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Thesis for the Degree of Ph. D‘
MICHQAN STATE UNIVERSITY
J. SCOTT DOWNENG
1959

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thesis entitled

"Manifolds Which are Homology Doubles"
presented by

J. Scott Downing

has been accepted towards fulfillment
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ABSTRACT

MANIFOLDS WHICH ARE HOMOLOGY DOUBLES

By

J. Scott Downing

The basic question considered is under what conditions a
manifold contains a subset which is in some sense equivalent to its
complement. A closed (compact, connected, and without boundary)
n-manifold M is a double if it is the adjunction space N U N

l h 2’

where N1 and N2 are two copies of the same manifold with
boundary and h is the identity between their boundaries. Gen-
eralizing this concept, M is a t-double (for twisted) if M con-
tains a submanifold N which is homeomorphic to the closure of its
complement. If R is a principal ideal domain, M is called an
R-homology double, or simply an R-double, if there exists a compact
subset A in M for which there is an isomorphism
H*QA;R) ; H*(M2A;R) for singular homology. If A can be taken as
a PL Subspace of the combinatorial manifold M, then M is a
PL R-double. It is shown that if M is a PL R-double then it
contains 3 PL submanifold N of dimension n satisfying
H*(N;R) E H*(M:N,R). The subset A or N above is called a
t—half, PL R-half, etc., for M.

In chapters 11 and III necessary and sufficient conditions
for a manifold to be a generalized double are studied. It is seen

that a closed manifold is a t-double or a PL R-double only if its

Euler characteristic is even. Conversely, a closed 2-manifold is a

 

J. Scott Downing

double if its Euler characteristic is even, and every 3-manifold is
a PL t-double. In higher dimensions, every closed, combinatorial
manifold of odd dimension is shown to be a PL R-double for any
principal ideal domain R. For a combinatorial manifold M of
even dimension a similar but weaker result is proved. If R is

a field, this states that if M is orientable over R and the
Euler characteristic of M is even, then M is a PL R-double.
Two miscellaneous results on homology doubles are that the product
of a manifold with an R-double is an R-double, and that the con-
nected sum of two PL R-doubles is a PL R-double.

There are numerous examples showing that a closed n-manifold
may have many different R-halves. However, a PL R-half is shown
to be homologically unique if its homology modules are those of a
Space of low dimension (i.e. < [n/2]), and in this case the modules
of the half are determined by those of M.

In the last chapter compact, combinatorial manifolds with
boundary are considered. Such a manifold M is 3 PL t-double
if it contains a PL submanifold N such that N 5 M:N, or a PL
R-double if N satisfies H*(N;R) ; H*(M:N;R). Every compact 2-
manifold with boundary is seen to be a PL t-double where N can
be taken as a disc, and it is proved that every compact 3-manifold
with boundary is a PL t-double. For higher dimensions, certain
compact, combinatorial manifolds (in odd dimensions, all those whose
boundary components are spheres; in even dimensions a subset of

these) are shown to be PL R-doubles.

MANIFOLDS WHICH ARE HOMOLOGY DOUBLES

BY

JSISEott Downing

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1969

€3’7QCI
?-3--4‘7

ACKNOWLEDGEMENT
I would like to thank Professor Patrick Doyle, my thesis

advisor, for suggesting the topic of investigation in this

dissertation and for many helpful conversations.

ii

TABLE OF CONTENTS

Chapter Page
I. INTRODUCTION AND PRELIMINARIES ........................ 1
II. DEFINITIONS AND GENERAL RESULTS ....................... 9
III. CLOSED, COMBINATORIAL n-MANIFOLDS, n 2 3 .............. 19
IV. COMPACT, COMBINATORIAL MANIFOLDS WITH BOUNDARY ........ 36
BIBLIOGRAPHY .......................................... 46

iii

11

LIST OF FIGURES

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OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

iv

CHAPTER I
INTRODUCTION AND PRELIMINARIES

An important method of constructing examples of manifolds
has been to paste two homeomorphic manifolds together by a homeo-
morphism between their boundaries. Manifolds with boundary can be
obtained by identifying certain subsets of the boundary of one
manifold with homeomorphic subsets in the boundary of another. In
this thesis I consider the converse idea. More specifically, the
question is asked: Under what conditions does a manifold decompose
into two pieces which are homeomorphic, or which at least have
isomorphic homology groups?

The basic definitions are introduced in chapter II, and a
model for the type of theorem I am concerned with is proved:
roughly, that a closed, connected, 2-manifold separates into equi-
valent pieces if and only if its Euler characteristic is even.
Chapter III deals primarily with the same type of result for closed,
combinatorial manifolds of higher dimensions, and in chapter IV
compact manifolds with boundary are investigated. The remainder
of this chapter is devoted to basic notation and known facts which
I will be using.

An n-manifold M is a paracompact Hausdorff space with the
property that each point has a neighborhood which is homeomorphic
to Euclidean n-space En or to %En = {(x1,...,xn) E En: xn 2 0}.
Also, all manifolds in this paper will be connected unless otherwise
indicated. The set of points which have En neighborhoods is called

1

the interior of M, intM, and the set M-intM is the boundary, bdM.
M is with or without boundary depending on whether bdM is not
empty or empty. A closed manifold is one that is connected, compact,
and without boundary.

If a manifold N is a subSpace of a manifold M we avoid
any confusion between the interior and boundary of N as a manifold
and as a subspace by letting intMN and bdMN represent the latter
ideas. If A C M then A. denotes the closure of A in M.

Most of my results are for combinatorial manifolds, the main
reference here being the mimeographed notes of E.C. Zeeman [13]. In
the following paragraphs some basic definitions and facts from
combinatorial topology are summarized.

An n-simplex A is the convex hull of n+1 linearly in-

dependent points a .,an, called vertices, in some Ep. It will

0".
be convenient to denote the set of vertices by a Greek letter, say
a, and we write A = Iol for A spans a. A simplex B spanning

a subset of a is called a face of A, written B < A. The geo-

metric center

b(A) = (a +...+an)/(n + l)

O

is called the barycenter of A. If the vertices of two simplexes

 

A and B are linearly independent in Ep we call the simplex
spanning them the join A*B.

A simplicial complex K is a collection of simplexes

 

satisfying:
a. If A E K then all faces of A are in K.

b. If A,B E K then A n B is a common face or empty.

The dimension of a complex is that of its largest simplex if this

exists, and the r-skeleton Kr of a complex K is the subcomplex

 

consisting of all simplexes of dimension 5 r. If A E K is an
arbitrary simplex then the subcomplex stQA,K) = {B E K : A < B}
is called the star of A in K.

A topological space X is called a polyhedron if it is

 

homeomorphic to a subspace of some Ep which is the union of a
collection of simplexes forming a simplicial complex K. For con-
venience it is assumed that X is actually the subspace of Ep,

and we say that X is triangulated by K, written X = IKI. A

 

simplicial complex L is called a subdivision of the triangulation

 

K if ILI = IKI and each simplex of L is contained in some
simplex of K. For the definition of two special types of sub-

division, the r-th derived and the barycentric, the reader is re-

 

 

ferred to [13, chap. I, p. 4]. I note here only that the vertices
of each simplex of the barycentric first derived subdivision of K
are barycenters b(Al),...,b(An) of simplexes of K which can be
ordered so that A < A <...<An.

l 2
If X is a polyhedron, a subSpace Y is a PL subspace (for

 

piecewise linear) if Y is triangulated by a sub-complex of some
triangulation of X. A continuous map f : X 4 Y from one poly-
hedron to another is called a PL map if there exist triangulations
K and L of X and Y respectively with respect to which f is
simplicial; that is, f maps vertices to vertices and simplexes

linearly to simplexes.

A PL_n-ball is a polyhedron which is PL homeomorphic to an
n-simplex, and a PL n-sphere is a polyhedron PL homeomorphic to the
boundary of an (n+l)-simp1ex. A manifold is said to be a combina-

torial n-manifold if it is a polyhedron IKI with the property

 

that Ist(v,K)| is an n-ball for each vertex v E K. If X<I M
is a compact PL subspace of a combinatorial n-manifold, then a

regular neighborhood of X in M is a neighborhood of X which

 

is a compact n-manifold (PL subspace) and collapses to X (see
[13, chap. 111]). Some important facts about a combinatorial n-

manifold M = IKI which I will need are:

1.1 The star Ist(A,K)I is an n-ball for each simplex A in K.

[13, III, p. 2].

