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Thesis fur the Degree of Ph. D.

HQHRGAN STATE UNNERSITY

RGUAND LAVERNE SWANK
1969

”aim:

 

gut-“fa?“ ‘ ' '

9‘ -.“"0' ""V i E?” ‘v'
J. —. 3. 5"} 2‘5— -‘ 3"“ "

Michigan State
Universtty

     
     

  

This is to certify that the
thesis entitled
COLLAPSING OF SIMPLICIAL COMPLEXES
CROSSED WITH THE UNIT INTERVAL

presented by

ROLLAND LAVERNE SWANK

has been accepted towards fulfillment
of the requirements for

Ph.D Mathematics

degree in

 

 

0-169

 

ABSTRACT

COLLAPSING OF SIMPLICIAL COMPLEXES
CROSSED WITH THE UNIT INTERVAL

BY

Rolland Laverne Swank

In this thesis we consider the problem of collapsing a pro-
duct space L X I, where L is a simplicial complex and I the unit
interval.

By developing the concept of a simplicial relation, we find we
are always able to collapse L X I onto an intermediate complex
called the sewn complex. This leads us to examine some prOperties
of the sewn complex which will allow us to conclude when it is
collapsible. We find that a proper fold factorization insures
collapsibilityif the first complex of the factorization is collapsible.

A second technique for collapsing the product space is intro-
duced, which, in Special cases, is related to our first method. In
studying this relationship, we introduce the concept of an unfoldable
star. We examine, in the final section, a class of complexes possessing
unfoldable stars and find that for L in this class, L X I is always

collapsible.

OOLLAPSING OF SIMPLICIAL COMPLEXES
CROSSED WITH THE UNIT INTERVAL

BY

Rolland Laverne Swank

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1969

ACKNOWLEDGMENTS

I am indebted to Professor P. H. Doyle for his helpful guidance

during the preparation of this thesis.

ii

Section
I.
II.
III.

IV.

VI.
VII.

VIII.

INTRODUCTION

CONTENTS

RELATIONS AND SIMPLICIAL RELATIONS .. ......... ........

EMBEDDINGS AND SIMPLICIAL‘TIDT EMBEDDINGS ............

THE SEWN COMPLEX ..... ...........

BEADED COMPLEXES ..........

FACTORIZATIONS OF RELATIONS

FURTHER TECHNIQUES ......

UNFOLDABLE STARS ..........

BIBLIOGRAPHY

iii

18

29

36

46

54

59

SECTION I

INTRODUCTION

We consider in this thesis the problem of collapsing the pro-
duct Space formed when a finite simplicial complex K is crossed with
the unit interval I. This space, K X I, is certainly a convex linear
cell complex and through a subdivision can be considered to be a
simplicial complex.

To fix some notation, if K is a complex, IKI will denote
the underlying point set of K. s E K denotes the fact that the

l

m-simplex (or cell) 3 lies in K, while lsllcz K will imply that

l
the underlying point set of 31 is contained in lKl. a*sn is the
join of the point a with 8“. If sm is a face of an, we will
denote this fact by 8m‘< s“. K' will denote a subdivision of K,

and we define the star of the subcomplex L in K to be
stK(L) = {s E Kls n |L| a! a}.

Collapsing in a simplicial complex was first defined by White-

head [3]. Let K and K be simplicial complexes. If

1 2
= 7"“
K1 K2 U a s and
K2 0 a*sn = a*s n
then Kl simplically collapses onto K2 (denoted Kls\\tK2). If
there is a sequence of simplicial collapses from K1 to L, i.e.,

if K18\K28\ K3 s\,t Kn = L, then K collapses to L and this

1
is again denoted by Kls‘ng. If L = <v>, then K1 is said to be

collapsible.

Whitehead's simplicial definition was later extended to the
polyhedral category where, for example, Zeeman [5] gives the following
definition of an elementary collapse.

If K1 and K2 are polyhedra, we say there is an elementary

collapse from K1 to K2 if there exists an n-cell s with an n-l

face t such that

K = KzlJ S and t = K (1 s.

1 2

If there is a sequence of elementary collapses from K to K2, we

1

1 collapses to K2 and denote this by KINKZ. If K2 is

a point, then K1 is said to be collapsible.

say K

Certainly if KISN K2, then K1\)K2, and it is well known,
see [5], that if Kl‘xtKQ, then there exist subdivisions of K1 and

K2, say Ki and K5, such that Kis\1 K5. For this reason, the
collapsing used primarily in this thesis is of the second type, where
it is understood that (if it is needed) simplicial collapsing could
be used through the proper subdivisions.

The motivation for the investigations undertaken in this
dissertation can be found in Zeeman's paper [4], where several pro-
perties of the dunce hat are considered. Of particular interest for
us, is Zeeman's proof in that paper that while the dunce hat D is
not collapsible, D X I is collapsible. Generalizing this leads us

to investigate the collapsing of product spaces L X I, where we will

take L to be a simplicial complex.

SECTION II

RELATIONS AND SIMPLICIAL RELATIONS

Definition 2.1: A relation R on K is a set of ordered
tuples of the form (s,s', h), where s and s' are m-simplexes of
K,and h is a linear homeomorphism from ‘8‘ to ls'l such that
the following conditions are satisfied.

1. The first two co-ordinates of the tuples, considered as
ordered pairs form an equivalence relation R on the set
of Open simplexes of K. We denote the fact that two
m-simplexes s and s' are related by sRs'.

2. h is equal to the identity on '8' n '8']. For (s,s', h)

and (s', s, h') in the relation, we require that h' = h-l.

3. If two m-simplexes are related, then all their faces are

related each to each as follows. For the tuple (s,s',h),

if t < s, and t' < s' such that h(|t|) = It'l, then
th' and for the tuple (t,t',h1), h1 = h restricted to
t.

A relation on K is considered to induce an equivalence re-
lation R on the underlying point set K as follows, if a E ls],
and b e |s'| where sRs' and h(a) = b, then aRb.

The topological space IL‘ formed from a complex K. with a
relation R is the space ‘KI/R é lLl. t: K- L is the identification

function. Note that in general ‘Ll with the induced simplicial

structure may not be a simplicial complex. (The induced structure
L on lLl is given by the following: if s C K, then f(s) C L
and f(s) will be considered a "simplex" of L.) For example, if

K = ivl, w, v2, <W,v >, <w,v2>l, and the relation R identifies

1

v1 and v2 to a single vertex v, then L = iv, w, <w,v>1,

and we have two distinct one-simplexes sharing the same vertices.

<w,v>2},

However, the following theorem shows in what sense the space lLl
may be considered a simplicial complex.

Theorem 2.2: Let K be a simplicial complex with relation R,

 

and f: K- L be the induced identification map. If K" is the
second barycentric subdivision of K and L" the induced "simplicial"
structure on lL‘, then L" is a simplicial complex. This result will
follow from the lemmas discussed below.

Essentially, we are asking why the "simplicial" structure L
may not be sufficient to make lLl a true simplicial complex. It can
be seen to fail in possibly two senses. First, two distinct simplexes
in L may intersect in more than a single common face. Such a sit-
uation will be called an imprOper intersection. The second possibility
is that an "m-simplex" of L may have some of its faces identified so
that it is not homeomorphic to a standard m-simplex. We will show
that the structure L" has neither of these prOperties, so that ‘L‘
with this structure can be considered a true simplicial complex em-
bedded in some Rn.

Lemma 2.3: Let sn be a simplex of L. Then sn has at least
two faces identified iff there exists a "one-simplex" in sn with its

vertices identified. Such a one-simplex will be called a loop of type 1.

'nggf: If sn has faces 31 and sj identified, then the pre-
image simplex in K has a vertex v é f-1(si) and a distinct vertex
w E f-1(sj) such that under R, v = w. The one-simplex <v,w> will
under R be identified to a loop of type 1 in 3“. Conversely, if
sn 6 L has a loop of type 1, we know that at least two O-faces of
sn have been identified.

A loop of type 2 is defined to be two distinct m-simplexes with
their boundaries identified but not their interiors. Thus a one-
dimensional loop of type 2 consists of two one-simplexes with their
vertices identified. The "sphere" formed by identifying the boundaries
of two 2—simplexes would be a two-dimensional example.

Observe that if s U s is a 100p of type 2 in any dimension,

1 2

i.e., s U 82 is a "sphere", a first barycentric subdivision of

1

31 U 82 can contain no such loops.

Lemma 2.4: Let 3m and an be two "simplexes" of L of
dimensions m and n respectively, and suppose neither contains
' m n . . m n .
100ps of type 1. Then 8 0 s is improper iff s U 5 contains
loops of type 2.

m n . P

Proof: Suppose s n s is imprOper and let {vile be the

vertices of 3m that lie in sm 0 s“, and let {wilg be the vertices
n

of s that lie in the intersection. (Thus vi = w. for i = 0,1,...,p.)

