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LIBRARY

Michigan State
University

THESIS

 

This is to certify that the

thesis entitled
The Set of Generating Domains

for Certain Manifolds
presented by

Richard John Tondra

has been accepted towards fulfillment

of the requirements for

Ph.D. degree in Mathematics

Major pronsor
Date Wz

0-169

ABSTRACT
THE SET OF GENERATING DOMAINS FOR CERTAIN MANIFOLDS
by Richard John Tondra

Let X be a tapological space. A collection G* of non-
empty, connected tOpological spaces is called a set of
generating domains for X if each prOper domain (Open, con-
nected subset) of X is an Open, monotone union of some
element g(D) of 6*; that is, D - kglpk where each Dk is an
open set homeomorphic to g(D) and nk c.1111“.l for all k 6 2*.
The domain rank of X, denoted by DB(X), is the cardinal num-
ber of a set of generating domains for X that has a minimal
number of elements.

Let Mn denote a connected n-manifold, n g_2. The prin-
cipal theorems characterize those manifolds which have the
smallest possible domain rank. Let us say that Mn has
Euclidean compact subsets if for each pr0per, compact subset
c of as there is a homeomorphism h of the pair (c,c n in)
into the pair (tan, in), where tan . {x e an|xn g 0}. In
chapter II it is shown that if in - fi. then Da(nn) = 1 if
and only if up has Euclidean compact subsets. If in % fie
then it is shown in chapter III that DR(Mn) a 2 if and only
if up has Euclidean compact subsets. Chapter IV gives a
characterization of those up with hp an n-l sphere that have
domain rank 3. In the final chapter are found results con-

cerning the domain rank of spaces which are the cpen monotone

Richard John Tondra

union or the finite product of those manifolds considered in

chapters II through IV.

THE SET OF GENERATING DOMAINS
FOR CERTAIN MANIFOLDS

By

Richard John Tondra

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of mathematics

1968

ACKNOWLEDGMENTS

I wish to express my sincere thanks to all those per-
sons who have in any way helped to bring me to this achieve-
ment. I wish to thank especially Professor P. H. Doyle for
his sincere kindness and consideration to me during the past
three years. Finally, a special thank you to my wife, Rose,
for helping in the preparation of this thesis.

11

To
Mom and Dad

iii

CONTENTS

CHAPTER I. INTRODUCTION . . . . .
l. Notational conventions. . .
2. Sets of generating domains .
3. Collared manifolds . . . .

4. Piecewise linear manifolds .

CHAPTER II. CONNECTED MANIFOLDS WHICH HAVE A

GENERATING DOMAIN . . . .

1. Characterization. . . . .

2. Compact, connected n-manifolds with

o 0
Mn # a and DR‘Mn) =3 1 o o o o 0
CHAPTER III. CONNECTED MANIFOLDS WITH BOUNDARY WHICH

HAVE DOMAIN RANK 2 . . . .

1. Characterization. . . . .

2. Some special manifolds of domain rank 2 .

CHAPTER IV. MANIFOLDS WITH COMPACT BOUNDARY WHICH

HAVE DOMAIN RANK 3 . . . .

l. A generator for a certain dominion of a

compact, punctured n-sphere .

2. Characterization. . . . .

CHAPTER V. MONOTONE UNIONS AND PRODUCTS.

BIBLIOGWHY. O O O O O O O 0

iv

#- Co +4 l4

18

24
24

32

at
31+
42

#7

47
50
55
58

Figure 1.1

Figure
Figure
Figure

Figure

2.1
2.2
3.1
3.2

LIST OF FIGURES

17
28
29
38
1+0

CHAPTER I
INTRODUCTION

In [3], Morton Brown proved that any Open monotone
union Of an Open n-cell is an Open n-cell. This thesis is
concerned with a problem quite Opposite to that considered
by Brown - proving the existence of certain n-manifolds from
which many non-homeomorphic spaces can be Obtained as Open
monotone unions. In particular it is shown that there ex-
ists a connected n-manifold Mn such that each Open connected
subset Of Euclidean n-space can be Obtained as an Open mono-

tone union Of Mn.
1. Notational conventions

Let A, B, and X be sets. If A is a subset Of X, then
this will be denoted by A c.X; if A c.X but A i X, then this
will be denoted by A £3X. If A # fl and A f X, then A is
called a prOper subset of X. If A and B are subsets Of X,
then A - B will denote the set theoretic difference Of A
and B in X.

