
and
1» v^
= 0.
LCv"")
i f
LCx"' r\
xJ
,
= 1,
j
-=
,
T
1 «
= p,
j
= T^
x^"*"^)
U
in the
p-1,
x
.... m.
Thus H. is i-connected.
like ours, but depends on augmenting
K
C
trlangulatlon of
Kj^
C
•
v. = 1
for
.
.
,
there must be
Since
m.
. ,
T^"*"^,
1,
is
the intersection
for
such that
Since
x
j
(v
T^
= p-1,
v
,
..., m.
}
does not
®
His is algorithmic
by adding another set of vertices
and using a lexicographic labelling rule.
the triangulatlon
= T
X
must contain
This proof is different from that of Gale [6].
to
B, by Lemma 13
such that
1 € T
T^
x^
v
Thus there is a sequence of vertices
is a 1-simplex in K and
exhausts
A(0) contains only one 0-simplex.
Since
m.
if and only if
x^"*""""
J
since
is i-connected by exhibiting the desired sequence.
of labels of any two consecutive
contain
to
C
be the path of adjacent simplices from
Tj,j=l
X
T^=
A(T)
T}.
x € B, there exists an
Since
B.
e
vex*
a number
0=x,...,x =x
Let
of Part I.
to
6 8, since
,
is finite there exists at least one other element of
K
Since
a V-complex.
We have
N.
if
for
is i-connected.
He also restricts
C
to be
(see Todd [18]), whereas this proof is valid for any
- 16 -
Our final lemma of this section is Ruhn*s Cubical Lemma of [10].
Let I » {y e IR
-
Ijtj^
or
Cubical Lemma (Kuhn [10]).
i - 1,
1,
Let
((•):K
-^
We have:
n}.
. . . ,
such that
I
V- =
implies
X^(v)
•=
v^ - 1
implies
jlj^(v)
= 1.
0, and
Let L(v) equal the number of leading zeroes of
Then there exists an odd number of simplices
£(v)
x
K
€
veK
for each
.
L(x) =
such that
{0,1, ..., n}.
PROOF:
1
= {T
We first construct a V-complex.
C {0, ...,n-l}Io<i
implies
T
A(0) = {0,{e}}, and for
We define
We then define
restriction of
A(T)
f T €
K,
J
,
Let us now examine the set
Q ^ X € B, where
all
j
= 1,
and some
x = {v
and
T^^
....
i€
{l,
a contradiction.
'3
,
,
..., n}, and let
then is the collection
{0,1,..., n-1}.
T = {0,...
,
m}
for some
to
C
It is simple to verify that
L(x) =
...
1,
to be the pseudomanifold corresponding to the
i>mfl}.
{x € c"|x^ = 1 for
and
J
i-1 € T}.
{Q}, {0,1}, {0,1,2},
0,
m<n.
6
N = {6,
Let
m
x€
A(.), and N define a V-complex.
v™}
9'A(Tj^).
and some
..., m}.
We know that
B.
for some
x€
Since
i € {1,
.
. .
m.
,
m}
Then
T
= {0,
9'A(Tj^), either
,
or v^ = 1
Suppose the former is true.
If the latter is true, then
Suppose
B.
€
i
. .
vj =
for all
Then
.
,m-l}
for
j
= 1,
...,
m
i-1 ^ L(x)
d L(x), which is a contradiction
- 17 -
1 - m.
unless
x € A({0, ..., in-2}), so that
But then
Therefore
a contradiction.
B
m<n,
or
m
<
n-1
and
L(x)
3
T
for all
j
Let
contradicts s ^ m+l.
Therefore
n
G consists precisely of those
x
- {0,
..., m}
L(x) - {0, 1,
..., m, s).
. . . ,
v
}.
L(v^) <mfl
But then
L(x) = (O,
Since
...,
for which
T
" {O,
C
6,
..., m),
This
for all j.
n}..
L(x) = (O,
x € G.
.... n).
Therefore
By Lemma
has an odd number of elements, which proves the lemma.
the path starting at
...,n}
Tn+l
,
An algorithm for finding an element of
element of
L(x) = (O,
L(x) = {0, ...» n}, then clearly
Furthermore, if
13 of Part I, G
x = (v
T
Therefore either
.
such that
.... m+l).
(O,
€
mf 1
Then
x € G.