1

Let t1 be the p-face of 8m that Spans {vi}: and let ti be the

p-face of s‘1 that spans {wilg. Since we have assumed an imprOper
intersection, t1 # ti. Now if the boundaries of t1 and ti are
identified, we have a p-dimensional loop of type 2. If this is not

and t5, 8 p-l

the case, there exists t2, a p-l face of t1,
face of ti, such that the vertices of t2 are identified with the

vertices of t5 but t2 # t5.

Again we either have a loop of type two or can find two non-
identified lower dimensional faces. An inductive argument, assuming
no higher dimensional loops of type 2 are found, leads us finally to
two one-simplexes with their vertices identified, obviously a loop of
type 2.

For the other direction, it is immediate that the existence of
type 2 loops implies an improper intersection.

If' L contains loops of type 1, L' can contain no such loop
types, but could contain loops of type 2. However L", the second
barycentric subdivision of L, will break up all possible loops of
either type. This concludes the proof of Theorem 2.2.

The map f: K" ~ L" has the property that the image of an m-
simplex of K" is an m-simplex of L", and the map can be considered
to be between two simplicial complexes.

This map f induces an equivalence relation R, called a
simplicial relation, on K" defined by relating two m-simplexes s

1
and s of K" iff f(sl) = f(sz). If s and s are related

2 l 2
by R, we will denote this by issz. The simplicial map f also
induces an equivalence relation, again called R, on the underlying
set lK‘ defined by relating two points p and q of ‘KI iff
f(p) = f(q), and we will denote this fact by qu. (In fact, any
simplicial map from one complex onto another that has the dimension
preserving property can be considered to induce a simplicial relation
R on the domain complex.)

Our results thus far show that we really need only consider

simplicial relations and maps between simplicial complexes, rather

than the more general concept of relation. The following definitions,

however, are given (where possible) in terms of relations rather than
simplicial relations to allow more freedom for some later considerations.
Definition 2.5: The set of simplexes of K which are related

to a given simplex s by a relation R will be denoted by SS

1 1'

Thus $31 = {s E K: sRsl}.

Definition 2.6: The set of points of |K| which are related to

 

a given point p by a relation R will be denoted by Sp. Thus
Sp = {x c ‘Kl: xRp}.

Definition 2.7: The order of the relation R on a simplex s

 

l

of K is the cardinality of $5 The order of the relation is the

1.
maximum of the simplicial orders, taken over all simplexes of K. We
consider only relations of finite order.

Definition 2.8: The set of points of lKl related to more
than one point is called the seam S of K. Thus 8 = UlSql, where
the cardinality of Sq is greater or equal to two. S, the closure
of S, is called the closed seam of K.

Definition 2.9: In a relation R, if s and t are simplexes
of K such that th and 8 O t = r, a non-empty face of each, the
relation is said to fold 3 across the crease r onto t.

Note: if the relation R folds 3 onto t and relates no
other simplex to the crease r, then the points of ls-rl are in S
and the points of |t-r| are in S, but the points of lr‘ will not
be in S as they are related only to themselves. However, lrl will
be contained in S, for if all the points in the interior of a simplex
are contained in a closed set, all its faces must also be in the closed
set and r is a face of both 5 and t.

Note, too, that if two simplexes are related, they are either

disjoint or one of them folds onto the other. In either case, all the

points of their union will lie in S; These observations lead
immediately to the following lemma.

Lemma 2.10: S = UlSs‘, where the union is taken over all

 

simplexes related to more than one simplex of K; and S, with the
induced simplicial structure of K, is a subcomplex of K.

Lemma 2.11: If er, R a simplicial relation, and if the

 

vertices of s are <v0,v1

<wo,w1,...,wm>, and if viRwi for all i, then any point p in s

,...,vm> and the vertices of r are

having barycentric co-ordinates (d0,d1,...,dm)S is related to that
point q in r having the "same" barycentric co-ordinates. That
13, q = (d0,d1,ooo,dm)ro

Proof: Note that f is a simplicial map inducing the relation
R and thus f(s) = f(r). By the prOperties of simplicial maps,

since p and q have the same barycentric co-ordinates, f(p) = f(q)

and hence qu.

SECTION III

EMBEDDINGS AND SIMPLICIAL TIDT EMBEDDINGS

We consider in this section embeddings of K in K X I or,
more precisely, embeddings of lKI in lK X ll. As a simplification
we will avoid, where possible, using lKI for the underlying subspace
and instead use K for both the simplicial structure and for the
underlying space. The nature of the discussion will make clear which
interpretation is being used, although in most cases we are considering
the underlying point set.

An embedding is denoted by 1: Ken K X I, and iK denotes the
image of K under i. K X I is considered a square with base K
and vertical 1. The "level" function 1: K X I H I is defined as
follows. For any 2 = (x,t), x E K and t E I, define 1(2) = t.

The point z is said to have level t in K X I iff 1(2) = t.

Definition 3.1: A point z of K X I is said to be above a
point w of K X I if 1(2) 2 1(w). A point 2 is said to be over
a point w if 2 is above w and if p(z) = p(w), where p: K X I a K
is the projection map onto K.

Definition 3.2: The vertical distance between two points z
and w in K X I is the absolute value of 1(2) — l(w).

Definition 3.3: A slice at level t for some fixed t E I, is
denoted by Kt and is the set of all points z of K X I having

level t. Thus Kt = {2 E K X I: 1(2) = t}.

10

Definition 3.4: An embedding iK of K in K X I is called

 

a projective embedding if for each x E K, p(i(x)) = x.

Definition 3.5: An embedding iK of a simplicial complex K

 

with a simplicial relation R in K X I is said to be relation-
sliceable, or R-sliceable, if, for all t in I, lp(Kt fl iK)| con-
tains no pair of points related by R.

For projective embeddings of K in K X I the following
notation is used for points of iK. Since a point x in an open

0
m-simplex '3‘ of K. has barycentric co-ordinates (do’dl’d2’°'°’dm)s

and since the point ix is over x at some level t, ix is denoted:
ix = [(d0,d1,...,dm)s, t] e x x 1

A projective embedding iK of K in K X I can be determined
by specifying the level of each vertex iv where v E K, then extend-
ing the map linearly between the images of the vertices. Thus, if s

is a simplex of K having vertex set <v ,v1,v2,...,vm>, then i(s)

0

will be its image, and if x G s has barycentric co-ordinates

(d d1,ooe,dm)8, then IX 3 [(d0,d1,ooo,dm)s, d + d t +°°'+ dmtm],

0’ 0‘0 1 1

where 1(i vj) = tj for each j.

This type of projective embedding will be called a simplicial

 

tilt embedding, and the image of any point x E K is completely deter-

 

mined by the image, iv, of the vertices v of K.

Unless specified otherwise, all embeddings discussed from now
on are simplicial tilt embeddings, and we will assume that for any
vertex v of K, l(iv) # 0 or 1. It is obvious that iK é K with
the homeomorphism given by the projection from K X I onto K

restricted to iK.

11

Let i be a simplicial tilt embedding of K in K X I and
let i(s) be the image of a simplex s of K. s X I is the subset
of K X I consisting of all the points above the simplex 3. Obviously
i(s) is contained in s X 1.

Definition 3.6: Let 2 and w be two points of s X I. Then

 

E = t(z) + (l-t)(w) denotes the maximal line segment in s X I running
through the points 2 and w. Note that the line E is not just the
set of points "between" 2 and w, but that it is defined to be the
whole line segment out to the "edges" of s X 1. Note, too, that as
i(s) is convex in s X I, if z and w are points of i(s), the line
E will lie entirely withh1i(s).

Lemma 3.7: The level l(r) of any point r E E, where E, is
the line through 2 and w and r = t*(z) + (l-t*)(w), for some fixed
t* a real number, is given by l(r) = t*(l(z)) + (l-t*)(l(w)).

"ggggfz It is obvious.

Corollary 3.8: If the points 2 and w in s X I are at the
same level, then all points on the line segment E through 2 and w
are on the same level.

Lemma 3.9: If Kt n i(s) is an m-cell, where s is an m-
simplex of K and i a simplicial tilt embedding, then i(s) is
contained in K .

t

‘groof: Let x 6 int Kt n i(s), and select an m+l spherical

 

neighborhood '1‘x about x in s X I such that TX 0 K = UX, an m-

t
spherical neighborhood contained in the m-cell Kt n i(s). Let iv
be any embedded vertex of s and consider the line segment E through

iv and x. Since i(s) is convex, there exists y E E n Ux’ y # x,

and y is on the same level as x. The line segment

12

E' = t(y) + (l-t)(x) contains iv since in fact E = E'. Thus, from
the corollary above, l(iv) = l(x) and hence, iv 6 Kt' Since iv was
an arbitrary vertex, all embedded vertices of i(s) are contained in
Kt’ and it follows that i(s) is contained in Kt.

Corollary 3.10: If, in a simplicial tilt embedding, the vertices
of an m-simplex, m > O, are embedded each at distinct levels in K X I,
then Kt n i(s) is a cell of dimension strictly less than m for all
t in I.