Let X be a tOpOlogical space, A CLX. IntxA, Cle, and
FrxA will denote the interior, closure, and frontier Of A
in X respectively. The set A - IntxA will be denoted by
deA and is called the edge Of A in X. Note that if
A = Cle, then FrxA = deA. When there is nO possibility

of ambiguity, the subscript "X" will be omitted.

If X is a tOpological space, we will denote by R(X)
the set of all homeomorphisms of X onto itself. CH(X) will
denote the subset of H(X) consisting Of all those homeomor-
phisms which are the identity on the complement Of some
prOper compact subset of X.

If X and I are tOpological spaces, a homeomorphism of
X into X will be called an embedding. If there is a hO-
meomorphism f Of X onto I, we will write X I I.

Let X and Y be disjoint tOpological spaces, and let
X + Y denote the disjoint union Of X and Y with the weak
tOpology. Suppose that A c:X is closed, A # fl, and that
f is a continuous function from A into I. X Uf I will de-
note the space Obtained by attaching X to I by f where
p : X + I r X Uf Y is the identification.

The follOwing notation will be used for certain sets
and tOpological spaces:

Z = {nln is an integer};
2+ = {n 6 Zln > 0}; and
Rn = {xlx a (x1,...,xn), an n-tuple Of real numbers,
n E Z+}.
Rn is assumed to have the tOpology determined by the Eu-
clidean metric dn on Rn. The subsets
aRn a {x 6 RnIxn g 0};
En - {x e Rnldn(x,0) 5 1};

En(r) - {x E Rnldn(x,0) < r, r a real number > 0k and

3

Sn‘1 a {x E Rnldn(x,0) - 1} are assumed to have the

subspace topology induced by the tOpology Of Rn.
2. Sets of generating domains

Throughout this section, X will denote a fixed non-

empty tOpological space.

Definition 1.2.1 A non-empty set D CLX is called a domain

Of X if D is Open and connected.

Definition 1.2.2 A non-empty set D* of subsets Of X is
called a dominion of X if each element Of D* is a prOper

domain Of X.

Definition 1.2.} Let 0 c:X be a non-empty Open set. A
topological space g(O) is called a generator Of 0 if there
exists a countable collection Of sets {0k}kez+ such that
O a‘kglok and such that for all k e 2*,

1) ok CLX is Open and homeomorphic to g(O) and

11) 0k C °k+1°
If g(O) generates 0, then 0 is called an gpgn, homogeneous

monotone union of g(0).

Definition 1,2.u Let K(X) a {DID is a proper domain or X}
and suppose that K(X) # g. .A non-empty set B* Of nonpemp-
ty, connected topological spaces is called a §§2_g£_gggg;r
Eggng domains for X, if for each D E K(X) there is some

B E B* such that B is a generator Of D.

If B* is a set of generating domains for X, then each
B 6 B* is homeomorphic to a prOper domain Of X. Therefore
a set of generating domains with a minimal number Of ele-
ments can be found among the dominions Of X. This consid-

eration leads to the following definition.

Definition 1.2.5 Suppose that K(X) é a and let 6* a {B*IB*
is a dominion Of X which is also a set Of generating dO-
mains for X}. The domain Eggk Of X, denoted by DR(X), is
defined by DR(X) a g.1.b.{IB*|IB* e c*} where |B*| denotes
the cardinal number Of the set B*. If DR(X) a 1, then X

is said to have a generatigg domain. If K(X) - fl, we define
DR(X) a 0.

The following theorem is an immediate consequence Of

the foregoing definitions.

Theorem 1.2.6 If I c:X is a domain of X, then DR(X) is
less than or equal DR(X).

3. Collared manifolds

Definition 1.341. A tOpological space X is called an 222$?
mensional manifold, n E ZI, and is denoted by up, if X is a
separable metric space and for each x 6 X there is an Open
neighborhood Ux or x such that U1 3 an or Ux I in“. A
O-dimensional manifold no is defined to be an at most

countable, descrete tOpological space.