1.2 Any derived neighborhood [13, III, p. 14] of a compact PL sub-
Space XC M is regular, and in particular every such subspace has

a regular neighborhood.

1.3 Because a collapse preserves homotopy type, a regular neighbor-

hood of X in M has the same homotopy type as X.

Standard references for algebraic concepts I will be using
are [12] or [3]. Some of those ideas used in this paper are stated
next.

Throughout this thesis R will represent a principal ideal
domain, and the symbols 2 and Zn are reserved for the integers
and the integers modulo n respectively. If X is a topological
space then H1(X;R) and Hi(X;R) represent the i-th singular

homology and cohomology modules respectively of X with coefficients

5

*
in R, H*(X;R) and H (X;R) representing the corresponding graded

R-modules. We write H*(X;R) E H*(Y;R) to indicate that there is

~

an R-isomorphism Hi(X;R) Hi(Y;R) for each integer i, and
similarly for the graded cohomology modules. Similar notation is
used for the relative homology and cohomology modules H*(X,Y;R)
and H*(X,Y;R) of a topological pair (X,Y). It will often be
convenient to abbreviate H*(X;R) by H*(X), and the coefficient
module will then be clarified in the text.

If Hi(X;R) is a finitely generated R-module, then it is
known that Hi(X;R) g Fi(X;R) e Ti(X;R), where F1 and Ti re-
present the free and torsion submodules and C> indicates the

direct sum over R. From the Universal Coefficient Theorem for

cohomology we have [3, p. 136]:

1.4 If Hi(X;R) is a finitely generated R-module for each i,

then H1(X;R) E Fi(X;R) e Ti_1(X;R).

If K is a simplicial complex there are simplicial homology
modules Hi(K;R) defined, and H1(K;R) ; H1(Kr;R) if i < r. Since
the simplicial homology is isomorphic to the singular homology of
the polyhedron X = IKI [12, p. 191], the notation H*(X;R) = H*(K;R)
should cause no confusion.

Let M be a compact, connected, n-manifold where bdM may or
maynot be empty. Then Hn(M,bdM;R) is either R or O [12, p. 302]

and we say that M is orientable over R if the former holds, other-

 

wise that M is non-orientable over R. An orientation of M is a
choice of generator in Hn(M,bdM;R) and a homeomorphism h between

two oriented manifolds M and N is said to be orientation preserving

if the induced isomorphism h* : Hn(M,bdM;R) ~ Hn(N,bdN;R) sends the
chosen generator to the chosen generator. For orientable manifolds

some important duality theorems hold:

1.5 (See [12, p. 296]). If M is a closed, combinatorial n-manifold
which is orientable over R and if N is a closed, compact, PL sub-
manifold of dimension n, then there are isomorphisms:

a. Hi(M,N;R) g Hn-1(M2N;R),

Hn'1(M,MeN;R).

"2

b. Hi(N;R)

Note that here M-N is also an n-manifold [13, chap. III, p. 20].

1.6 (Poincare Duality [12, p. 297]). If M is a closed n-manifold
orientable over R, then for each i there is an isomorphism

~ n-i
Hi(M;R) = H (M;R).

1.7 (Lefschetz Duality [3, p. 186]). Let M be a compact n-manifold
with boundary which is orientable over R. Then there is a sign-

commutative diagram of R-modules (all coefficients in R)

.. ~ Hq‘1(M) a Hq’1(bdM) a Hq(M,bdM) e Hq(M) a ...
I I 1 I

a 4 ~ bd a ...
H (M,bdM) Hn_q(bdM) Hn_q(M) Hn_q(M, M)

noq+l

where the rows are the exact Mayer-Victoria sequences and the

vertical arrows are isomorphisms induced by the cap product.

Another useful result, based on the idea of relative homeo-

morphism (see [12, p. 202]» is:

1.8 If M is a closed, combinatorial n-manifold, N1<Z M is a

compact PL submanifold of dimension n, and N2 = M-N1 is the

complementary manifold, then for any R
H*(M,N1;R) = H*(N2,bdN2;R).

If A is a finitely generated R-module, then rankA denotes
the number of elements in a basis for its free submodule. The follow-

ing lemma [3, p. 100] will be needed.

1.9 Given an exact sequence of finitely generated R-modules

0-+A1-+A2-+...-+Ar-+O,

then

rankA1 - rankA2 + + (-1)r+1rankAr = 0.

For any space X such that H*(X;R) is a finitely generated

graded R-module, the Euler characteristic of X is defined as the

 

finite sum
’ i
1(X) = Z (-l) rankHi(X;R).
i
Although the rank of the i-th homology module may depend on the co-
efficients R, it follows from the Universal Coefficients Theorem
for homology that I(X) is independent of R [3, p. 103]. Some

useful results involving the Euler characteristic are listed:

1.10 Let M be a closed, combinatorial n-manifold and N a

compact, n-dimensional PL submanifold. Then
I(M) = I(N) + I(M-N) - I(bdN).

Proof This follows by applying 1.9 above to the Mayer-Vietoris

sequence of the couple {N,MPN}.

1.11 The Euler characteristic of a closed, odd dimensional manifold

is 0.

Proof [11, p. 246] or: since any manifold is orientable over 22,

apply Poincare duality (1.6) and 1.4 obtaining Hi(M;Zz) 3 Hn-i(M;22)°
I conclude this chapter with the statement of the important

Newman-Gugenheim homogeneity theorem [4, p. 32].

1.12 Let B and B be two n-balls which are PL subspaces of

l 2
the closed, combinatorial n-manifold M which is oriented over Z,
and let P<Z M-(B1 U B2) be a compact polyhedron which does not
disconnect M. Let h : B1 d B2 be an orientation preserving
homeomorphism, where the orientation on the B1 is induced by that
of M. Then there is an orientation preserving PL homeomorphism

H of M onto itself which is PL isotopic to the identity and such

that H B1 = h and H P is the identity.

CHAPTER II
DEFINITIONS AND GENERAL RESULTS

In the literature (see, for example, [10, pp. 4, 56]) a closed
manifold M is a double if M = N1 Uh N2, the adjunction space of
two copies of the same manifold with boundary, h being the identity
map on their boundaries. I will call each Ni a half of the mani-
fold M.

In this chapter the weaker concepts of a twisted double and

a homology double are defined. The relationship between these ideas

is discussed and some general results and examples are presented.

Definition 2.1 Let M be a closed n-manifold and R a principal
ideal domain. M is said to be an R-homology double, or simple an
R-double, if it contains a compact subset A such that the follow-

ing isomorphism for singular homology holds:
Huh (ARR) E H*(M‘ASR) -

The set A will be called an R-half of M.

In the special case that M is a Z-double (and hence, by the
Universal Coefficient Theorem for homology, an R-double for any R)

I may also refer to M as a homology double.

Definition 2.2 The combinatorial manifold M is called a PL R-double

 

 

if there exists an A as above which is a PL subspace of M.

10

The following facts result immediately from the Universal Co-
efficients Theorem for cohomology [12, p. 243] and from the corres-

ponding theorem which expresses homology in terms of cohomology

[12, p. 248].

2.3 If A is an R-half for the manifold M, then
* ~ *
H (A;R) = H (M-A;R).

2.4 If M is a closed n-manifold, A is a compact subset such that
* N * *
H 0A;R) = H (M:A;R), and H QAgR) is a finitely generated R-module,

then M is an R-double with half A.

In the case of a PL R-double more can be said about a half.

Proposition 2.5 The closed, combinatorial n-manifold M is a PL

 

R-double if and only if M has an R-half which is an n-dimensional

PL submanifold.

Proof If the condition holds then M is a PL R-double by definition.
If M has an R-half A which is a PL subSpace, let N be a regular
neighborhood of A in M. Then N has the same homotopy type as

A (1.3) and similarly for M-N and M-A. Thus H*(N;R) ;H*(M-N;R).