Egggfz This follows from the above lemma, for if the inter-
section were an m-cell for some t, then all the vertices of i(s)
would have level t, but as the vertices are at distinct levels, this
cannot occur.

Lemma 3.11: Given a complex K with a simplicial relation R,
there exists a simplicial tilt embedding iK of K in K X I such
that for each slice Kt and for each Sq, either i(Sq) n Kt = i(Sq)
or i(Sq) n Kt = $-

2522;: Divide the set of vertices of K into disjoint subsets
Bl’BZ"'°’Bk’ where each B1 is an equivalence class of related vertices.
Then define for all v E Bj’ i(v) = [v,j/k+l]. Thus, related vertices
are embedded at the same level, while unrelated vertices are embedded
at distinct levels. The mapping is then extended linearly to a
simplicial tilt embedding i. Now note that, for two related m-simplexes

s and r, if a E s and b E r such that aRb,

' = d ... , d + d +...
1a [(do, 1, ,dm)s 0tO 1t1 + dmtm] and

. = d .00 d ' + d ' 000 '
1b [(do, 1, , m)r, doto 1t1 + + dmtm] and

as ti - t; for all i, then the levels of a and b are the same.

13

Hence for each point q of K, Sq is embedded at the same level, and
it follows that Kt fl i(Sq) = T or i(Sq).

Definition 3.12: Given two distinct simplexes s and r of K

 

such that er and a simplicial tilt embedding iK of K in K X I,
the m-simplex i(s) is said to be above the m-simplex i(r), written
i(s) 2 i(r), if for each pair of distinct related vertices vj E s and
E r, 1(iv

w ) > l(iwj). If the relation is a fold of 5 'onto r, then

J .1

some of the vertices of s are the same as some of the vertices of r,

and for these common vertices, say v = wj, of course l(ivj) = l(iwj).

1
If all the vertices of s are embedded strictly higher than their re-
lated counterparts in r, then s is said to be strictly above r,

and this is indicated i(s) > i(r).

Theorem 3.13: A simplicial tilt embedding of K in K X I
is R-sliceable iff for all pairs of related simplexes r and 3,
either i(s) 2 i(r) or i(r) 2 i(s).

:EEEEE‘ The "if" direction will be proved by induction. Note
that if 2 and w are related O-simplexes for the relation to be
R-sliceable either l(iz) > l(iw), or l(iw) > l(iz), and the theorem
holds for 0-simplexes.

Consider now two distinct l-simplexes s and r, s = <w,x>
r = <y,z>, where wRy and sz.

Case 1: If the relation is a fold, i.e., suppose x = 2, then
either l(iy) > l(iw) or l(iw) > l(iy), for if the levels were equal,
we would contradict the theorem for 0-dimensional simplexes. Thus
either i(s) 2 i(r) or i(r) 2 i(s).

Case 2: If all four vertices are distinct, we note first that

the images of related vertices must each be at different levels, or we

14

have a contradiction to the theorem for O-simplexes. Without loss of
generality, let iw be the point at the lowest level. We show that
i(s) < i(r).

We know that l(iw) < l(iy), and it remains to be shown that
l(ix) < l(iz). Suppose by way of contradiction that l(ix) > l(iz).
Then the number a = l(iw) - l(iy) is negative and so too is the

number b = l(iz) - l(ix). Consider the following statement:
* d0(1(iW) - l(iy)) + d1(1(iX) - 1(12)).

where d1 = l - dO’ and O < di < l 1 = 0,1.

If this statement is equal to O for some choice of the di,
we know that the point ig 6 i(s) with these di's as barycentric
co-ordinates will be at the same level as the point ih E i(r) with

the same barycentric co-ordinates. This follows for if ig has bary-

centric co-ordinates do and d1, that is if

is = £<d0.d1>s. douche) + dluuxm

then the level of ig, l(ig) = d0(l(iw)) + d1(l(ix)), and similarly for
ih 6 l(r) with the same barycentric co-ordinates dO and d1,
1(ih) = d0(1(iy)) + d1(l(iz)). If the levels are equal, i.e., if

l(ig) = 1(ih), we then get the following equation:
d0(l(iw) - l(iy)) + d1(l(ix) - l(iz)) = 0

Hence we will show our assumption that l(iw) < l(iy) and
l(ix) > l(iz) implies that * has a solution for the di's, which implies
that the relation is not R-sliceable, a contradiction.

Statement * becomes after substituting l-d for do, setting

1

the result equal to zero and solving for d1,

15

= l(iz)—l(ix)

** d
1 l(iw)-l(iy) + l(iz)-l(ix)

If 0 < d < l, we know the relation is not R-sliceable. Our

1

assumptions on the levels of the vertices imply that

 

where a and b are both negative. Thus o.< d1 < l, and we have
arrived at a contradiction.

Hence l(iz) > l(ix) and for the l-simplexes r and s in
a simplicial tilt embedding, i(s) < i(r).

Suppose now that u and v are two related m-simplexes in
the R—sliceable relation and that neither i(u) s i(v) nor i(v) s i(u).
Suppose, without loss of generality, that a vertex iw of i(u) has
the lowest level of all the embedded vertices of both simplexes. Hence
if wRy, l(iw) s l(iy). Since our assumption is that i(u) i i(v),
there exists a pair of related vertices x of u and z of v such
that x # z and l(ix) > l(iz). Consider now the one simplexes <w,x>
and <y,z>. These are related, as they are the faces of related
simplexes, and their images in i(u) and i(v) must be R-sliceable.
But we have shown that for two related R-sliceable one-simplexes, one
of them must be embedded greater or equal to the other. Hence, our
assumption that l(iw) s l(iy) and l(ix) > l(iz) implies a con-
tradition.

Thus for the related m-simplexes u and v, i(u) s i(v). This
concludes the proof of the theorem in one direction.

Suppose now for each pair of related simplexes u and v that
either i(u) s i(v) or i(v) S i(u). Without loss of generality,

suppose that for the related simplexes u and v that i(u) S i(v).

16

We must show that the relation is R-sliceable through these embedded

simplexes. To fix some notation, let

U=<W ,W

0

,w2,...,wm> with l(iwj) = tj, and

l

V=<Z Z Z
,1,

0 ..,2m? with l(izj) = t3, and

2"

for all j let w R2,.
j J
Consider the point a in the interior of u, a with bary-
centric co-ordinates (d ,d ,d ,...,d ) , with 0 < d < l for all i.
0 l 2 m u 1
We know by a previous lemma that a is related to the point b in v
with the same barycentric co-ordinates. To show R-sliceability, we

must show that l(ia) # l(ib). From previous discussions we note that

oto 1 1

= t t
l(ib) doto + dltl +.. + dmtm.

If, by way of contradiction, l(ia) = l(ib), then
-'+ -'+... -' =.
do(t0 to) d1(t1 t1) +dm(tm cm) 0

But this clearly cannot occur because ti-ti a 0 for all i
with at least one term strictly negative, (as u and v have at least
one pair of distinct related vertices), and for all i, di > 0. Hence
we have shown that if i(u) s i(v) the relation is R-sliceable through
the embedded simplexes. This concludes the proof of the theorem.

Corollary 3.14: Given an R-sliceable simplicial tilt embedding
and two simplexes u and v such that uRv. If for the vertices

x 6 u and y E v, ny and l(ix) < l(iy), then i(u) < l(iv).

Proof: Immediate from the theorem.

17

Corollary 3.15: In an R-sliceable simplicial tilt embedding,

 

if u and v are two related simplexes with i(u) > i(v) and r
and s are two distinct related simplexes such that r O u # m and
s n v # ¢, then i(r) 2 i(s).
2522;: Immediate, as we observe that there exists a vertex w
of r and a related vertex 2 of s such that l(iw) > l(iz). Note

if r n s = ¢, then i(r) > i(s).

SECTION IV

THE SEWN COMPLEX

In the following section we assume that iK is an R-sliceable
simplicial tilt embedding of a complex K in K X I, where the
simplicial relation R is induced by a simplicial map f: K'« L. A
complex called the sewn complex, denoted by sK, is constructed in
L X I which contains a homeomorphic image of iK. This sewn complex
will play a special role in studying the collapsing preperties of
L X I and of L.

Consider for the point x E S (recall S is the seam of K)
the set Sx. Since for y and 2 in Sx with y f 2, iy and i2
are at distinct levels in K X I (assume that iy is above i2), it
is possible to erect a one-cell in K X I such that one end of the
cell is at iy, the other end of the cell is at the 12321 of i2, and
the whole cell lies in the vertical above y. We will denote this
cell by yIz. Likewise, from the point iz, a one-cell 21y can be
constructed with one end at iz, and the other end at the level of iy.