5

Definition 1.3.2 Let Mn be an n-dimensional manifold,

n E Z+. The set fin a {x 6 Mnlx has an Open neighborhood
homeomorphic to an} is called the interior Of Mn. The set
in 2 Mn - g“ is called the boundary Of Mn. If an - H

[Mp i fl], then Mn is called a manifold without [Eigh]
boundary. An n-manifold without boundary is called closed
if Mn is compact; otherwise, Mn is called 222g.

Definition 1.3.3 A space X which is homeomorphic to

En, En, or Sn is called an gzggll, 222p.n:ggll, or nzgphggg

respectively.

Definition 1.2.4 Let Mn be an n-dimensional manifold,
n e 2*. An (n-l)-manifold Ln'l c.Mn is said to be 22;:
lgggd [bi-collared] in up if there is an embedding h Of
Ln'l x [0,1) [Ln‘l x (-l,1)] into M? such that

i) h(x,0) s x for all x 6 Ln"1 and

n-1

ii) h(Ln"1 x [0,1)) [h(L x (-l,lmlis Open in Mn.

Definition 1,2,5 Let an be an n-dimensional manifold with
boundary, n e 2*. The n-manifold Mg . Mn x [0.1) 00 up
where c : Mn x {0} ~ an is defined by c(x,0) - x for all

x 6 Mn is called an abstract collarigg of ME.

MP and Mn x [0,1) will always be considered as embed-
ded in m: in the usual way under the identification map
p : Mn x [0.1) + Mn e n3.

Definition 1.3.6 Suppose that up and Ln are n-manifolds,

6

n E Z+, such that Ln # fl and Ln c Mn. Ln is called a £21:
sills ME.n-manifold if
1) Ln n T“ i a,
ii) Ln n Mn is an (n—1)-manifold, and
iii) an a Ln - Inth is empty or an (n-l)-manifold.
Note that if Ln is a relative Mn n-manifold, then the
boundary Of (Ln 0 Mn) a boundary Of an. Also, since Ln is
an n-manifold contained in Mn, an = EdLn 2 Ln - Inth and
boundary EdLn - EdLn n h“.

2 2

Example. Let M2 = %R2 and L a [-1,1] x [0,1). Then L
is a relative M2 2-manifold with rL2 - {-1} x [0,1) u

{1} X [0:1).

pefinition 1.3.2 Suppose that Mn, Ln are n-manifolds,
n 6 2*, such that in w ¢ and Ln c h“. Ln is said to be
collared in Mn if Ln c 3“ or Ln is a relative up n-mani-
fold, and if there is an embe ding h Of the pair
(EdLn x [0,1),(EdLn n En) x [1,1)) into the pair (Mn,fin)
such that

i) h(x,0) - x for all x e EdLn and

ii) h(EdLn x [0,1)) n Ln - h(EdLn x {0}).
The set h(EdLn x [0,1))is called a collar Of Ln in up and
is denoted by an. The set CLn - Ln U an is called a 22;:

laring Of Ln in Mn

Definition 1.3.8 Let ME, L? be n-manifolds, n e 2*, such
n

that L1 C.M?, i - 1,2. A homeomorphism h Of L? onto L2 is

called a relative homeomorphism if h induces a homeomor-

n n 'n n n '
phism of the pair (L1,Ll 0 M1) onto the pair (L2,L2 n Mg).

Lgmma 1.3.9 Let ME, L? be n-manifolds, n E Z+, such that
n

L1 is collared in M? and let CL? be a collaring Of L? in

Mg, 1 = 1,2. If h is a relative homeomorphism of L? onto

Lg, then h extends to a relative homeomorphism Of CL? onto
n
CLZ'

Proof. Assume that EdL? # D; otherwise, the required
extension of h is h itself. Note that both Ln and an are

i 1
closed in CL? and that L? n cL? - EdLE, i = 1,2. Let
n a n n
f1 : EdL1 x [0,1) M? give a collar cL1 of L1 in Hi,

i - 1,2. If y 6 0L?, then y - f1(x,t) for a unique pair

(Iat) 6 EdL? x [0,1), 1 a 1,2. Define g1 : cL: * ch by

81(y) - 81(f1(1.t)) = f2(h(1).t); that is. 81(y) .