Corollary 2.6 A closed, combinatorial n-manifold M is a PL R-double

 

if and only if M contains an n-dimensional PL submanifold N with

~

H*(N;R) lad-FER)-

Proof If N is as in the theorem, then M-N is an n-manifold with
interior M-N. Because the boundary of a manifold is collared, MrN

and MrN have the same homotopy type.

11

If a closed manifold M contains a submanifold N which is
homeomorphic to MrN, then N is a homology half for M. in a
stronger sense than that of definition 2.1. This motivates the

following:

Definition 2.7 A closed manifold M. will be called a-t-double (for
twisted) with t-half N if it contains a submanifold N such that
N and M:N' are homeomorphic. If N can be taken as a PL sub-
manifold of the combinatorial manifold M then M will be called

a PL t-double.

 

Considerable study has been made of t-doubles, especially in
the case of 3-manifolds. Heegard splittings [11, p. 220], [5, pp.
40, 83] and toroidal manifolds [1, p. 99] are examples. It is clear
that every double is a t-double, but the toroidal manifolds, for
example, show that a t-half of a manifold may not be a half in the
stronger sense. The key here is the way in which the t-halves are
pasted together on their common boundary. The toroidal manifolds
also show that a given manifold with boundary may be a t-half for
different closed manifolds, whereas the double of a manifold with
boundary is topologically unique. That the concept of a t-double

is stronger than that of a homology double, even in the PL case,

is shown by the following:

Example 2.8 Let M be a closed, combinatorial Poincare space; that

 

is, M is a closed 3-manifold which has the homology groups (co-
efficients in Z) of a 3-sphere but which is not a sphere (See, for

example, [11, p. 226]). Let A<: M be a 3-ba11 (PL subspace) and

12

set B = MsA. B cannot be a 3-ba11 for then M would be a sphere,
and thus A is not a t-half of M.

0n the other hand, from 1.5b and because the reduced homology

H*(A) = 0.
111(3) E H3-i(M,A) E fi3’i(M)

for each i. Thus H*(A) ; H*(B) and M is a Z-double with z-half A.

The following theorem gives a necessary condition for manifolds

to be certain types of doubles.

Theorem 2.9 If the closed n-manifold M is a t-double or a PL R-

 

double for some R, then the Euler characteristic of M is even.

nggf Since every odd dimensional manifold has Euler characteristic
0 (1.11), we consider only the case that n is even. By hypothesis
there is some principal ideal domain R and some submanifold A<Z M
with H*(A;R) E H*(B;R), where B = EEK, and A n B = bdA. Applying
1.9 to the Mayer-Vietoris sequence of the couple {A,B} (coeffi-

cients in R)

. -+ Hi+1(M)-° Hi(bdA) -+ Hi(A) @ Hi(B) -* Hi(M) -' ...,

it follows that
I(M) - (I(A) +-I(B)) + I(bdA) = 0. (1)

Since bdA is a closed, odd-dimensional manifold and 1(A) = 1(3),

we have I(M) = 21(A).

Corollary 2.10 The boundary of a compact n-manifold has even Euler

 

characteristic.

13

Proof We need only consider the case that A is an odd-dimensional
manifold with boundary. Letting M be the double of A, formula

(1) still holds but now I(M) = 0 and I(bdA) = 21(A).

From theorem 2.8 we obtain manifolds which cannot be homology

doubles.

Example 2.11 a. The homology groups of real projective n-space Pn
are well known (See, for example, [6, p. 137]). In particular, if
n is even I(Pn) = 1. Thus in every even dimension we have a non-
orientable, closed, combinatorial manifold which cannot be a PL
homology double.

b. If CPn is the complex projective space of dimension 2n,
then it is known [6, p. 143] that I(CPn) = n + 1. Thus if n is
even, CPn is an orientable 2n-manifold which cannot be a PL homology

double.

Much of chapter III will be devoted to the question whether
the converse of theorem 2.9 is true. For closed 2-manifolds the

answer is easy to obtain.

Theorem 2.12 A closed 2-manifold is a double if and only if its

 

Euler characteristic is even.

Proof If M is a closed 2-manifold whose Euler characteristic I
is even, then by the classification theorem [7, p. 33] M is the
connected sum of g = (2-1)/2 tori in the orientable case or

g = Z-I projective planes if M is non-orientable (g is the genus

of the surface).
If M is orientable then it is the (O 0
double of a disc with g holes punched out

(Figure 1). Figure l

 

14

If M is non-orientable there is a circle which separates M
into halves each of which is the connected sum of g/2 projective

planes with one hole. For example, if g = 4,

 

 

 

 

 

a
has half aII e Ib
//
b‘

Figure 2

The decompositions above are not unique. For example, an
orientable surface S of genus 2m contains a circle which
separates S into two halves each of which is an orientable sur-

face having genus m and one boundary circle (Figure 3).

Figure 3

In chapter III all the R-halves obtained will be manifolds
with connected boundary. Thus it is interesting to ask whether every
closed 2-manifold with even Euler characteristic has a PL R-half
which is a 2-manifold with connected boundary.

Suppose A is a compact, orientable (over 2) 2-manifold
with one boundary circle. It follows from [7, chap. 1, section 10]

that H1(A;Z) is a free abelian group of rank 2g, where g is the

15

genus of A. Hence for any R, rankHICA;R) = 2g. Now suppose such
an A is a PL R-half for the closed, orientable 2-manifold M.

IM:A is also an orientable 2-manifold with one boundary component
so its genus is also g, and it follows that M is a surface of
genus 2g. Thus although every closed, orientable surface has even
Euler characteristic, for those with odd genus the answer to the
above question is no. The best we can do in this case is to find a

half with two boundary components, as indicated in Figure 4.

 

Figure 4

A compact, orientable 2-manifold is characterized by its
homology groups and number of boundary components. QAgain see [7,
chap. 1].) Applying proposition 2.5 to the orientable case, theorem
2.12 is equivalent to the statement: A closed, orientable 2-mani-
fold M has a PL Z-half (not necessarily a manifold) if and only if
I(M) is even. The following example of P.H. Doyle shows the impor-

tance of choosing A to be a PL subspace here.

Example 2.13 Every closed 2-manifold has a decomposition as a dis-

 

joint union M = E2 U P, where P is the one-point union of p

circles [2, p. 74]. If we replace P by P', the one-point union
~ 2

of p closed sin(l/x) curves, it is still true that M-P' = E .

~ ~ 2
Since we are dealing with singular homology, H*(P';Z) = 0 = H*(E :2),

15

genus of A. Hence for any R, rankHIQA;R) = 2g. Now suppose such
an A is a PL R-half for the closed, orientable 2-manifold M.

REA, is also an orientable 2-manifold with one boundary component
so its genus is also g, and it follows that M is a surface of
genus 2g. Thus although every closed, orientable surface has even
Euler characteristic, for those with odd genus the answer to the
above question is no. The best we can do in this case is to find a

half with two boundary components, as indicated in Figure 4.

 

Figure 4

A compact, orientable Z-manifold is characterized by its
homology groups and number of boundary components. QAgain see [7,
chap. 1].) Applying proposition 2.5 to the orientable case, theorem
2.12 is equivalent to the statement: A closed, orientable 2-mani-
fold M has a PL Z-half (not necessarily a manifold) if and only if
I(M) is even. The following example of P.H. Doyle shows the impor-

tance of choosing A to be a PL subspace here.

Example 2.13 Every closed 2-manifold has a decomposition as a dis-

 

joint union M = E2 U P, where P is the one-point union of p
circles [2, p. 74]. If we replace P by P', the one-point union
2

of p closed sin(l/x) curves, it is still true that M—P' E E .

~ ~ 2
Since we are dealing with singular homology, H*(P';Z) = 0 = H*(E :Z),

l6

and P' is a homology half for M. Any closed 2-manifold has such

a half, regardless of its Euler characteristic.
I conclude this chapter with two miscellaneous results.

Theorem 2.14 If M and N are closed manifolds and one of them

 

is a t-double (resp. R-double), then the manifold M X N is a t-

double (R-double).

g£22£ a. If A is a t-half of M, then A X N is a t-half of
MXN.

b. If M is an R-double, then there is a compact A<Z M
such that H*(A) E H*(M2A). (All coefficients are in R.) Then in
M X N, A X N is an R-half as the following application of the

KUnneth formula [3, p. 198] shows:

Hk(A X N) .