Since the embedding is R-sliceable, the process can be carried
out at every point in the embedded seam i(S). A cell from iy, yIz,
which lies below iy is called a root, and a cell which lies above iy
is called a stalk. If we consider two related simplexes uRv, we note
that if a point of i(u) is above its related point in i(v), then all

points of i(u) are above their related points in i(v). Thus each

18

l 9

point of i(u) has a root, and each point of i(v) has a stalk. We
note, too, that if u is an m-simplex, i(u) united with all its roots
and stalks is an m+1 cell. A given simplex can have roots to one
related simplex and stalks to another, or stalks to two other related
simplexes with two stalks or roots on the same point overlapping. (See
Figure 1.) In any case, the embedded simplex united with all its roots
and stalks is a cell one dimension higher than the simplex. Figure 2
shows the situation we are discussing for three related points.

Definition 4.1: We call iK united with all roots and stalks

 

the sprouted complex and denote it by rK.

We now extend the relation R on K to a relation RI on
K X I by defining (x,t)RI(y,t) iff ny, where x and y are re-
lated points in K, and t is a point of I. Let f': K X I
eKXI/RIPLXI.

Definition 4.2: f'(rK) = sK<: L X I, and we call sK the sewn
complex.

Note that sK contains a homeomorphic image of iK, namely
f'(iK), with additional cells "sewn" across the images in iK of re-
lated simplexes of K. For example, if uRv, where u and v are
m-simplexes of K, and if in iK, i(u) 2 i(v), then the roots from
i(u) down to the level of i(v) and the stalks above i(v) up to
the level of i(u) will, under RI’ be identified to form in SK 3
prismatic m+l cell with base f'(i(v)) and top f'(i(u)). (See
Figure 3.)

If u,v, and w are three related m-simplexes of K and if

i(u) 2 i(v) 2 i(w), then in sK there will be a "path" of two m+l

cells with top f'(i(u)), bottom f'(i(w)), and f'(i(v)) embedded

KXI

20

 

yIz

 

1y

12

zIy

 

 

Figure 1

 

 

 

\u

 

 

12 \\\\\‘

 

 

Figure 2

21

 

W
2 "3 W
1:"2

 

 

 

 

 

 

 

 

 

 

 

allflllllll S K

 

 

 

 

 

 

 

 

 

 

 

 

22

in the middle, separating the two cells. (See Figure 4.)

Definition 4.3: If u and v~ are two related simplexes in
K, uIv will denote the prismatic cell in sK between f'(i(u)) and
f'(i(v)).

Definition 4.4: C contained in SR is the union of all the

 

prismatic cells in sK. Thus, C = SJ. silsj, where Si and sj are
related simplexes of K. ,J

Theorem 4.5: Let iK be an R-sliceable, projective embedding
of K in K X I. If sK is the sewn complex in L X I formed by
the relation RI’ then L X I‘SgsK;

2322:: Order the simplexes of K in decreasing dimension. If
31 is a simplex not contained in the closed seam S, then i(sl) is
an m-simplex embedded in K X I on neither the base K X 0 nor the
top K X 1. Hence there will be two m+l cells in K X I, one above
i(sl) with i(sl) as its floor and one below the embedded simplex
with i(sl) on its top. (See Figure 5.) Under the identification
RI’ these two cells are sent by f' to corresponding m+1 cells in
L X I, as identifications can only occur on the boundaries of these
cells. Thus both can be collapsed onto f'(i(sl)) and onto the
"walls" of the cylinder in L X 1, since f'(slxl) and f'(sle) will
be free faces.

Thus in L X I collapse out all cells which are over or under
embedded simplexes of this type (i.e., simplexes not contained in S)
proceeding down from the highest dimensional cells.

Next consider each cell path in sK, slISZI"°ISm’ where the 81

are m-simplexes in S; Above f'(i(sl)) there is an m+l cell in

L X I, and below f'(i(sm)) there is a similar m+l cell. (See Figure 6.)

23

 

 

f (i(u)) I‘F

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11-4.1

f'(i(v))

 

 

 

 

 

 