Let
L(x)^J
s >
Suppose the latter is true.
v^^^ = 1
m+2
G.
,
and there is an
.»., in-2},
{0}.
Next we examine the set
for some
- {0,
T
®
consists of following
G
and terminating at its other endpoint, an
G.
All of the above four results are closely related to Brouwer's Fixed
Point Theorem on the cube, stated below.
Brouwer
s
Fixed Point Theorem on the Cube
Then there exists
v € C
such that
n
.
Let
f(v) = v.
f:C
• Cn
be continuous.
®
In [6], Gale shows that the Hex Theorem implies and is implied by
Brouwer's Theorem.
In [10], Kuhn shows that his Cubical Lemma implies
Brouwer's Theorem, and that Brouwer's Theorem implies a weaker version of
his (strong) Cubical Lemma.
We also have the following results:
- 18 -
Lemma
implies Brouwer's Theorem
1
PROOF ;
Let
.
be continuous.
f:c"-»-c"
Define
L(.):K°-»{1,
....
if
||f(v) -
v|L
- fj^(v) - v^,
v^
If
l|f(v) -
v|L
= v^ - f^(v),
v^ / 0.
n,-l,...,-n}
1.
,<
L(v)
f.;
If there is more than one choice for
index.
in some simplex
K
of
x
for which
be the smallest such
L(v)
satisfies the hypothesis of Lemma 1.
L(')
of K
L(v), let
Thus there exist
L(v') = -L(v").
v', v"
Using the typical
limiting argvment as we take a sequence of finer triangulations, we obtain
a point
in
y
c"
f(y) = y.
such that
®
A similar argument establishes that Lemma
where we define
r
i
if i|f(v) - v||^ = vi - fi(v)
-i
if ||f(v) - v||^ = fi(v) - Vi
\
\
If there is more than one choice for
ensuring that if
implies Brouwer's Theorem,
by
L(')
L(v) =
2
v 6 3C
L(v), choose the smallest such index,
the hypothesis of Lemma 2 is satisfied.
,
Finally, we show the equivalance of Lemma 1 and the Hex Theorem.
The Hex Theorem implies Lemma
be as described in Lemma
Define
1.
1
.
L(«)5K *
Let
L'(v) = |L(v)|
•••» n, -1,
Q-t
and let
..., -n}
= {v £ K
H
|
L'(v) = i}, i = 1,
. .
.
Thus there is a sequence
j
- 0,
m
v^ =
and
L(v
j
€
..., m, and
)
=
{0,
1.
i,
By the Hex Theorem, at least one
n.
,
v
{vj.j^,
..., v
,
vj
}
is a
By the definition of
so that L(v
.... m-1) with
)
=
i
and L(v
L(v^)
of vertices such that
1-simplex,
L(').
)
j
= 1,
= -i.
L'(\r) = 1,
Also
..., m.
we have neither
is i-connected.
H.
L(v
o
)
=
= -i, nor
Thus there must be some
= -i, L(v^"^^) = i.®
v.
by
- 19 -
Lemma
1
and define the sets
Hj^
L(«)!K
Let
implies the Hex Theorem .
- {v € K |l(v) - 1}.
{1»
An element
.... {v""
Define
,
v"}
K
in
L'(v):K°-^
{1,
for which
-vH
€
H^,
is
1-simplices
{v
of
,
v
},
m, v^ = 0, and v = v".
=
j
be given
n}
E^
defined to be base-connected if there is a sequence of
v
. . . ,
as follows:
.... n ,-l,...,-n]
L(v)
if
V
is base-connected.
-L(v)
if
V
is not base-connected.
L'(v) -
Note that if
= 0, we cannot have
v
L'(v) = -1.
L' (v) = 1.
L'(') satisfies the hypothesis of Lemma 1.
Thus
exists a 1-simplex
{v', v"} €
of generality, assume
{v'
,
v"}CH^, v"
= 1, we cannot have
v
Then if
Suppose the Hex Theorem is false.
K
such that
L'(v') = 1
is base-connected because v'
contradiction, since L'(v") = -1
< 0.
L'(v') = -L'(v").
Since
> 0.
Therefore there
{v'
is a l-simplex and
v"}
,
Without loss
is base-connected.
This is a
Therefore the Hex Theorem must be true.
^
Our two final results of this section are:
Brouwer's Theorem implies Lemma
PROOF
for
:
Let
V € K
as in Lemma 1.