1.

m
u C)x‘

k
(Hi(A)® Hk-i am e is: (Hi(A>*Hk_i_1(N)>

O O

k
O‘Hi(M'A>®“k-i(N)) e i€O(Hi(M-A)*Hk-i-1(N))

m
Ch:

0

1

"2

Hk((M-A) X N).

In the following discussion all maps and subspaces are PL.

The connected sum M1#‘M2 of two closed, combinatorial,

 

oriented (over Z) n-manifolds is obtained by removing the interior
of an n-ball from each and then matching the resulting boundaries by
an orientation reversing homeomorphism (See [8, p. 1]). This
operation is independent of the choice of balls by the homogeneity
theorem (1.12) and is uniquely determined up to orientation pre-

serving PL homeomorphism.

17

A related idea for manifolds with boundary is that of the disc
sum. The disc sum N1 A N2 of two combinatorial, oriented, n-mani-
folds with connected boundary is obtained by taking an (n-1)-ba11 in
the boundary of each and matching them by an orientation reversing
homeomorphism. Because the boundary of a combinatorial manifold has
a PL collar [13, chap. V, p. 13], the homogeneity theorem shows that
the disc sum is also independent of the balls chosen. To see this,
let B and B be two balls in bdN and H : bdN X I d bdN X I

l 2

be the isotopy such that H1 is the identity on bdN and
HO(Bl) = B2. Since bdN X I is (homeomorphic to) the collar, H

extends by the identity to an orientation preserving homeomorphism

on all of M.

Theorem 2.15 The connected sum M1#M2 of two PL t-doubles (resp.

 

PL R-doubles) is a PL t-double (PL R-double).

Proof a. For i = 1,2, let Ni be aPL t-half of M and triangulate

i
M. by K1 so that Ni is triangulated by a subcomplex. Since in

1
forming the connected sum arbitrary balls can be chosen, choose an

(n-l)-simplex o. in bdN. and let B, = |st(a.,x.)| be a ball
1 l 1 1 1.

(1.1) in Mi' Note that

 

18

Now choose an orientation reversing homeomorphism h : de1 r de2

satisfying
h(de1 0 N1) = de2 n N2

and M1#M2 = (M1 - lntBl) Uh (M - intBZ).

2

This can always be done by first choosing an orientation reversing
homeomorphism between the (n-2)-spheres bdNi n dei and then extend-

ing to a homeomorphism between the (n-1)-balls dei n Ni and between

\

de. n M-N.. Then
1 1

(N1 - lntBl) Uh (N2 - inth) = N1 A N2

0")

- 1ntB2) = Ml-Nl A M -N .

and ((M1 - N 2 2

l) - intBl) Uh ((M2 - N

2)

Since the disc sum is well defined (Although bdNi = bd(M-Ni)

may not be connected, this is no problem since the connecting discs

are from the same boundary component.), these two spaces are homeo-

morphic and (NI-intBl) Uh (NZ-inth) is a PL t-half for M1#M2.
b. Suppose for i = 1,2, Ni is a submanifold of M1 which

is a PL R-half. Proceeding exactly as in case a, we obtain (*).

From the reduced Mayer-Victoria sequence of the couple [N1,N2] in

N1 A N2 it follows that (with coefficients in R)
H*(N1 A N2) 3 H*(N1) e H*(N2).
Similarly
H*(-M_-N1 A F132) 2 Hui/1‘31) o 11*6131'2).
Thus (N1 - intBl) Uh (N2 - inth) is an R-half for M1#M2.

CHAPTER III
CLOSED, COMBINATORIAL n-MANIFOLDS, n 2 3.

In this chapter all manifolds will be closed, combinatorial
n-manifolds, where n 2 3, and all maps and subspaces will be PL.
In considering the converse of theorem 2.9, we will show that for
odd dimensions every such manifold is a PL homology double. For
even dimensional manifolds with even Euler characteristic a slightly
weaker result is obtained. The last theorems concern the uniqueness
of these double decompositions.

We first develop some notation and lemmas concerning combin-

atorial n-manifolds.

Definition 3.1 Let K be a combinatorial triangulation of a closed

 

manifold M, and let K' be its first barycentric Subdivision. For
each simplex A in K, the dual cell A* is defined as the union
of all simplexes of K' of the form |{b(Aj),...,b(Am)}|, where

A < Aj<...<Am. The collection of cells KI = [A*: A E K] is called

the dual cell decomposition of K.

 

Note that if A is a k-simplex of K then A* is an (n-k)-
dimensional cell which has a simplicial triangulation by a subcomplex
of K'. In this triangulation the (n-k)-simplexes of A* are pre-
cisely those (n-k)-Simplexes of K’ having b(A) as first vertex.
It is known that KI is a cell complex [11, sect. 67] and that if

*r

*
K denotes the r-skeleton of K (which has a simplicial tri-

angulation by a subcomplex of K'), then for each i < r,

19

20
~ *r
H,(M;R) = H.(K ;R).
1 1.

Definition 3.2 Let K be a combinatorial triangulation of the n-

 

manifold M, and let L<I K be a subcomplex. The complementary cell

 

* *
complex L <2 K is defined as
* * *
L = {A e K : A e K - L}.

That is, the cells of L* are the duals of those simplexes of K
which are not in L.

Observe that every vertex of K' is in either |LI or |L*|
For if there is such a vertex b i |LI, then b = b(A) where A E L
and thus b E A*<Z IL*|. Also, the notation L' is appropriate for
the subcomplex of K' which triangulates ILI and similarly for

*I
L (I K'.

Lemma 3.3 Suppose K is a combinatorial triangulation of the mani-
fold M, L<Z K is a proper subcomplex, L* is the complementary cell
complex, and K" is the second barycentric subdivision of K. Let
N = N(|LI,K"), N* = N(IL*|,K") be the derived neighborhoods. Then

* ___.
N = MAN.
3599; (Cf. [13, chap. 111]) First note that if x E M - (ILI U |L*|),
then there is a unique choice of a e ILI, a* e |L*|, and o < 1 < 1
such that x = is + (1-A)a*. (Recall that M can be considered
imbedded is some Euclidean space.) To see this, let A be the unique
simplex of K' such that x E inuA. Each vertex of A is in lLI

or |L*I, but not all the vertices of A are in one of these, since

*
L' and L are full subcomplexes of K' [13, III, p. 13]. Thus

21

* * *
A is the join B*B , where B 6 L', B E L , and x = a + (l-l)a
* *
uniquely for some a E B, a E B .
Now it follows that there is a unique continuous map
f : Mlfi [0,1] which is linear on simplexes of K' and such that
-l -1 * , *
f (O) = ILI, f (1) = IL I. From the definitions of N and N ,
- * - *
N = f 1[0%], N = f 1[%3l]. Clearly then, N U N = M_ and
* -1 1 *
bdN = bdN = f (E) = N O N . This is equivalent to the conclusion

of the lemma.
We can now prove the main result on odd dimensional manifolds.

Theorem 3.4 Let M be a closed, combinatorial manifold of odd

 

dimension 2m + 1 2 3. Then M is a PL homology double.

Proof Let K be a combinatorial triangulation of M, let L = KW

*
be the:m-skeleton,and let L be its complementary cell complex.
* * *
Note that L is the m-skeletonof K . Let N and N be the
*
derived neighborhoods of |L| and IL I respectively in K", as
*
in lemma 3.3. Since N collapses to IL] and N collapses to
*
IL I, it follows from the remarks preceeding definition 3.2 that for

any R,

~ ~ *
Hi(N;R) = Hi(M;R) = Hi(N :R), if i < m,

and (1)
~ ~ *
Hi(N;R) = 0 = Hi(N ;R), if i > m.

Thus to Show that N is a homology half for M it remains only to
check the modules in dimension m.
We first consider the case R = 22. Since all manifolds are

then orientable, there is, by 1.7, a Sign commutative diagram

22
(coefficients in Z2):

..~ Hm(N,bdN) ~ Hm(N) a Hm(bdN) a Hm+1(N,bdN) a Hm+1(N) a...