Figure 4

~~~~f'(ix3)

 

 

 

f'(1X4)

24

a)

 

 

 

 

 

b)

 

 

 

 

e)

 

 

 

 

 

 

Figure 5

26

In the vertical cell path contained in L X I, collapse down onto the
top embedded simplex of the cell path in SK, in the order of decreas-
ing dimension. This is always possible as a free face is found at
the top of L X I. The same procedure is followed from the bottom of
L X I, collapsing upward onto the base of a cell path. The end result
gives a method of collapsing L X I onto sK.
Theorem 4.6: If L is collapsible then sK is collapsible.
2522:: Consider the projection map p: L X I e L. As SR
is embedded in L X I consider p restricted to 3K, i.e., p: sK-a L.

By the nature of the embedding, p is a strongly pointlike map, as

p (x) is a single point if x is not in f(S) and p-1(x) is a

chain of one cells if x is in f(S). In either case p (x) is
collapsible. Thus by the characterization of strongly pointlike maps
[2] we get that if L is collapsible, then 8K is collapsible.

The following is an example of how theorem 4.5 can be used to
show that a given L X I is collapsible. The complex L is con-
structed from the two-cell K = I X I in the following manner. (See

Figure 7a.) We identify the "diagonal" elements <v ,v2> and

l

<v3,v4> with the boundary of K. That is to say <v is broken

1"’2>
into 4 one-cells, say 1', 2', 3', and 4', and we then identify 1'

with the edge <vl,v3>. 2' is identified with the edge <v >, 3'

3 "’5
>. Likewise diagonal element

with <v5,v > and 4' with <v6,v

6

<v3,v4> is broken into 4 one-cells, and the first is identified with

l

<v3,v5>, the second with (v5,v

these identifications is L.

6>’ etc. The complex that results from

We assume that K. is suitably subdivided such that the mapping

f: K a L is a simplicial mapping.

v6

27

 

 

V4

 

 

 

 

 

 

 

Figure 7

 

A trianpular cell
sewn onto K across
the diagonal (V1,v2)
and the boundary

of K.

28

Now it is obvious that L is not collapsible, as L has no
free edges from which a collapse may be started. However, we shall
show that L X I is collapsible.

First we will show that the relation on K is R-sliceable.

This is obvious if we consider the following projective embedding of
K in K X I. We embed the boundary of K at level %, and the diagonal
element 4<v1,v2> linearly such that iv is at level 3/4. The second

2

diagonal element is "bent downward", i.e., we embed iv4 at level %.

(See Figure 7b.)

This embedding iK of K is obviously R-sliceable. Now we
consider sK. It can be seen that sK will be homeomorphic to K
with two additional "triangular" two-cells sewn on in the following
manner. The tap cell is sewn on in such a way that one edge of the

"triangle" is identified with i<v1,v >' and the second edge with the

2
boundary of K. The remaining edge is free in sK. (See Figure 7c.)
The other two-cell is sewn onto K in a similar manner but across the
4> and the boundary of K. It is obvious that sK\ iK

as these two added triangular cells can each be collapsed, starting

edge <v3,v

from the free edges. Thus 1L X I\ sKNiKNO, as K is a two—cell.

SECTION V

BEADED COMPLEXES

The techniques we have introduced thus far lead us to consider
in this section the following problem. ‘We have been studying the
collapsing of L X I by looking at a complex K which maps onto L,
then forming from K a sewn complex sK. ‘We can ask.what types of
complexes K. we can find for a given L. In particular, could we
always find a collapsible complex K such that the relation induced
on K by the simplicial map f: K-4 L is always R-sliceable?

Definition 5.1: A principal simplex s in a complex K with
the property that for each distinct principal simplex t in K,

s n t = ¢, or s n t = v, (v a vertex), is said to be vertex joined
in K, and v is called a joining vertex. V will denote the set of
joining vertices of such a simplex.

Definition 5.2: A principal simplex s of K is proper if:

1. It is vertex joined in K.

2. It meets at most two distinct principal simplexes
in K.

3. If s meets exactly two distinct principal simplexes,
then the set V of joining vertices contains exactly
two distinct points.

Definition 5.3: In a prOper simplex with joining vertex set
V = {v,w}, the one-simplex <v,w> is called the distinguished face of
a. If V - {w}, then the vertex w is the distinguished face of s.

29

30

Note: if V = ¢, then K = s, i.e., K contains only one
principal simplex and in this case an arbitrarily chosen vertex of s
will be called the distinguished face. Thus a prOper simplex in a
complex K has one distinguished face, which is either a vertex or a
one-simplex.

Definition 5.4: A complex K is a beaded complex if all

 

principal simplexes of K are prOper and if the union of all their
distinguished faces in K' is either:
1. the homeomorphic image under a map g: I a K of the
unit interval, or
2. the union is a single point. (Such a complex will
be called a degenerate beaded complex.)

Notation: bK denotes a beaded complex. (See Figure 8.)

Definition 5.5: A principal simplex in a beaded complex is
called a bead. For non-degenerate beaded complexes, the image of I
under g is called the string of the beaded complex, and it well
orders the principal simplexes of K as follows. For 3 and t
two beads in K with joining vertex sets V and V' reapectively,
s precedes t, written 3 S t, if there exists v E V such that
g-1(v) is less than or equal to g-1(w) for all w E V'. As a
beaded complex is connected and finite, this well orders bK.

Lemma 5.6: If bK is a degenerate beaded complex, either bK
contains only two distinct principal simplexes s and t such that
s n t = v or bK = s for some simplex s.

Igggggz Obvious.

A degenerate complex's beads can be well ordered trivially.

Definition 5.7: An extreme bead in a beaded complex is a bead

with vertex set V containing only one point. A beaded complex

32

(non-degenerate) contains exactly two extreme beads,and all other
beads will have two points in their joining vertex set.

Definition 5.8: Given a beaded complex bK containing two or
more beads, a £22 divides bK into two disjoint beaded complexes bK

1

and bK2. This is done by selecting any two beads s and t of K

sudh that s n t = v, then setting bK homeomorphic to the subcomplex

l
of bK consisting of all beads that precede s, (which includes 8
itself), while sz is a complex homeomorphic to the subcomplex of K
consisting of t and all beads that follow t. Two disjoint beaded

complexes of dimension greater than 0, say bK and bK , may be

1 2
joined to form a beaded complex bK containing a homeomorphic image
of each by selecting extreme beads of each, say 3 of bK1 and t of
bK2, and identifying a non-distinguished vertex of s with a non-
distinguished vertex of t. (A non-distinguished vertex of a bead is
a vertex not contained in a distinguished face.)
Lemma 5.9: A beaded complex is collapsible.
2522;: Obvious.
Theorem 5.10: If L is a connected complex, there exists for
each chosen "starting" vertex v of L a beaded complex bK and a
simplicial map 3: bK- L such that:
l. g is surjective
2. g preserves dimension (hence induces a simplicial
relation on bK)
3. bK is made up of exactly one n-bead for each
principal n-simplex of L, n 2 2, and an arbitrary

number of one-beads which map into the l-skeleton of

L.

33

4. v is the image of a non-distinguished vertex of an
extreme bead of bK.

‘ggggf: This theorem will be proved by induction on the number
of simplexes in L. If L contains one simplex, then L is a beaded
complex as L r {v}, and the identity map on L can be taken for the
map g. Suppose any complex containing k or fewer simplexes can be
"covered" by a beaded complex which satisfies conditions 1-4.

Let L contain k+l simplexes. If L is a complex of
dimension 1 and L is not a tree, there exists a principal one-simplex
s of L such that L-s is connected and contains k simplexes.
Hence L-s can be covered by a beaded complex by the inductive
hypothesis, and we select the starting vertex w to be a face of

l
s in L-s. If L is a tree, there exists a one-simplex s =1<w1,w2>
of L with the property that L-s-{w2} is connected and contains
k-2 simplexes. Again L-s-{wz} can be covered by the inductive
hypothesis with selected starting vertex wl.

If dim(L) = n, where n is strictly greater than one, there
exists a principal simplex s of L with dimension n; and L-s,
containing only k simplexes, can again be covered by a beaded com-
plex with starting vertex w1 a face of s in L-s.

Thus in all cases, given an L containing k+l simplexes, we
have removed a principal simplex s and covered L-s with a beaded
complex, i.e., we have a bK and a g: bK-a L-s such that bK
satisfies conditions 1-3 with the starting vertex w1 a face of s
in L-s. Now in bK select the extreme bead t whose non-distinguished
vertex x is mapped onto wl, i.e., g(x) = wl. As t is an extreme

bead in bK, it has a single joining vertex 2, and we modify bK to

form a new beaded complex to cover L as follows.

34

Select a copy 8 of the principal simplex s in L that was

removed, and let h: s a L be a simplicial map such that h(s') = s.

Select the vertex h-1(w ) of s' and attach s' to bK by identify-

1

ing h-1(w1) with x = g-1(w ). This forms a new beaded complex

1
bK.U s', which covers L by extending the map g: bK a L-s onto
bKlJ s' by setting g(s') = h(s'). Finally, replace 2212 the beads
in bK. which map onto the faces of s by their distinguished faces.
Note that this final beaded complex satisfies conditions 1,2, and 3.

For condition 4, a "tail" of one simplexes (i.e., a beaded
complex of dimension 1) can be added onto bKlJ s' at a non-dis-
tinguished vertex u of s' such that the tail maps into the one-
skeleton of L and forms a path that connects g(u) with any selected
vertex v of L.

Definition 5.11: A beaded complex satisfying the conditions of
the above theorem will be called a covering beaded complex of L.

Corollary 5.12: Any beaded complex mapped onto a complex L
by a dimension preserving map can be modified to consist of only one