K
Define
f(v),
as
r
f(-)
(vj^,
.... v^_^, 0, v^^^,
(vj^,
.... v^_3^,
....
v^)
if
L(v) =
-KO
if
L(v) =
i>0
<
[
L(*)
.
be a labelling of
L(-)
f(v) =
and extend
1
1, v^^j^,
...,
In a PWL fashion over all of
in Lemma 1, we cannot have
f (v)
= v
for
Vjjj)
C
.
By the definition of
v € K
.
Since
f
is continuous
- 20 -
and maps
Into Itself, there exists a fixed point
C
be a simplex of
C
corresponding to
(T.
y
containing
L(x)
D
y, and let
for some
{-1, 1}
cannot be a fixed point of
f.
x
y
of f .
Let
(f
be the abstract simplex
iff {1
n}, otherwise
(X)
Similarly, we have
Brouwer's Theorem Implies Lemma
PROOF
;
2
The proof Is analogous to the one above, except we define
(v
f(v) =
..., V
,
,
^-1
^
'j
....
(v]^,
Vj^_j^,
0, V
i+1
,
1, Vj^^j^,
..., V
)
If
1>0.
L(v) =
"
..., v^) If
L(v) = -1<0.
Q.E.D.
®
The Interrelationships shown in this section are summarized in
Figure
IV.
3.
Tucker's Lemma and Antipodal Point Theorems
In this section we present a proof of Tucker's Lemma [19] on the cube
and show its relationship to two antipodal point theorems.
Let
Let
C
p
be a positive integer and let
C^ = {x € IR"
I
-pe<x<pe}.
be any locally finite centrally symmetric triangulation of C
that refines
P
the octahedral subdivision, for example J,, see Todd [18].
set of vertices of
X € K, we denote by
and
C
-x
K
Let
K
be the
the pseudomanifold corresponding to
the simplex
{-v|v € x}.
By symmetry,
C.
For
-x € K.
A function L(.) whose domain is K° is called odd if L(v) = -L(-v) for all v
Our first result is:
€
K"
.
LEMMA
HEX THEOREM
1
^^
KUHN S STRONG
CUBICAL L£MMA
LEMMA
2
I
Figures
- 21 -
Tucker's Combinational Lemma
-1
PROOF
;
and let
?*
of
such that L(.) is odd on
..,, -n)
(v', v"
a 1-slmplex
T
<£
(see Tucker [19]).
,
K
of
}
-=
t[
7
,
{TC N|i€
let
-1^
implies
T
sK
-
{1, ..., n,
L(v) - -L(v").
Let
N - {1, .... n, -1
T}.
Let
-n}
A(0) = {0, {0}}, and for
to the region
C
for
|>
A('). and N define a V-complex.
Let us now examine the set
ment. Suppose
L(-x) = {-ill € T
},
and in fact
Thus we see that except for
Therefore
B
L(x) = T
Also
Therefore, x € 3K.
€ 6, so
B.
x € 3'A(T
Then
X € B.
j'
T_
0, B
if neither 1 nor -1 € T}
=
x
1 € T, and
K,7,
It is simple to verify that
=
has at least one ele-
B
^ T €*[
For any
).
{-l|i€ T
}
.
3'A(T)
,
C 3K.
-x £ 3K, and
Furthermore,
.
X
Therefore,
consists of pairs of the form
has an odd number of elements, and so must
-x € B.
x, -x.
G, by Lemma 13 of
I.
Thus there is an element
say
)
be the pseudomanifoid corresponding to the restriction
A(T)
{x e IR^Ilx.
Part
L(-
Then there exists
SK.
such that
We first construct a V-complex.
Let
v'
and
elements of
x £ G.
L(v') = -L(v").
v", such that
is a 1-simplex of
x, {v', v"}
An algorithm for finding a pair
path that originates with
If it is an element
x
Thus there are two vertices of
of
this fashion, an element of
And since
K.
v', v"
v'
and v"
®
consists of following the
If its endpoint is an element of
0.
B,
G
reinitiate the path at
will be found.
are
-x.
G,
stop.
Continuing in
For a complete description
x,
- 22 -
of the pivot rules for this algorithm, see Freund and Todd [5].
assert that there are an odd number of l-slmplices (v'
,
v"}
We cannot
with L(v') = -L(v").