I I I I I

..—o '0 d -’ "" "’ "'...
Hm+l(N) Hm+l(N’b N) Hm(bdN) Hm(N) Hm(N,bdn)

Ill

Because L is an m-dimensional complex, Hm+1(N) E H (N) 0.

m+l

Also, since all Z -modu1es are free, 1.4 gives Hm(N) E Hm(N),

2
and it follows that Hm(bdN) E Hm(N)<D Hm(N). An identical argument

~

* * *
Hm(N )IO Hm(N ). But bdN = bdN , implying

*
shows that Hm(bdN )

~ *
that Hm(N;Z2) = Hm(N :2 Thus N is a Z -half for M, and in

2)' 2
particular, since the Euler characteristic of M is independent
*
of the coefficient module, I(N) = I(N ).
Now considering coefficients in Z,
2m+l i
I(N) = 8 (-1) rankHi(N;Z).
i=1
*
and similarly for I(N ). From equations (1) above it follows that
* *
ranka(N;Z) = ranka(N :2). But since L and L are m-dimensional

complexes, these two Z-modules are free and hence isomorphic. This

*
completes the proof that H*(N;Z) E H*(N :2).

It should be pointed out that while every closed, combinatorial
manifold of odd dimension 2m + 1 has a PL homology half, the half
N exhibited in the theorem is by no means unique. Even if we in-
sist that the half N satisfy conditions (1) for R = Z, Hm(N;Z)
may be almost any free abelian group. Examples and a discussion of

uniqueness follow later in this chapter (pp. 31-33).

23

For 3-manifolds a result stronger than theorem 3.4 is possible -
namely, that every closed 3-manifold is actually a t-double. Seifert
and Threlfall have this result for orientable 3-mainfolds [11, p.

219], and it is not too difficult to extend it to the non-orientable
case. Since any 3-manifold has a combinatorial triangulation [9],
we will in fact Show that every closed 3-manifold is a PL t-double.

Recall that all maps and subspaces are PL.

Definition 3.5 ([11, p. 219]) Let B be a PL 3-ball, let

 

1,... n 1,. “’Dn be mutually disjoint discs in de, and let
h. : C ~ Di be a PL homeomorphism, i = 1,...,n. Then the quotient

Space H obtained by identifying each x 6 Ci with hi(x) E Di’

i = 1,...,n, is a 3-manifold with boundary called a handle body of

 

If B is given an orientation then this induces an orientation

d . . . .
on each Ci an Di The homeomorphism h is said to be of type 1

i

if it is orientation reversing and of type 2 if it is orientation

preserving. If n1 of the hi are of type 1 and n are of type 2,

2

then H is said to have n1 handles of type 1 and n2 handles of

type 2.

As in the case of the disc sum (p. 17), the homogeneity theorem
shows that the construction of H is independent of the choice of
the discs C1 and Di' 0n the other hand, de is a closed surface

of genus n which is orientable if and only if all the hi are of
type 1. Thus H is an orientable 3-manifold with boundary if and
only if all the hi are of type 1. Hence an orientable handle body

is determined up to PL homeomorphism by its genus, and this is

24

determined if de is known. The following lemma establishes the

same result for a non-orientable handle body.

Lemma 3.6 A non-orientable handle body H of genus n can be rep-

resented as a handle body with n handles of type 2.
Proof Let hi': Ci fl Di’ i = 1,...,n, be the homeomorphisms between

discs in de defining H. Since H is non-orientable, at least

one of the hi’ say h is of type 2. Suppose that k 2 1 of the

1’

hi’ and in particular hz, are of type 1. The lemma will follow in
a finite number of steps if we can Show that H can be represented
as a handle body with k-l handles of type 1.

If B is represented as the suspension of a 2-simp1ex A,
with suspension points p1 and p2, then B is the disc sumlof

= *
the balls B1 p1 A and B2

place on A. If ACIB1 is denoted by

c: .
c0 and A B2 18 denoted by DO’

h : C d D the identity map, then

= p2*A, with the identification taking

.with

B = B1 Uh B . We can assume that

 

c: -
D1,D2 de2 DO,

1 0

 

and that Ci,Dj<: de - C for

i = 1,...,n, j = 3,...,n. (See figure

Figure 6

6)
*
Now let B = B1 Uh B2, which is formed by cutting B through
1

*
A and repasting by h1 : C1 * D1. »We wish to orient B , and since
h1 is orientation preserving we must change the orientation on one

of the pieces, say on B Now for i = O,2,3,...,n, hi induces a

2.

25

*
homeomorphism gi between discs in de . Because the orientation

on B2 is reversed, g0 and g2 are now of type 2. 0n the other

hand, since all the remaining discs are in B is of the same

1’ gi
*
type as hi for i = 3,...,n. Thus H results from the ball B

where the number of identifications of type 1 has been reduced by

one, completing the lemma.

Theorem 3.7 Every closed 3-manifold is a PL t-double.

 

{Eggpf Let K be a combinatorial triangulation of M, and let N
be the second derived neighborhood in M of the l-skeleton of K
as in theorem 3.4. In this case N and N* = M:N are handle

bodies with the same boundary [11, p. 219]. By lemma 3.6 and the

remarks preceeding it, a handle body is determined by its boundary,

*
and it follows that N and N are PL homeomorphic.

For even dimensional manifolds I have not been able to obtain
a result having the full strength of theorem 3.4. In order to obtain

a slightly weaker result the concept of a weak R-double is introduced.

Definition 3.8 A closed, combinatorial n-manifold M will be called

 

a weak R-double if it contains an n-manifold with boundary N (PL

 

subspace) such that for each i, rankHi(N;R) = rankHi(MrN;R).

Note that if R is a field, then a weak R-double is in fact
a PL R-double.

The goal is to Show that if an even dimensional manifold has
even Euler characteristic (Cf. theorem 2.9) and is orientable over R,
then it is a weak R-double. Some notation and lemmas are needed. In

the following discussion all homology modules have coefficients in

26

R, and rank refers to rank as an R-module.
Suppose M is a polyhedron of dimension 2 m, with a cell

1"°°’Ct' Let L0 be the (m-l)-

8k81€tOHof the cell decomposition, and let L1 = Li-l U {Ci}’

i=1,...,t. Thus Lt is them-skeletonof M and ILiI is

decomposition having m-cells C

obtained from ILi_1I by attaching the m-cell Ci via the identity
map on its boundary. The following facts are known (See [3, pp.
89, 101]):

A. Hm(Li) is a free R-module, for i = 0,1,...,t.

 

B. Hj(Li) = Hj(M) for jt< m - l, i = 0,1,...,t.

C. Hj(Li) = 0 for j > m, i = 0,1,...,t.

D. For i = 0,...,tel,_precisely one of the following holds:
1. Hm(Li+l) = Hm(Li) and ranka_1(Li+1) = ranka_1(Li) - l.
2. Hm(Li+l) = Hm(Li)<D R and ranka_1(Li+1) = ranka_1(Li).

Lemma 3.9 Let M be as above. Then the m-cells of the cell de-

composition can be ordered so that the following hold:

E. H (L,) 0 for i = 0,...,r.
m 1

F. Hm(Li) Hm(Li-l) @R for 1 = r+l,...,t.

G. ranka_1(Li) = ranka_1(M) for 1 = r,...,t.

Moreover, having chosen C °’Cr’ the ordering of Cr+ .,C

l"' l’°° t

is arbitrary.
Proof Let S = [01,...,Cr} be a maximal set of m-cells having the
property that Hm(Lr) = 0; that is, for any additional m-cell C,
Hm(Lr U {0}) E R. Order the remaining m-cells arbitrarily.

Now suppose for some i > r, Hm(Li) = Hm(Li-1)° By [3, p. 89],

kernale_1(f) = o, where f : dei 4 IL. is the identity map and

1-l|

27

Hm_1(f) is the induced homeomorphism on homology. But f factors as
f' j
dei ILrI ILi_1I ,

where f' is also the identity and j is inclusion. Since
= ' ' ' = =
Hm_1(f) Hm-1(J)Hm-l(f ), kernale_1(f ) 0. But then Hm(Lr U {Ci}) 0,
contradicting the maximality of 8.
Therefore conditions E and F hold, and condition G holds be-

cause of D above.