bead covering each principal simplex in L of dimension greater than
one and connecting one-beads which map into the one-skeleton of L.

Iggggf: The part concerning the principal simplexes can be met
by selecting one bead of bK to cover each principal simplex of L
and then replacing all other simplexes of bK, i.e., all of those not
selected as covering a principal simplex with their distinguished
faces. The modified complex will then satisfy the conditions of the
corollary.

Corollary 5.13: Every connected n-complex is the simplicial

image of a collapsible n-complex.

35

Proof: Obvious, as a covering beaded complex is collapsible.

Theorem 5.14: Given a beaded complex bK covering a complex

 

L by a simplicial map f, then the simplicial relation R induced on
bK by f is such that bK has an R-sliceable simplicial-tilt em-
bedding in bK X I.

nggf: Let n equal the number of beads in bK, and let

81 s 82 s...s an be the ordering of the beads. Divide the interval

[1/n+2, n+1/n+2] up into n subintervals, I1 = [i/n+2, i+1/n+2],

i - 1,2,...,n. Now 8 is embedded projectively in s X I such

1 1
that its distinguished vertex is a level 2/n+2 and all other ver-

tices v1 of 81 are at levels chosen arbitrarily but such that

they lie strictly within the interval I The second distinguished

1.
vertex of 32 is embedded at level 2/n+2, as the first is shared

with 81

ing vertices of s

and has already been embedded at level 2/n+2. The remain-
2 are embedded projectively at levels strictly

between 2/n+2 and 2/n+2. A simple inductive argument embeds 31
in bK X I between levels i/n+2 and i+l/n+2, i = 1,2,...,n.

This embedding of bK in bK X I is R-sliceable and pro-
jective, since any two related simplexes in the seam S can not both
he faces of some 31, they must lie on distinct beads and their images

in bK x I lie one strictly above the other by the nature of the

embedding. Hence by Theorem 3.13, the relation is R-sliceable.

SECTION VI

FACTORIZATIONS OF RELATIONS

We begin with a lemma which provides the motivation for the
investigations undertaken in this section. This lemma leads us to
ask questions concerning the homological relationships that exist
between K, 8K and L; and these considerations lead in turn to the
concept of fold factorizations. After developing the factorization_
concept, we will conclude with some theorems on "folding" and homology.

Lemma 6.1: Let L be a two-complex, and let K be a two-
complex mapped onto L by a simplicial map f. Let the closed seam

‘S of K contain only two one-simplexes, say 5 and 82’ which are

1
identified by f. If K is homologically trivial and L is homo-

logically trivial, then the identification must be a fold, (see 2.9).

nggf: Let d(Hi(K)) be the Betti number of the ith homology
group of K. Then. X(K) = d(HO(K)) - d(H1(K)) + d(H2(K)) = 1-0 + 0 = l,
and likewise X(L) = dO(HO(L)) - d(H1(L)) + d(H2(L)) = 1-0 + 0 = 1.

Let v1 be equal to the number of ith dimensional simplexes in K,

i = 0,1,2 and Vi represent the number of ith dimensional simplexes

in L, i B 0,1,2. It is well known that X(K) = VO - V1 +-V2 = l, and

likewise for the Vi. Now as f identifies no two simplexes, Vé = V2.

If the relation were not a fold, i.e., if 81 n 82 = ¢, then V6 3 VO-Z

and Vi = Vl-l, since disjoint one-simplexes and their vertices are
0 I _ U U = _ _ _
identified. This implies that V0 V1 +“V2 (V0 2) (V1 1) +-V2

36

37

= 1-1 = 0, a contradiction. Thus the relation must be a fold, i.e.,

31 n 32 is a crease point.

Attempting to generalize this result by allowing more simplexes
in ‘S, hoping to arrive at a "sequence" of folds, proved fruitless.
However, approaching from the "converse" direction leads to the follow-
ing:

Definition 6.2: Let K be a simplicial complex with a relation

R. A factorization is a sequence of complexes" K and maps fi

'51 f2 1
such that K = Kl-~ K.2 -H K.3 a...» K.n = L = K/R, where K1+1 is
formed from Ki by identifying "simplexes" in Ki and f1 is the
identification map. We require, of course, that fn_lofn_20...of1 =

f: K-» L. Certainly f: K- L is a sequence which forms a trivial
factorization. For notation, 'Ki’fi}: denotes a factorization.

If s 6 K1 and f;1(f1(s)) = s, we will consider fi(s) = s,
to simplify notation. If f;1(fi(s)) = s U t U...U p, we will say
that fi(8) - s = t 8 f1(t), and be careful to note what stage of a
given factorization we are in. Thus in the above cases, in Ki’ 8 f t,

but in K = t, as they have been identified by f1.

i+1’ '3
Lemma 6.3: Given a factorization {K1,f1}?, there exist "sub-
divisions" Ki" of each K1 such that the resulting factorization
can be considered to be a sequence of simplicial complexes and
simplicial maps, or a simplicial factorization.
‘Egggg: This follows by a simple inductive argument, using the
second barycentric subdivision of K and the results of Theorem 2.2.

If {Ki’fi}l is the factorization, then the fact that

'Kl’fi}l is a simplicial factorization is just the conclusion of 2.2.

38

Let us assume that if 'Ki’fi}I is a factorization,then

{Ki’fi}l’ where K" is the second barycentric subdivision of Ki’

1
is a simplicial factorization.
n+1 . it u .
Let {Ki’fi}l be a factorization. Then K1 a K2 is a

simplicial factorization of length 2, and {K2,f1}2+2 is a simplicial
factorization of length n by the inductive hypothesis. Thus
{Kl’fi}I+1 is a simplicial factorization.

We will assume for the next theorem that K and L are n-
complexes with n greater or equal to 2 and that f: K'e L induces
a simplicial relation on K such that dim(S) is less than or equal
to n-l, i.e., no n-simplexes in K are identified. (Note: we can
always cover L with a beaded complex K such that no n-simplexes
of K are identified by 5.12.)

Theorem 6.4: Let {K1,f1}: be any simplicial factorization.
If Hn(L) = 0, then Hn(Ki) 8 O for all i.

.EEQEE‘ Suppose for some i that Hn(Ki) f 0, then it is easy
to see that Hn(Ki+l) f 0. Consider the definition of H“. As K1
and Ki+l are n-complexes, Hn(Ki) = ker(Cn(K1)-g Cn-1(Ki))’ where
Cn is the free group generated by all n-simplexes, Cn_1 the free

group generated by all n-l simplexes and d the usual boundary

identifies no n-simplexes, f in-

operator. Now as f1: Ki'fl K 1

1+1

duces an isomorphism from Cn(Ki) » Cn(Ki+l)' Now consider the follow-

ing commutative diagram.

‘ f. f

1
0n- l (Ki) fiJ> C'n- l (Ki-+1)

 

i+l

 

39

If d: Cn(Ki) » c (K ) is not 1 : 1, i.e., if Hn(Ki) s o,

n-l i

then d: cn(K ) a Cn-1(Ki+l) has to have a non-zero kernel as:

1+1

H (K ( ))

n 1+1 ) d C

v
I

” ker(d‘ Cn(Ki+l n-l K1+1

= ker(fiod: Cn(Ki) 4 Cn_1(K ))

1+1

ker(dofi: Cn(Ki) e Cn- (K ))

1 1+1

r 0

But Hn(K ) # 0 implies inductively that Hn(L) # O, a con-

i+l
tradiction.

Lemma 6.5: If f: K,a L induces an R-sliceable relation on K

and {K1,f1}: is a simplicial factorization, then the relation Rj

induced on K by f o...of K!» K.+ is R-sliceable.

iOfJ-l 1‘ J 1
Proof: There exists an R-sliceable embedding, say iK, of K

in K X I, and it is obvious that iK will be R-sliceable for the

relation R as R is "essentially" a subset of the relation R.

J 1
Definition 6.6: Let 'Ki’fi}I be a factorization.

fi: Ki'e K1+1 is called an m-fold of n-simplexes, or an (m,n) fold,

if the "closed seam" in Ki induced by fi is a set {5} of m
n-simplexes which intersect in a common p-face, (the crease), and which

are identified by f to one n-simplex in K The faces of a fold

i i+l°

simplex s E 88 not contained in the crease will be called the free

faces of 8. If K1 and Ki+1 are simplicial complexes, the fold is

a simplicial fold.

Theorem 6.7: Let K be a simplicial complex and let f: K-4 L
= > g .
1 <w,v1 onto 82 <w,v2> If
lst(v1)| n lst(v2)| = w, then L is a simplicial complex.

be a fold of the simplex s

40

nggf: From Theorem 2.2 we know that if L contains no loops
of type 1 or 2 then L will be a simplicial complex with the structure
induced from K. The only possibility for a 100p of type 1 to occur
is if there exists a one-simplex in K of the form <v1,v2>. But
since the intersection of the vertex stars contains only w, no such
one-simplex exists.

The only place for a 100p of type 2 to occur, since K is a
simplicial complex, is if it is of dimension 1 and is of the form

<v,x>' U <x,v>2, where v = v = v in L and x is some other

1 2

vertex of K, x # w. Again the intersection of the stars implies

1

that x = w, and this loop cannot occur.

Corollaryj6.8: If K is a simplicial complex and f: K!» L
is a (m,1) fold of the simplexes si =«<w,vi>, i = 1,2,...,m, onto
a single one-simplex and if for all pairs 1 and j,

|st(vi)‘ n |st(vj)| = w, (i i j), then L is a simplicial complex.

Theorem 6.9: If K is a simplicial complex and f: K a L is

a (2,1) fold of 81 = <w,v1> onto 82 = <w,v2> and if L is not
a simplicial complex, then there exists a subdivision K? of K and
a simplicial factorization K? a K; a K; a K2, where lKZI = |Ll and

II __. ll .
each stage Ki K1+1 is a simplicial fold.
Proof: First take the 2nd barycentric subdivision of K and

call it KY. Let us assume that 81 is broken into the four one-

simplexes {<xi,xi+1>}:, and 3 into the four one-simplexes

2
4 — — -
{<yi,yi+f>}1, where x1 y1 - w, x5 v1 and y5 v2.

Now K; is formed from K: by folding <xi,x. > onto

+1 1+1

(yi’yi+1>' Note that K2 is a simplicial complex and that lKZ‘ = IL"|

by 2.2. It follows that each stage K? is also a simplicial complex

41

for if some K; were not, it would contain 100ps of type 1 or 2, and

it is obvious from the nature of the subsequent identification that

K"

1+1 would also contain such 100ps.

cannot occur 0

Corollary 6.10: If {Ki’fi}? is a

 

Since

K"

4 is a complex, this

(mi,l) fold factorization

but is not a simplicial factorization, then there exists a simplicial

one-fold factorization iK¥,filT where m

i = 1,2,...,n.

Theorem 6.11:

 

If an R-sliceable relation on

= 4n and lKil = 'Kzi"

K has a simplicial

factorization as a sequence of (mi,l) folds, say {Ki’fi'l’ and if

1. for each one-simplex s of

of some mi fold, Ss
one-simplexes, and
if for each v an element

free face of some m1

m1 points,