n -l n
f2(h|EdL1,id)f1 (y). Since h|EdLl

Of (EdL§,EdL§ n a?) onto (EdLg,EdL§ n Mg), 31 is a relative

induces a homeomorphism

homeomorphism of 0L; onto an. If y 6 EdLg, then

81(y) - f2(h(y),0) - h(¥). Define 8 by

h(y). y 6 L?
g(y) - n .
31(y). y 6 cL1

Then g : CLn r CLn

1 2 is the required extension of h.

Definition 1,3.10 Let X be a metric space, A and B subsets
Of X with A C-B. Suppose that f and g are bounded contin-
uous functions from B into R1 such that f(a) 5 g(a) for all

a E A. The prism on A determined pz_f and g,is denoted by

8

P(f,g;A) and defined by P(f,g;A) = {(x,t) e x x Rllx e A
and f(x) $_t 5.g(x)}. If f(a) < g(a) for all a E A, then
TP(f,g;A), is defined by TP(f,g;A) = {(x,t) e X x R1|x 6 A
and f(x) g_t < g(x)}. If f(a) n c, c a constant, for all
a 6 A, then 0 will denote the function f. The graph of g
restricted to A will be denoted by G(g;A). Note that

G(s;A) = P(s.s;A).

The following lemma is a summation of the remarks

found on page 556 Of [13].

Lemma 1.3.11 Let X be a metric space, A c:B ch, and let
f1, f2 be bounded continuous functions from B into R1 such
that f1(a) 5_f2(a) for all a 6 A. Suppose that 81 and 32
are continuous functions from A into R1 such that
i) f1(a) 5,g1(a) S_f2(a) for all a 6 A, i - 1,2,
ii) f1(a) - g1(a) if and only if f1(a) a g2(a), and
iii) f2(a) - g1(a) if and only if f2(a) = g2(a).
Then there is a homeomorphism h[f1,f2;gl,g2] Of P(f1,f2;A)
onto itself such that
iv) h[f1,f2;gl,g2](a,t) a (a,t) for all
(a,t) e (G(f1;A) u G(f2;A)) and
v) for each a 6 A, h[f1;f2;gl,g2] carries the seg-
iment P(f1,gl;a) linearly onto the segment
P(f1,g2;a) and the segment P(gl,f2;a) linearly
onto the segment P(g2,f2;a).

The following rather complicated lemma is used to
establish the existence Of certain nice collarings Of a

collared manifold.

Lemma l;3.12 Let X be a metric space such that there is an
embedding Of X into Rn for some n E Z+. Assume that X is
embedded in Rn and that A is a prOper subset Of X such that
i) A is locally compact,
ii) Cle is compact and C - Cle - A is either empty
or closed in X and hence compact,
iii) there is an embedding h Of A x [0,1) into X such
that h(a,0) - a for all a E A.
Then there is a continuous function f : Cle a [0,5) such
that f(x) = 0 if and only if x 6 C and the following hold:
iv) there is an embedding h1 of P(0,f;Cle) into X
A and

X
hl(x) - h(x) for all x E P(O,f;A) and

such that hl(y,0) a y for all y 6 Cl

v) if G is Open in X, A c:G, then there is a homeo-
morphism g Of h1(P(0,f301xA)) onto itself such
that g(hl(TP(O,%f;A))) c:G and g restricted to
h1(G(0;Cle) U G(f;Cle)) is the identity.

1292;. Let d be the Euclidean metric on a“; then d|X
is a metric equivalent to the metric Of X. For r 6 (0,»),
D .4 a, and D c: a”, let B(D,r) s {x (- Rn|d(x,D) < r}. Then
CIRnB - {x e Rn|d(x,D) g r}, where B = B(D,r).