Lemma 3.10 Let K be a combinatorial triangulation of the closed
2m-manifold M, 2m 2 4, which is orientable over R. Suppose L<Z K

is a subcomplex of dimension S m satisfying Hm(L;R) = 0. Then
L*
ranka_1( ,R) - ranka_1(M,R).

*
Proof Let N and N be the derived neighborhoods of |L| and
*
IL I respectively in K", as in lemma 3.3. From 1.5 and 1.4,

m

* ~ ~ :1 :
Hm(M,N ) = H (N) = Fm(N) o Tm_1(N) - Fm(L) o Tm_1(L) - T (L).

m-l

*
Similarly, Hm_1(M,N ) = Fm+1(L)<D Tm(L) = 0. Then in the long exact

*
sequence of the pair (M,N ) we have
L H N* M) 0
" Tm-l( ) m-1( ) Hm-l( "

*
It follows that the torsion-free submodules of Hm_1(N ) and

Hm 1(M) are isomorphic, completing the lemma.

Theorem 3.11 Let M be a closed, combinatorial manifold of even

 

dimension 2m and orientable over R. Then M is a weak R-double

if and only if its Euler characteristic is even.

28

[2529; Necessity and the case 2m = 2 were proved in theorems 2.9
and 2.12.

Suppose that I(M) is even and 2m 2 4. Let K be a
combinatorial triangulation of M and K? the dual cell decom-

position. Denote by L the (m-l)-ske1eton of K and by

0

01,...,Ct the m-simplexes in K. Using the notation established

above and lemma 3.9, we can order these m-simplexes so that con-

ditions A through G hold.

* I

i-l
* *

by attaching the dual cell 01' In particular, Lr is obtained

*
Note that for i = 1,...,t, IL is obtained from ILiI

*
from Lt (which is the dual (m-l)-skeleton) by attaching

* 9c

*
Ct’Ct-l"°°’cr+l in that order. Applying lemma 3.9 to the poly-

*
hedron ILrI, these dual cells can be ordered so that

* *
E.H(L,)=0 for SSiSt,
m 1

* * *
F. Hm(L,)@R=H(L

f s ’ < s
i m 1+1) or r 1 ’

and conditions A through G still hold for this ordering. Note also
* * * *
that analogous conditions A , B , and C hold for each ILiI. By

lemma 3.10

*
rankH (L ) = rankH (M),
m-l r m-l

*
and from F , D, and G it follows that

H. rankH (L*) = rankH (M) = rankH (L,), for r S i S s.
m-l i m m-l i

-l
*
AS in lemma 3.3, for i = r,r+l,...,s, let Ni and Ni be
*
the derived neighborhoods in K" of ILiI and ILiI respectively.

*
Since Ni 0 N1 = bdNi is an odd dimensional manifold, 1.10 and 1.11

give

29

1(M) = I(Ni) + I(N:) I(Li) + I(L:).

*
Consider in particular the case i s. From B, H, and E it follows
that
m-l m
1(M) = 2 2 (-l)qranqu(M)) + (-1) ranka(LS).
i=0

By Poincare duality (1.6), this implies that
ranka(LS) = ranka(M),

which is an even number since I(M) is even. Now because F holds,
ranka(LS) = s-r, and thus r+s is even. Let p = (r+s)/2. Then

*
from F and F ,
kH (L — /2 — k L*)
ran m p) - (s-r) - ran Hm( p a

N *
and because these R-modules are free, Hm(Lp) = Hm(Lp). Letting
* *
N = Np, N = Np, we summarize:

Hj(N) = Hj j

*
ranka_1(N) = ranka_1(M) 8 ranka_1(N ).

~ *
(M)=H(N) for j<m-l.

~ *
Hm(N) = Hm(N ), both free R-modules with rank ='%ranka(M).

H (N) = 0 = H (N*) for j >'m.

J 1
Thus N is a weak R-half for M, completing the proof.

It is not difficult to construct an example of an even
dimensional manifold having a weak Z-half which is not a PL Z-half.

Example 3.12 Let L = L(p,q) be a lens space and M = S1 X L.

 

Then the homology groups (coefficients in Z) of L are known [6,

p. 148], and those of M are calculated from the KUnneth formula:

30

2 if i = 0,3,
Hi(L) = zp 1f 1 = 1, H (M) =

0 otherwise.

0 otherwise.

Let B be a PL 3-ball in L and C = L-B. Then the homology
groups for C can be calculated using 1.4, 1.5, and the fact that

H1(L,B) E H1(L) for each i:

if i = O,
Hi(C) = Zp if i = 1,
otherwise.

Thus B is a weak Z-half of L, and to obtain a weak half of M,

l * 1
let N = S X B, N = S X C. Then:

Z if i = 0,
Z if 1 = 0,1, 2 + Z If 1 = 3
~ * ~
Him) = Him ) =
0 otherwise. Zp if i = 2,
0 otherwise.

Note that this example does not show that no PL Z-half of
S1 X L exists. In fact, combining theorems 3.7 and 2.14, S1 X L
has a PL t-half. Also, the N and N* above do not satisfy the
conditions Summarized at the end of the proof of theorem 3.11. I
have not been able to find a counter-example to a stronger conclusion
in theorem 3.11, or even find an example in which the specific con-

struction in the proof gives a weak R-half which is not actually an

R—half.

31

With an additional hypothesis, however, theorem 3.11 can be

strengthened.

Theorem 3.13 Let M be a closed, combinatorial 2m-manifold which

 

has even Euler characteristic and is orientable over R, and suppose

Hm(M;R) = 0. Then M is a PL R-double.

*
Proof All coefficients will be in R. Let N and N = M-N be
weak R-halves as constructed in theorem 3.11. Since Hm(M) = 0

(1.6), it follows from the cohomology sequence of the pair (M,N)

that there is a monomorphism Hm(N)<: Hm+1(M,N). In this case

Fm(N) = 0, implying (1.4) that Hm(N) E Tm 1(N), and by 1.5,

111+

1 ~ * *
= ' ' C
H (M,N) Hm_1(N ). Thus there is a monomorphism Tm_1(N) Tm_1(N ).

(N*) .

* a
Reversing the roles of N and N gives T (N) = T
m-l m-l
*
In theorem 3.11, N and N were constructed so that their

homologies agree except possibly in their (m-l)-dimensiona1 torsion

~ *
modules. Thus in this special case, H*(N) = H*(N ).

In order to discuss the uniqueness of an R-half in a closed

manifold, the following concept will be useful.

Definition 3.14 A polyhedron X is said to have cohomology dimension

 

S k if H1(X;Z) = 0 for all i > k. The cohomology dimension of X

 

is the smallest integer with this property.

Observe that if X has cohomology dimension k, then for any
R and any i > k, Hi(X;R) = 0 [12, p. 246]. Then it follows (1.4)
that Hi(X;R) = 0 for i > k and Tk(X;R) = 0.

If M is a closed, combinatorial manifold of dimension 2m + 1,
then the PL Z-half of M found in theorem 3.4 has cohomology dimension

S m. Similarly, if M is a closed, combinatorial 2m-manifold with

32

even Euler characteristic and orientable over R, then the weak
R-half of M found in theorem 3.11 has cohomology dimension S m.
The following examples show that the cohomology dimension of a half

may be large.

Example 3.15 Let Y be the compact 2-manifold with boundary formed

by cutting an open disc from the torus. Let N = Y X Sk. Then

 

k+ ~ '
H 1(N;Z) = 29 z, and H1(N;Z) = 0 for i > k + 1. If M is the
double of N, M is a closed manifold of dimension n = k + 2, and

N is a half for M having cohomology dimension n - 1.

These examples suggest that the only hope for obtaining re-
sults on the uniqueness of a half is by placing restrictions on its

cohomology dimension.

Theorem 3.16 Let M be a closed, combinatorial n-manifold which is

 

orientable over R, and let m be the greatest integer in n/2.
If M has a PL R-half N of cohomology dimension k1< m, then

Hi(M;R) E Hi(N;R) for i s 111.

Proof Assume that N is a PL submanifold of dimension n (2.5)

*
and let N = M-N. Then for i S m,
N n-i *
Hi(M,N;R) = H (N ;R) = 0,

by 1.5. It follows from the exact sequence of the pair (M,N) that
Hi(M;R) S Hi(N;R) for i < m.