then SK\ iK.
Proof: We will show this result by
the factorization. If K2

lation on K consists of a single (m,l)

of m-l 2-cells sewn across the fold.

in K

of S

fold,

S where s in an element

consists of exactly 1111
such that v is a

Sv consists of exactly

induction on the length of

is the final stage, then the entire re-

fold, and sK will consist

Between any two of the one-

simplexes in the fold, one embedded directly above the other in K X I,

there will be, in

through the free faces of the two one-simplexes.

a collapse of the two-cell. Similarly all

be collapsed, leaving us with iK.

Assume

ditions 1 and 2 that sK can be collapsed

that in a fold factorization of length n

sK, a triangular two-cell having a free edge running

This free edge allows

of the m-l two cells can

meeting con-

back to iK, and let us

42

suppose that {K1,fi}:+1 is a fold factorization. The mn one-
simplexes identified on the last fold will be such that through their
free faces will be found, in SR, a sequence of free edges which will
allow the triangular two-cells sewn across these one-simplexes to be
collapsed. The resulting complex is homeomorphic to the sewn complex
which would be formed by mapping K onto K.n through an n-step fold
factorization using the original R-sliceable embedding (see 6.5) and
omitting the last fold in the given n+1 stage factorization. This
complex is collapsible by the inductive hypothesis onto iK so
sK\)iK.

Definition 6.12: A fold factorization is called proper if for
each y a point in some (mi’ni) fold (y not in a crease) Sy has
order mi.

Note: we have with the above definition "abstracted" conditions
1 and 2 of Theorem 6.11 into higher dimensions.

Theorem 6.13: If an R-sliceable relation on K induced by a
map f: K- L has a simplicial factorization as a sequence of proper
(mimi) folds, then sK\iK and if K\;o then LX 1N0.

nggf: Dimension 1 is just a restatement of 6.11, and the higher
dimensional cases would be proven in a similar manner. Since K ; iK,
the final conclusion is obvious, as it follows from 4.5.

To illustrate the above theorem we will consider an example. We
will construct a complex L through a sequence of proper l-folds from
a collapsible complex K. Corollary 6.10 implies that the initial

proper fold factorization can be subdivided into a simplicial fold

 

factorization, and it is obvious that this resulting factorization is
also prOper. Thus we will satisfy the hypotheses of 6.13. (We should

make note of the fact that 6.10 does not appear to have higher

43

dimensional analogs, at least the second barycentric argument does not
carry through. One must be careful in applying 6.13 to insure that the
complexes considered satisfy all conditions described.)

Let K be the two-cell I X I, (see Figure 9), and let L be
formed from K by identifying the n diagonal elements <wo,vi>
i = 1,2,...,n, with the boundary of K in the following manner. For
each i, <w0,vi> is broken into 4 l-cells, say 1;, 2;, 3;, and 4;;

and identify 1; with <w ,w1>, 2; with <w ,w

1 >, 3; with <w ,w >

0 2 2 3

f ' >.
and 41 Wlth <w3,wO

An R-sliceable embedding of K in K X I is given by embedding
the boundary of K at level l/n+2 and the vertices vi at levels
(i+l)/n+2, i - 1,2,...,n. This embedding of K will be R—sliceable

for any subdivision of K.

Now a proper fold factorization of f: K<~ L is given by let-

ting K1 = K and then identifying all the 1' cells with. <w0,w1> to
get K2. K.3 is formed from K2 by identifying all the 2' cells with
<w1,w2>, and K; is formed from K3 by identifying the 3' cells. The
4' identification yields K which is L. Each of these stages is

5
obviously a proper fold and 6.10 implies we can find a proper simplicial

fold factorization. Thus L X I is collapsible.

We conclude this section by returning to some homological con-
siderations examined in light of our folding results.

Theorem 6.14: If f: KH L is a strongly pointlike map [2],
i.e., a map between simplicial complexes such that for all y in L,
£"(y) is collapsible, then u(x) = H(L).

.nggg: Since for each y E L, f-1(y) is collapsible, the con-

ditions of the Victoria mapping theorem [1] are met, and thus f induces

44

 

 

 

 

 

Figure 9

45

an isomorphism from H(K) to H(L).

Corollary 6.15: If f: K-» L induces an R-sliceable relation

 

on K, then H(sK) = H(L).
Proof: Immediate as the projection p (see 4.6) is a strongly
pointlike map from 8K to L.

Theorem 6.16: If f: K a L is an (m,n) simplicial fold,

 

then H(K) = H(sK).

.3E22E3 Any (m,n) fold induces an R-sliceable relation on K
and is certainly proper; thus sK is defined. Since sK‘SgK, i.e.,
sK‘\siK which is homeomorphic to K, SK and K have the same homo-
topy type and thus H(sK) = H(K).

Corollary 6.17: If {K1,fi]? is a simplicial fold factorization
of an R-sliceable relation, then H(sK) B H(Ki) = H(L), for all 1.

Proof: Immediate from the above results.

SECTION VII

FURTHER TECHNIQUES

We will develop in this section some new techniques for
collapsing L X I. However, similarities between these new methods
and the methods of the previous sections will become apparent, and
we will conclude with an investigation of a few of these relationships.

Definition 7.1: A simplicial covering of a complex L is a

 

finite set of complexes K1 and maps fi such that fi: Ki 4 L is
n

a simplicial homeomorphism into L and U fi(Ki) = L. We will let
1

[Ki’fiJI denote a simplicial covering of L.

Lemma 7.2: Any complex L has a simplicial covering [Ki’fiJI

such that for all i, Ki\)0.

Proof: The set of principal simplexes {$1}: of L (considered

as a disjoint set) together with the embedding maps fi: 31 a L such

that f(si) = <2 L, constitutes such a covering.

Si
Definition 7.3: If L has a simplicial covering [K1,f1]$
such that for all i, Ki\0, we will call the covering of L a
collapsible covering.
Definition 7.4: For 5 a simplex of L, 88 = [Ki: s<Z fi(Ki)].
We can well order the set of complexes in 83 by saying that if Ki

and K are in 88, K .< Kj iff i < j.

1
Lemma 7.5: If [Ki’fiJI is a covering of L, then L X I‘\9cK,

.1

where cK is defined as follows:

46

 

Cl

47
n -1
ex = (U(f.(K.) x i/n+1)) U (U (p (f.(K.) n
1 1 1. l,j l l

fj(Kj) n L x [i/n+l, j/n+l])

‘nggf: Order the simplexes of L in decreasing dimension.
Then proceeding from the highest dimensional simplexes, collapse cells
of the form 8 X [0,i/n+1]\\,s X (i/n+l) U s X [0,i/n+l], where K1
is the first complex in Ss and collapse cells of the form
5 X [1, j/n+l]‘\is X (j/n+l) U s X [1,j/n+l], where Kj is the last
complex in 88. The remaining set is cK.

Lemma 7.6: If [Ki’fi]l covers L and cK is defined as in
the preceding lemma, then there exists a map m: cK a M, where M is a
one-cell complex defined in the proof below.

nggf: First we consider Lt fl cK for t = i/n+l, i = 1,2,...,m,
and at each level, shrink the connected sets lying in Lt 0 CK to
points. Now we note that between any two levels i/n+l and i+l/n+l
in the so modified cK we will have a structure that consists of
points on the "ends", i.e., at the levels i/n+l and i+l/n+l and
connecting segments running between the two levels. Now if a point on
level i/n+l is connected to a point on level (i+l)/n+l, then the
connecting segment is shrunk to a one-cell between the two points. We
will consider this shrinking to preserve "levels". Thus, the inverse
image of the point halfway between the endpoints of the one-cell will
be a set lying on a segment in the modified cK all of whose points
have leve l ( i+%) / n+1 .

It is conceivable that two points on different levels may have
more than one distinct connecting segment, so in the final result, the

two points will be connected by as many one-cell segments as there are

 

,...ng'

48

"connecting" segments in the modified cK.

The end result is a one-cell complex M, and the map m: cK « M
is merely the composition of all the shrinking steps. Because of the
construction, we can meaningfully speak of the level of a point belong-
ing to the cell complex M.

Lemma 7.7: If [K1,fi]: is a collapsible covering of L such
that for all t, l.t 0 CK is the disjoint union of collapsible complexes,
then m: cK- M is strongly pointlike.

nggf: m-1(b), where b is a point of M at level t, is
'collapsible, as it is one of the collapsible sets in the disjoint union

of l.t fl cK.

 

Theorem 7.8: If H(L) = 0 and [Ki’fi]l is a collapsible
covering of L such that for all t, 1% n cK is the disjoint union
of collapsible complexes, then L X I‘\90.

25999 L X I\cK and m: cK-aM is strongly pointlike. By
6.14, as H(L) = 0, H(cK) = 0 and H(M) = 0. Now as M is a one-
cell complex with zero homology, M is a tree; and thus it is collapsible.
Hence cKPSsO by the characterization of strongly pointlike maps [2].

Corollary 7.9: If [K1,fi]: is a collapsible covering of L

 

such that Kit] K1+1 is collapsible for all i = 1,2,...,n-l, and
Ki n Kj = ¢ if j # i+l, then L X I is collapsible.

.ggggf: The map m: cK n M is a strongly pointlike map onto a
chain of n l-simplexes. .

An application of Corollary 7.9 is given below. Let Lj be a
copy of Bing's house with two rooms, and let L be made of 6 copies of

Lj arranged as pictured in Figure 10b, (Bing's Hotel perhaps).

50

We divide L into 4 sets Ki’ i = 1,2,3 and 4,by cutting L
with planes perpendicular to the x,y plane and parallel to the y
axis passing through the points x = 1,2, and 3. Thus each K1 is

collapsible, and Kilfl K1+. is either homeomorphic to a 2-cell or to

l
the object shown in Figure 10d. In either case the intersection is
collapsible, and thus Corollary 7.9 implies that L X I\0. (Note
that the same argument can be applied to an m X n array.)

Suppose we now consider the following situation. Let L be
a non-collapsible two-complex and suppose L can be divided into two
disjoint collapsible subsets L1 and L2 such that L1!) L2 is a
tree. Corollary 7.9 would then imply that L X I is collapsible.
However, we will now show that L X I is also collapsible by the
methods of the previous section; i.e., we will find a collapsible