Suppose that c i a. Define A1 = {x e Cle|d(x,C) > A}
and for k 6 2*, k _>_ 2, define

lO

A.k - {x e CleIl/(k+1) < d(x,C) 5 1/k}. Then ClcleAk c: A
and therefore Pk - ClAAk - A n ClcleAk = ClcleAk for
k 2.2. Thus Pk is compact and we have that
A =k§1Ak 2 A1 U (kngk). Let d* be the metric on
ole x R1 defined by d*((x,t),(y,u)) a d(x,y) + It - ul.
Since Fk is compact for (k-l) E Z+, there is a real number
ak, o < ak < g, such that if x 6 Pk, then
d(h(x,0),h(x,t)) < l/k for 0 S,t g ak. Therefore it is
possible to construct a sequence {bk}kez+ of real numbers
such that O < bk+1 < bk < % and bk+1 < ak+1 for all k 6 2+,
and tip bk = 0. Define g1 : A ~ [0,%) by g1(x) - bk if
x 6 AK. Then if r e 31, the set L(r) .. {x e AIg1(x)_<_ r}
equals A or kgpAk for some (p-l) 6 2+. Therefore L(r) is
closed in A and thus g1 is lower semi-continuous and posi-
tive on A. It follows from a theorem due to Dowker (see
page 170 Of [9]) that there is a continuous function
32 a A ~ [0.5) such that o < g2(a) < gl(a) for all a e A.

Define f t Cle e [0,1) by

a) f(x) = i for all x E Cle if C = fl and

O, x 6 C
b) f(x) ={ }ifC#¢.
min(d(x,C),g2(x)), x 6 A

Since C is compact, f(x) - 0 if and only if x E C. If
C a C, then f is clearly continuous. Suppose that C # fl.
If x E A, then f is continuous at x since A is Open in

Cle and fIA is clearly continuous. Suppose that x E C and
that E > 0. Let y 6 Cle such that d(y,x) < 6. Then

11

|f(y) - f(X)| = |f(y)| - f(y) s d(y,C) s d(y.X) < 6.
Therefore f is continuous at x and hence on Cle.

Define hl : P(0,f;Cle) e X by

( ) x, x 6 C
h X,t - o
l {h(xst): (1915) E P(09flA)}

If C = fl, hl is certainly continuous. Suppose that C # fl.
Since C is compact, P(O,f;C1xA) - C x {O} - P(O,f;A) is
Open in P(0,f;Cle) and therefore h1 is continuous at
points (x,t) 6 P(0,f;A). Suppose that c E C and that

o < e < b Choose k 6 2* such that 1/k < s6. Then

2.
U(c,€) = ((B(c,l/k) n Cle) x [0,bk)) fl P(O,f;ClXA) is
an Open ileighborhood Of (c,O) in P(0,f;C1xA). Suppose
that (y,t) 6 U(c,6). Since d(y,C) 5.d(y,c), y 6 A3 for
some 3 g k _>_ 2. Therefore d(hl(c,0),hl(y,t)) _<_
d(h1(c,0),hl(y,0)) + d(hl(y,0),h1(y,t)) $,d(c,y) + l/J <
l/k + 1/3 5 2/k < 6. Therefore h1 is continuous at (c,0).
Since h t A x [0,1) a X is an embedding and h(a,0) - a
for all a e A, c n h(A x [0,1)) = fl. Therefore h1 is in-
Jective. Since X is Hausdorff and P(O,f;Cle) is compact,
hl is an embedding.

Now suppose that G is Open in X and that A c:G. Since
C is compact or C a C, G - C a G1 is Open in X and A c:Gl.
Let P - P(O,f;Cle), and define G2 = G1 n h1(P) and
F2 a (x - 91) n hl(P). Then hi1(Gz) a G3 is open in P,

hi1(F2) a P - c - F3, A x {0} c.03, and c x {o} c F3.

3
Let f3 : ole ~ R1 be defined by f3(x) = d*((x,0),F3).

12

Then f3 is continuous and f (x) = 0 if and only if x 6 C.

3
Let gl a min(f3,%f); then gl is continuous. Furthermore,
if 0 S,t < gl(y) where y 6 A, then (y,t) 6 G3. Since

gl(y) a 0 if and only if x 6 C, it follows from 1.3.11
that there is a homeomorphism 32 = h[0,f;%f,gl] Of P onto
itself such that g2|(G(O;Cle) u G(f;ClXA)) = id and 52
carries P(O,%f;x) linearly onto P(O,g1;x) for all x E Cle.