If n is odd, n = 2m + 1 and Hm+1(M,N;R) E Hm(N*;R) = o,
implying Hm(M;R) E Hm(N;R) = 0.

If n = 2m is even, then I(N) = %‘I(M) as seen in the proof

of theorem 2.9. From Poincare'duality it follows that ranka(M;R) = 0,

33

and also that

~

T (M;R) 1 (M;R) E T (N;R) = 0.
m m-l m-l

Thus Hm(M;R) = 0 = Hm(N;R), completing the proof.

Theorem 3.16 says that a closed, combinatorial n-manifold M
has a PL R-half of cohomology dimension less than m = [n/2] only
if Hm(M;R) = 0, and that in this case the homology of such a half is
uniquely determined by that of M. The examples below Show that M

may have many different halves of cohomology dimension m.

 

Example 3.17 a. Let Xt be the one-point union of t circles

imbedded as a PL subspace in SB. Then by Alexander duality [12,

p. 296],
2 if i = O,
3 .. ~ t . .
Hi(S - Xt) — Hi(xt) $12 if 1 — l,
0 otherwise.

b. Let Yt be the one-point union of t projective planes

imbedded as a PL subspace in SA. Againuapplying Alexander duality,

2 if i = 0,

4 a z: . - ..
Hi (8 - Yt) Hi(Yt) % 22 if i l,
0 otherwise.

If R is a field and M is an even dimensional manifold,
note that theorem 3.13 is the converse of theorem 3.16. The last

theorem in this chapter gives a similar result for odd dimensions.

34

Theorem 3.18 Let M be a closed, combinatorial (2m + l)-manifold

 

which is orientable over a field F. Then M has an F-half with

Hi(M;F) if i S m,

Hi(N;F)
0 if i > m.
ngpf Let all modules have coefficients in F. The technique here
is similar to that in the proof of theorem 3.11, except that now
there are no torsion submodules.

Let t be the rank of Hm(M) as an F-module, let L0 be
the m-skeleton of a combinatorial triangulation K of M, and let
L; be the complementary subcomplex in the dual cell decomposition
KI. Then Hm(LO) ; Hm(L;) (theorem 3.4), and we denote by r the

rank of these modules. As in lemma 3.9, we can adjoin s = r - t

(m+l)-simplexes to L0’ obtaining LS satisfying:

Ill

0 o <
Hi(Ls) Hi(M) if i m,
Hm(Ls) has rank t, and hence Hm(LS) = Hm(M),
= . . > .
Hi(Ls) 0 if i m
* * .
Let N and N be the derived neighborhoods of LS and LS in
K", as in lemma 3.3.

* * *
Since the m-skeleton L of K is obtained from L8 by

0
* ~ *
adjoining m-cells, Hi(N ) = Hi(M) for i < m - l and Hi(N ) = 0
for i > m. Because H1(N) = 0 for all i > m and applying 1.5

and 1.6,

III

m+
H

Ill

* m+2 2 -
Hm_1(N) H (M,N) (M) =Hm_l(M).

35

* *
Thus adjoining s m-cells to L8 gives L0

*
(m - l)-dimensiona1 homology. Thus by condition D, p. 26, Hm(Ls)

with no change in the

~ *
has rank r - s = t. Therefore HiCN) = Hi(N ) for all i,

completing the theorem.

CHAPTER IV
COMPACT, COMBINATORIAL MANIFOLDS WITH BOUNDARY

In this chapter compact, combinatorial manifolds with non-
void boundary are considered. As in chapters 11 and III, the question
considered is: Under what circumstances does such a manifold de-
compose into two pieces in some sense equivalent? All compact mani-
folds with boundary of dimension 2 or 3 are shown to be PL t-doubles
(definition below). In higher dimensions, homological results for
manifolds whose boundary components are spheres are obtained by
applying theorems from chapter III. As in chapter III, all sub-

spaces and maps will be PL.

Definition 4.1 (Cf. 2.6 and 2.7) A compact, combinatorial n-mani-

 

fold with boundary, M, is a PL t-double if it contains a PL submani-

 

 

fold N such that N E M-N. M is said to be a PL R-double if it
contains a PL submanifold N of dimension n satisfying
H*(N;R) 5 H*(MrN;R). The submanifold N is called a PL t-half or

a PL R-half of M.

Theorem 4.2 Every compact 2-manifold with boundary is a PL t-double

 

having a half which is a disc.

Proof It is known ([7, pp. 43-45]) that any compact 2-manifold with
boundary can be represented by attaching strips to a disc. Moreover,
as can be seen in figures 7 and 8, the strips can be attached

symmetrically so that cutting the manifold on the line of symmetry

36

37

separates it into two discs. Figure 7 shows an orientable Z-manifold
having genus 2 and 3 boundary components, while figure 8 represents

a non-orientable surface having genus 2 and 2 boundary circles.

 

 

 

Figure 7

_____ ____.____D-_

Figure 8

The following examples Show that in higher dimensions there
are many manifolds with boundary not having a homology ball as a PL

Z-half.

Example 4.3 If M is a compact n-manifold with boundary and H1(M;Z)

 

has torsion, then M cannot have a PL Z-half which is a homology

n-ball.

38

Proof If B1 and B2 = M-B1 are homology n-balls, then from the
reduced Mayer-Victoris sequence of the couple {B1,B2} we have

H1(M;Z) E HO(B1 O B2;Z), which is a torsion-free module.

Example 4.4 Let K be a non-trivial polygonal knot in 83, let N
be a regular neighborhood of K, and M = S3 - N. By Alexander
duality, H1(M:Z) is Z for i = 0,1 and 0 otherwise. On the
other hand, the fundamental group of M is the knot group of K

which is not cyclic.

 

Now suppose B1 and B2 = M - B1 are homology 3-balls. Then
H1(B1 n B2;Z) E H2(M;Z) = 0, and since B1 n B2 is a finite collection
of connected 2-manifolds with boundary it follows that B1 0 B2 con-
sists of disjoint discs only. Since two 3-balls meeting on a disc form
a 3-ball ([13, III, p. 4]), M, must be a handle body. But then
H1(M;Z) E Z implies that the fundamental group of M is also 2.

This contradiction shows that M cannot have a Z-half which is a

homology'3-ball.

Example 4.4 above is interesting in view of the following

result.

Proposition 4.5 Let M be a compact, connected 3-manifold with non-

 

void boundary such that H1(M:Z) is torsion-free for i = 1,2. Then
there is a manifold with boundary M' having a 3-ba11 as PL t-half

and satisfying H*(M';Z) E H*(M;Z).

Proof Let B beva 3-ball. In de embed a connected graph X
such that H1(X;Z) E H2(M;Z), which is possible since HZCM;Z) is

free. Let A be a regular neighborhood of X in de. If

39

rankH1(M;Z) = k, let D1,...,D be disjoint discs in de all of

k
which miss A. Now take B1 and B2 to be copies of B and h
a PL homeomorphism from A U D1 U...U'Dk in B1 to the corresponding
set in 32' Then M' = B1 Uh B2 is the required manifold as the

following discussion shows.

Since for each i (coefficients in Z),
' E E .
H104) Iii-1031 n 132) Hi_1(A U D1 U...U DR),

and the union is disjoint, it follows that HO(M') E Z, H1(M') is

N

free of rank k, H2(M') H1(A) E H2(M), and Hi(M') = O for all

Ill

other 1. Thus H*(M) H*(M').

Although not all compact 3-manifolds with boundary have a
half which is a ball, we will Show that every such manifold is a PL
t-double. The proof relies on theorem 3.7 and a technique for
modifying a compact 3-manifold with boundary into a closed manifold.

In the following discussion Y will always represent a PL

2
homeomorph of the set of all points (x,y) E E which satisfy

x e {0,1,...,k} and -1 s y s 1

or
IyI = l and o s x s k.

Also, X will represent a homeomorph of {(x,y) E Y : y 2 O}, with

xo,x1,...,xk being the and vertices of this tree. Note in particular

that Y = X1 U X2, two copies of X attached at the points

x0,x1,...,xk, as illustrated in figure 9.