This is because not all such pairs are subsets of elements of
G.
That Tucker's Lemma implies both the Borsuk-Ulam Theorem (see [13])
and the Lusternik-Schnirelman Theorem (see [13]), and that the Borsuk-Ulam
Theorem is implied by the Lusternik-Schnirelman Theorem, have been shown elsewhere, see Lefschetz [13] or Freund and Todd [5] for more familiar terminology.
Here we show that the Borsuk-Ulam Theorem Implies the Tucker Combinatorial
Lemma.
The former theorem can be stated:
Borsuk-Ulam Theorem
be continuous.
We have
;
Let
= {x 6 Ir"| ||x||2 = 1), and let
b"~
x* € B
Then there exists
f (•) tB""^
-^
IR""-"^
f(x*) = f(-x*).
such that
:
The Borsuk-Ulam Theorem implies Tucker's Combinatorial Lemma
PROOF
;
Let
c"^
= {x € IR"|-pe
x
<
<
and let
pe}
C
be a locally finite centrally
n
synunetric triangulation of C
to
C,
with vertex set
K
.
.
Let
K
be the pseudomanlfold corresponding
L(.):K
Let
-»
{1
n
n, -1,
..., -n}
For each
v €
be a labelling function which is odd on the boundary of C
K°, define f(v) = sign (L(v)).e 1^^^^', where
denotes absolute value, and extend
f(.)
|.|
in a piece-wise linear manner over
n
all of C
.
Note that
f(.)
is continuous,
n
f(.) is odd on the boundary of C
.
.
and since
C
is symmetric,
- 23 -
B° - {X €
Let
and let
Ir"""^!
- {x £ B |xj^^j^<0}.
B
- 1}. let
\\x\\^
.... Xjj)||(Xj,
p(xj^,
-
g:B"
Let
« {x €
b"*
b"|x^i>0},
be the following map:
C
.... Xj^)ll2
.
(xj^
,
.... x^)
»
vX,
»
• • • •
<^
.
g(x) \X_
• • • 9
y
X
^
I
X
n
J.
Note that
is bicontinuous and onto.
g (•)
(
h(x) =
f(g(x)),
x€
b"""^
-f(g(-x)),
x€
b"~
Is an odd continuous function from
Theorem, there exists
we may assume
.
x € b", let
<
I
h(.)
For
)
such that
x*
x* € B
Thus
.
X - g(x*), we see there exists
sign (L(v )).e'
i,
a pair of vertices
v
-^^
h(x*) = 0, whereby
x € C
v
^
such that
L(v
and
i,
^)
By the Borsuk-Ulam
IR°.
Without loss of generality,
f (g(x*)) = 0.
f(x) = 0.
such that
±j
v
into
h(x*) = h(-x*).
for appropriate
'
B"
> 0.
A
Setting
f(x) = Z X.
Thus there must be
^
i,
= -L(v ^)
,
proving Tucker's
Lemma. ^^
Final Comment
The algorithms presented herein for proving the Spemer Lemma, the Dual
Sperner Lemma, and Lemmas
1
and 2, all follow a path that is initiated at a
"corner" of the simplex or cube, i.e., the unit vector e^ or the origin,
respectively.
As is shown in [4]
,
it is possible to define a V-complex where
the initiation point (corresponding to A(0)) is an interior grid point of
the simplex or cube for the above lemmas if the triangulation is of a standard
form.
VHien the
V-complex is defined this way, the resulting algorithms
correspond completely to those of van der Laan and Talman [20]
BIBLIOGRAPHY
[I]
Brouwer, L.E.J. , tJber Abildung van Mannigfaltigkeiten,
Math Ann 71 (1910), 97-115.
.
[2]
Eaves, B.C., On the Basic Theorem of Complementarity, Mathematical
Programming 1 1 (1971), 68-75
[3]
Fan, K. , Fixed-Point and Related Theorems for Non-Compact Sets,
Game Theory and Related Topics Eds: 0. Moeschlin and D. Pallaschke,
North-Holland Publishing Company, 1979.
,
[4]
Freund, R.M. , Variable Dimension Complexes with Application, Doctoral
Dissertation, Department of Operations Research, Stanford University,
Stanford, C, 1980.
[5]
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c/\
J
^W^
c
073
Mil LieRARICS
3
TDflD QDM
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Date Due
Lib-26-67