Therefore g2(TP(O,%f;A)) c.c Define g : h1(P) ~ hl(P)

30
by g = hlgzhil. Then g is a homeomorphism Of
hl(P(O,f;Cle)) onto itself such that g(h1(TP(O,%f;A)))
is contained in G, and such that g restricted to

h1(G(O;Cle) U G(f;C1xA)) is the identity.

Now suppose that Mn is an n-manifold with boundary,
n 6 2*. If h” is not compact, then let x be the one point
compactification of DE, otherwise, let X 2 Mn. In either
case X can be embedded in Rp for some p 6 2+. Consider X
as embedded in RP and note that i) h? is locally compact;

ii) ClxMn is compact and C - Cl Mn - up is compact or emp-

X
ty; and iii) as a consequence Of theorem 2 Of [4], there is
an embedding h Of M? x [0.1) into x such that h(x,0) = x
for all x 6 Mn. Using 3.1.11 and 3.1.12, we can easily

establish the following well known results.

Corollary 1.3.13 Suppose that MT is an n-manifold with
boundary, n 6 Z+. Then there is an embedding h of

M? x [0,1] into Mn such that h(x,0) a x for all x 6 Mn
and h(hp x [0,1])is closed in Mn.

13

Corollary 1.3.14 Suppose that Mn is an n-manifold with
boundary, n 6 Z+. Then an abstract collaring M2 Of Mn is
homeomorphic to En.
pefinition 1.3.15 Suppose that Ln is collared in up. A
collaring CLn of Ln in up is called a tapered collaripg
Kipp support E, if given an Open set C, Ln C-G C Mn, then
there is a homeomorphism h of Mn onto itself such that

i) Ln c h(CLn) c.c and

ii) h(x) s x for all x 6 (Ln 0 (Mn - F)).

Theorem 1.3.16 If Ln is collared in HP, then Ln has a
tapered collaring CLn in HF. Furthermore, if the closure
Of EdLn in up is compact, then the support Of CLn may be
chosen to be compact.

2522;. Let Ln be collared in he and let A = EdMnLn.
We may assume that A i C, since otherwise CLn 2 Ln. If
CLMnA is compact, then set X a Mp; otherwise, let X be the
one point compactification Of up. In any case X can be
embedded in RP for some p 6 2*. Consider x and all sub-
spaces Of X as embedded in Rp. We note that i) A is local-

ly compact; ii) Cl A is compact and C a Cle - A is either

X
empty or compact (see p. 2&5 Of [9]); and iii) there is an
embedding h Of A x [0,1) into X such that h(a,0) = a for
all a 6 A. Thus there is a continuous function

f : Cle ~ [0,%) such that f(x) = 0 if and only if x e C

and iv) and v) of 1.3.12 hold. Suppose that G is Open in

14

Mn and that Ln C G. Since Mn is locally compact, Mn is em-
bedded as an Open subset in X and thus G is Open in X and
A C-G. Let h1 be an embedding Of P(O,f;Cle) into X such
that hl(y,0) - y for all y E ClXA and such that h1(x) =
h(x) for all x E P(0,f;A). Then there is a homeomorphism
g Of h1(P(O,f;C1xA)) onto itself such that
g(hl(TP(0,%f;A))) C-G and g restricted to

hl(G(O;Cle) U G(f;C1xA)) is the identity. Let F =
h1(P(O,f;ClXA)) 0 Mn. Then F is closed in we and FrMnF a
hl(G(O;C1XA) U G(f;Cle)) O Mn. Therefore g extends to a
homeomorphism g1 of up onto itself such that gl(x) a x

for all x E (Ln U (Mn - F)). The required tapered collar-
ing CLn of Ln is obtained by setting CLn -

Ln 0 h1(TP(O,%f;A)) . Ln U h(TP(O,%f;EdMnLn)). If CanA
is compact, then clearly F is compact and the theorem is

established.

Corollary 1.3,12 If Ln is collared in up and G is an Open

set such that Ln c-G C-Hp, then Ln is collared in G. Also,
n 0

if L is collared in up and the pair (Mp,Mp) is contained

in the pair (Q9,Qn), then Ln is collared in Qn.