40

 

 

 

 

 

 

 

 

 

 

Figure 9

Definition 4.6 Let H be a handle body with j handles and let X

 

as above be embedded as a PL subspace of H such that

X 0 de ={x0,x .,xk}, Let N be a regular neighborhood of X in

1""
H. X is said to be unknotted in H if H - intHN is a handle body

with j + k handles.

Now suppose that H is a handle body in a closed 3-manifold
M, and that X is embedded in de. We want to move X so that it
is unknotted in H.

Let K be a combinatorial triangulation of M with sub-
complexes K1 and K2 triangulating H and X respectively.
Let X' be the closed subspace of X homeomorphic to X and con-
sisting of all but the end l-simplexes of the second barycentric sub-
division K3,
derived neighborhood of X' in M. Note that N is a ball and that

as shown in figure 10. Now let N = N(X',K") be the

Figure 10

41

X is unknotted in N. Also N n H and N O M-H are balls meeting
in the common face N n de. Thus we can consider X embedded in

the standard (3,2) ball pair ([13, chap. IV]) as shown in figure 11.

 

 

 

Figure 11

By lifting slightly all except the and vertices in a triangulation
of X, we obtain a homeomorphism h of N onto itself which is the
identity on bdN and such that h(X) is unknotted in N n H.
Extending h by the identity to all of M gives a homeomorphism

of M onto itself such that h(X) is unknotted in H and

h(xi) = x1 for i = 0,1,...,k.

Theorem 4.7 Any compact 3-manifold M with boundary is a PL t-

 

double.

ngpf Case 1: M has connected boundary. In this case bdM is a
closed 2-manifold and there is a handle body H with de E bdM.

Let M' be a closed 3-manifold formed by attaching H to M by a
homeomorphism between their boundaries. Let Y (see figure 9) be
a spine of H and K a triangulation of M' such that Y<I IKII,
the 1-skeleton. If N is a regular neighborhood of IKII in M',

it is a handle body and by theorem 3.7 is a t-half for M'. Writing

42
Y = X1 U X2 (figure 9), we need:

Lemma 4.8 There is a homeomorphism g' of M' onto itself such

that g'(Xl) C bdN and g'(Xz) is unknotted in N.

Proof of lemma Since Y is contained in a 1-dimensional spine of
the handle body N, we can assume that Y is embedded in a 3-ball

with certain discs identified as Shown in figure 12. Let N' be

 

 

 

 

 

 

 

 

 

 

Figure 12. Y C: N c N'.

a regular neighborhood of N in M'. By bringing X Straight

1
forward (see figure 12) it is fairly eaSy to see a homeomorphism
g' of N' onto itself which is the identity on bdN' and which

satisfies the conditions g'(X1)<Z bdN and g'(X is unknotted

2)
in N. Extending g' to all of M' by the identity completes the

lemma.

Continuing with the proof of the theorem, NF_:END is a handle
body and g'(Xl) is contained in its boundary. By the discussion
immediately preceeding the theorem, there is a homeomorphism g" of
M' onto itself such that g"g'(X1) is unknotted in MIC—N and

g"|3'(xz) is the identity. Thus the homeomorphism g = g"g 0f

43

M' is such that g(Xz) is unknotted in N and g(Xl) is unknotted
in INT—EUR.

Let J be a triangulation of M' such that g(Y), N, and
FF—E—N are all triangulated by subcomplexes. Denote S B N(g(Y),J"),

S = S O M' - N, S = S n N. Then S is a regular neighborhood of

l 2 1
g(Xl) in MI - N and $2 is a regular neighborhood of g(Xz) in
N. Since N and M' - N are homeomorphic handle bodies, the un-

knottedness gives

. g —T-:—- _ .
N intNS1 M N lntfiT:fiSI.
‘1 . .
Now g (S) is a regular neighborhood of Y in M' so
there is a homeomorphism f of M' onto itself with fg-1(S) = H.

Letting h = fg-1 we have
. ‘7?— - - a . - . :
h(N lntNSl) U h(M N 1ntfifi'82) M intM,h(S) M.

Thus h(N - intNSl) is a PL t-half for M, completing case 1.

Case 2: bdM is not connected. In this case bdM is a
finite collection of connected, closed 2-manifolds. We sew a handle
body Hi into each boundary component to form a closed manifold M'.
Repeating the above argument for each Hi yields the homeomorphism

g of M' and the rest of the argument is identical.

In higher dimensions results can be obtained for a certain
set of manifolds with boundary which can be modified so that the
results of chapter III are applicable. The final two results are of

this nature.

 

44

Theorem 4.9 Let M be a compact, combinatorial manifold of odd
dimension n, each of whose boundary components is an (n-l)-sphere.

Then M is a PL Z-double.

Proof Let Sl"°"Sk be the boundary Spheres. We form a closed,

combinatorial n-manifold M' by attaching to each boundary component
a PL n-ball Bi through a homeomorphism dei E Si. For i = 1,...,k,

let pi be a point of M' in the interior of Bi' By theorem 3.4

there exists a PL submanifold N'<: M' which is a homology half for
M'. We can triangulate M' so that bdN' is triangulated by a sub-

complex L. Let A ,...,A be disjoint (n-l)-simplexes from L

l

(we can always subdivide so that this is possible) and for each i

k

choose qi<Z intT By homogeneity and the uniqueness of regular

1'
neighborhoods ([13, chap. III, p. 22]), there is a PL homeomorphism
h of M' onto itself such that for each i, h(pi) = qi’

h(Bi) = Ist(Ti,K)I, and (h(Bi),Ti) is an unknotted ball pair of
codimension l ([13, chap. IV]).

It follows then that h(Bi) n N' collapses across T1 to

h(dei) O N' for each 1. Thus
N'\ N' - (inth(B1)U...Uinth(Bk)).

Similarly, M' - N" 'M' - N' - (int h(Bl)U...Uinth(Bk)). Now let

N = h-1(N' - (inth(Bl)U...Uinth(Bk))). It is clear that

M - N = h'1(M' - N' - (inth(sli..lJinth(Bk))). Now N and N'

 

have the same homotopy type, as do M - N and M' - N'. Since N'
was a PL Z-half for M', N is a PL Z-half for M, completing the

proof.

45

Corollary 4.10 Suppose M is a compact, combinatorial manifold of
even dimension n, each of whose boundary components is a sphere.

Let M' be the closed manifold formed by sewing an n-ball into each
boundary component. If I(M') is even and M' is orientable over

a field F, then M is a PL F-double.

Proof The proof is identical to that above except that we apply

theorem 3.11 to obtain the F-half N' of M'.

BIBLIOGRAPHY

10.

ll.

12.

13.

BIBLIOGRAPHY

Bing, R.H., "Some ASpectS of the Topology of 3-Manifolds Re-
lated to the Poincare’Conjecture," in Lectures in Modern
Mathematics, Vol. 11, ed. T.L. Saaty, Wiley, New York, 1964,
93-128.

 

 

Doyle, P.H., Topics in Topological Manifolds, mimeographed
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Greenberg, M.J., Lectures on Algebraic Topplogy, W.A. Benjamin,
New York, 1967.

Gugenheim, V.K;A.M., "Piecewise Linear Isotopy and Embeddings
of Elements and Spheres (1)," Proc. Lond. Math. Soc., (3)3
(1953), 29-53.

 

Haken, W., "Some Results on Surfaces in 3-Manifolds," in
Studies in Modern Topology, ed. P.J. Hilton, Prentice Hall,
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Hu, 8., Homology Theory: A first Course in Algebraic Topology,
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Massey, W.S., Algebraic Tppologyig An Introduction, Harcourt,
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Milnor, J., "A Unique Decomposition Theorem for 3-Manifolds,"
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Moise, E.E., "Affine Structures in 3-Manifolds, V. The Tri-
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Munkres , J.R., Elementary Differential Topolpgy, Revised
Edition, Princeton University Press, Princeton, N.J., 1966.

Seifert und Threlfall, Lehrbuch der Topologie, Chelsea, New
York.

Spanier, E.H., Algebraic Topology, McGraw Hill, New York, 1966.
Zeeman, E.C., Seminar on Combinatorial Topology, (mimeograph),

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