complex K and a simplicial map f: K-» L such that the R-sliceable
relation induced on K has a prOper fold factorization. Thus L X I
is collapsible by Theorem 6.11, and, in this case at least, there is
a relation between the various collapsing methods.

Theorem 7.10: If L is a two-complex and L can be separated

by a tree T into two collapsible subcomplexes, L and L2, then L

l
is the simplicial image under a map f of a collapsible complex K

and f: Ki~ L has a factorization as a sequence of (proper) (2,1)

folds.
Proof: Select an extreme vertex v1 of the tree and define K
to be the complex formed by identifying L1 with L2 at this extreme

vertex. Certainly K is collapsible as each "half" can be collapsed

to the identified vertex v1. f: Kt» L is merely the identification

on the two copies of the tree that exist in both "halfs". The relation

51

is R-sliceable since we can embed all the vertices of one half at level

a and the remaining vertices in the second half (excluding the identified
extreme vertex) at level 3/4. The fact that we can realize the
identification through a sequence of folds follows from the construction
below.

Order the extreme vertices {v1};l of the tree with that vertex F 1
selected as the identifying point of the two "halfs" as the first
vertex v1 in this ordered set. Now select the unique path in the
tree that runs from the first vertex to the second vertex, the path

th
that runs from the first to the third vertex, the first to the i e

 

vertex, etc. Each of these paths is homeomorphic to a one-cell, and
any two paths intersect in a subcomplex of the tree that is also
homeomorphic to a one-cell and this subcomplex contains the first ver-
tex. These facts follow from the properties of trees.

Order the set of one-simplexes s of the tree T as follows.

81,82,83,...,s1 is the path from v1 to v2. Now 31+1 is the first

one-simplex in the path from v1 to v3 that has not already been

ordered. are the remaining simplexes in the path to v3.

814.23.00.81

1+1 is the first simplex in the path from v1 to v4 that has not

been ordered. A simple inductive argument orders the oneésimplexes of

the tree.
Now L1 contains a copy of the tree and so does L2. For no-
tation, 81 will denote an ordered simplex of the tree in L1, and s;

will denote a copy of the same simplex in L In this notation the

2.
first complex of the factorization, K1, has been formed by identifying

Inductively form K from K by

I l
of s with v of 8 1+1 1

v1 1 1 1'

folding 31 onto 8;. It is obvious that this construction gives a

52

fold factorization of K onto L.

We conclude by asking if, given a non-collapsible two-complex
L such that L x I is collapsible, the collapsing of the product space
could always be done through the techniques of this section, say by an
application of 7.9. A partial answer is obtained in the following
theorem, though, in general, the question is unanswered. We include
the theorem, however, because it leads into section VIII.

Theorem 7.11: The dunce hat is not separated by a (non-trivial)

 

tree.

2522:: Suppose the dunce hat D = KDJKZ’ where KanZ = T, a
tree. We consider the extreme points of T and note that for T to
cut D "in two", T must cut the neighborhoods in D of any point of
T "in two". In particular, consider the neighborhoods in D of the
extreme points of T. By the structure of D we note that the neigh-
borhoods of vertices v of D are of three possible forms. They are
either homeomorphic to a two-cell, a three book, or to a pair of cones
joined by a two cell. (See Figure 11.) Only one vertex of D can
have a neighborhood of the third type. I

Now since a neighborhood of an extreme vertex v of T must
be separated by a one simplex which terminates at v, and since there
is only one such neighborhood in D, namely the third one discussed
above, we have contradicted our hypothesis, for a non-trivial tree must
have at least two-extreme vertices.

This theorem shows that we cannot collapse D X I by using 7.9

with two collapsible subsets, even though we know D X I is collapsible.

53

 

 

 

0)

 

 

Figure 11

SECTION VIII

UNFOLDABLE STARS

In the proof of Theorem 7.11, we examined neighborhoods of
vertices belonging to the dunce hat. One of these neighborhoods had
the property that it could be separated by a single one-simplex. This
phenomenon and its relationship to the collapsing of product spaces is
examined in our concluding section. We will restrict ourselves to
two-complexes and show that the dunce hat belongsto a class of com-
plexes such that for any complex L in this class, L X I is
collapsible.

Let K be a simplicial complex containing a vertex v.

Definition 8.1: st(v) is said to be a proper star if
Ist(v)-v‘ is connected. A complex is prOperly connected if each
vertex has a properly connected star.

Definition 8.2: A properly connected star, st(v) is said to
be m-unfoldable if [st(v) - 3|, 3 E st(v) is not connected for 3
some m-simplex of K.

Theorem 8.3: Let K and L be properly connected 2-comp1exes

 

and suppose f: K-» L is a simplicial (2,1) fold. Then L has a 1-

unfoldable star.

Proof: Let the fold f be of 31 = <w,v1> onto 32 = <w,v2>.
That is f identifies s1 with 32. We know that ‘st(v1)| n
|st(v2)‘ = w, so let v = v1 = v2 in L and consider st(v)<: L.

54

55

We claim that |st(v) - <w,v>| is not connected. Since
lsth)‘ é ‘st(v1)l Uf ‘st(v2)|, it is immediate that |st(v1)‘ Uf
‘st(v2)‘ - ‘<w,v>‘ is not connected.

Lemma 8.4: If L is a two-complex, L can be subdivided such
that each two-simplex in the subdivision has the property that two of
its one-faces each lie in at most only one other two-simplex.

‘ggggg: A barycentric subdivision of L will have this pro-
perty. Note that if L is properly connected, so is L'.

We will next consider the class of pr0perly connected two
complexes with the prOperty that no one-simplex lies in more than 3
two—simplexes. From the above lemma, we can also assume that each two
simplex in such a complex has at most one l-face lying in more than
two 2-simp1exes. For notation let us call 2-complexes of this type
3-1 complexes.

Theorem 8.5: If L is a 3-1 complex with st(v)<: L, 1-un-

 

foldable, then there exists a 3-1 complex K and a simplicial map
f: K-4 L such that f is a (2,1) fold and K has at least one two-

simplex with a free edge.

Proof: For notation let |L - st(v)‘ = Z and Ist(v) - <w,v>| = Y.

As L is a prOper 3-1 complex we have at most 3 disconnected subsets
in Y.
Case 1: Suppose there are only two disjoint subsets in Y,

say x1 and X2, and [<w,v> is the face of only two 2-simplexes in

L. Then SI niz =<w,v>. Select two disjoint copies of i and It.

1 2’

1
_ > be the copy of <w,v> lying in W

say W1 and W2. Let <w1,v1

and likewise for <w2,v2> in W

1’

2. J01n W1 to W2 by identifying

the point w1 with w2. We can now form the complex K by attaching

the Space we have just formed to a c0py of Z along |lk(v)‘ in the

56

natural way since Wi U W now contains a copy of ‘lk(v)‘. By this

2

construction, f: K-» L is a (2,1) fold formed by identifying <w,v1>
with <w,v2>, and K has a two simplex with a free edge.

Case 2: Suppose Y contains two subsets X1 and X2 as before
but that <w,v> is a face of three Z-simplexes in L. Then two of
the 2-simplexes must lie in one of the pieces, say K1. Then Ké con-
tains only one two-simplex with the edge <w,v>. Proceed as in Case

1 to construct K.

Case 3: If Y contains three subsets X1, X , and X , then

2 3
<w,v> is a face of 3 two-simplexes, and one lies in K; for each i.

As Ki 0 Ki 0 i8 = <w,v>, let WI be a copy of K. U K. and W. be

1 l 2 2
a copy of i8° The construction of Case 1 again gives the complex K.

For our final results we note that a triangulated 2-cell L
has the prOperty that starting from any one-simplex in the boundary of
the cell, L can be collapsed onto its boundary less the starting
simplex.

We will let X denote the class of homologically trivial 3-1
complexes formed from triangulated 2-cells by l-cell identifications on
the boundary. For example, the dunce hat can be formed in this way by
identifying the edges of a triangle.

1 Figure 12 shows how some elements of X can be constructed.
Observe that starting from a two-cell and folding together two one-
simplexes on its boundary, we arrive at the situation pictured in part
a. Thus we can get any number of "diagonal-type" one-cells by folding

on the boundary of the starting two-cell. The remaining illustrations

in Figure 12 assume the diagonal cells have already been formed.

 

 

 

 

57

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

J V

v0 0

v
b) 4 v3
R\
\ \
V4
1 v
/ 1 a
/
l V
v2
\ \

/ ..
V1 v2

v

0) v2
~
\
\
V3
R v \
&
v0 v1

Figure 12

58

Theorem 8.6: If L C X and L has a l-nnfoldablo star, then

 

L X I is collapsible.

.Egggg: By 8.5 we can find a complex K with a free edge which
maps onto L by a fold. As K is also in X it can be collapsed
onto a one-complex, namely that one-complex formed from the boundary
identification of the original two-cell. H(K) = 0 implies that the
one-complex is a tree T and hence is collapsible. Thus
L X I\8K\\K\T\\O, using 4.5 and 6.11 for the first two collaps-
ings.

Corollary 8.7: If K1 is a two-cell and {Ki’fi}I is a

 

simplicial fold factorization forming a 3-1 complex L = Kn such that
the closed seam of K1 lies in its boundary, then L X I is collaps-
ible.

2523;: H(Kn) = O by 6.17 and K.n has an unfoldable star

left by the last fold in the factorization.

 

1.

BIBLIOGRAPHY

E. G. Begle, The Vietoris mapping theorem for bicompact spaces,
Annals of Mathematics, 51 (1950), 538.

P. F. Dierker, Strongly pointlike maps and collapsible complexes,
(unpublished Ph.D. dissertation, Department of Mathematics,
Michigan State University, 1966).

J. H. C. Whitehead, Simplicial Spaces, nuclei and m-groups,
Proc. London Math. Soc., 45 (1939), 243-327.

E. C. Zeeman, On the dunce hat, Topology, 2 (1963), 341-358.

E. C. Zeeman, Seminar on combinatorial t0pology (mimeographed
notes), Institut des Hautes Etudes Scientifiques, Paris, 1963.

59

 

WW".

3

”mm“:

234