The following lemmas lead to a theorem which gives
sufficient conditions for Ln to be collared in up.

Lemma 1.3.18 Let X be a tOpological space. There is a
homeomorphism h Of X X ([-1,0] X [O,1)) onto

X X ([-1,1) x [0,1)) such that h restricted to

15

(X x {-1} x [0,1) U X X [-l,0] x {0}) is the identity and
h carries X x {0} x [0,1) homeomorphically onto
X x [0,1) x {0}.

25292. There exists a homeomorphism g Of
[-l,0] x [0,1) onto [-l,1) x [0,1) such that g restricted
to ({-1} x [0,1) U [-1,0] x {0}) is the identity and g car-
ries {0} x [0,1) homeomorphically onto [0,1) x {O}. The
map h a X x ([-1,0] x [0,1)) * X x ([-l,1) x [0,1)) given
by h(x,y) = (x,g(y)), x 6 X, y 6 ([-l,0] x [0,1)) is the

required homeomorphism.

Lemma 1.3.19 Let Ln be a relative Mn manifold, n 2.29 such
that an = EdLn ; n. Then there is an embedding h of the
pair (EdLn x [0,1),(EdLn n Mn) x [0,1)) into (Mn,Mp) such
that

i) h(EdLn x [0,1)) CLn and

ii) h(x,0) s x for all x e EdLn.

2329;. Let Ln be a relative Mn n-manifold, n g_2,
such that EdLn ¢ ¢. Let En'1 a EdLn. If Sn'l - a, then
since En"1 c.Ln, the result follows easily from 1.3.13.

Now suppose that En-l # C and let Qn'l = Ln n Mn. Then
En'l = én-l. It follows from 1.3.13 that there is an em-
bedding gl : én-l x [-1,0] d En"1 such that

$102211“1 x [-l,0]) is closed in En”1 and g1(x,0) a x for all
x 6 En-l. Also there is an embedding g2 : én-l x [0,1) *

Qn"1 such that g2(x,0) a x for all x 6 En-1 = én-l. Let

n-l - ° -
P = En 1 U g2(En 1 x [0,1)). Note that since

16

n-l
P . It follows from 1.3.13 that there is an embedding

x [-l,0]) = F is closed in 3““ , F is closed in
g3 Of Pn“1 x [0,1) into Ln such that g3(x,0) = x for all
x e Pn‘l and g3(Pn‘1 x [0,1)) 0 Ln = g3(Pn"1 x {0}).
Let f be a homeomorphism Of énél x ([-1,0] X [0,1)) onto
EhIl x ([-1,1) x [0,1)) with the properties given in

1.3.18. Define g : én-l x [-1,l) * Ln by

g1(x,t), t 6 [—1,0]
8(X.t) = { }

g2(x,t), t 6 [0,1)
Define a homeomorphism hl of (g1 (énJl x [-l,03)) X [0.1)
onto (g(EnIl x [-1,l))) x [0,1) by hl a (g,id)f(gil,id),
where id is the identity map on [0,1). Note that h1(y,t) =
(y,t) if i) y e g1(En:1 x {-1}) or ii) if t . 0. Let
F1 2 En“1 - g]_(‘i7‘.n'1 x (-l,0]) and F2 8 gl(}.3:n'i1 x [-1,0]).
Then F1 and F2 are closed in En-l and F1 n F2 2 F3 -
g]_(En-‘1 x {-l}) which is also closed in En-l. Define
hz : En-l x [0,1) * Pn“l x [0,1) by

(xft)9 I 6 F1 -
h2(X,t) 3 { o
n-l
Then h2 is a homeomorphism onto P X [0,1), and h = 83h2

is the required embedding Of
(EdLn x [0,1), (EdLn n h“) x [0,1)) into (Mn,ne).

Remark. Note that it follows from 1.3.13 or 1.3.19 that
a collaring CLn Of Ln in up is an Open subset Of Mn.

Theorem 1.3.20 Let Ln and Mn be n-manifolds, n 2.2, such

17

that Ln is closed in an. Then Ln is collared in up if
1) Ln c.§n and Pn - Mn - Int Ln is an n-manifold or
ii) Ln is a relative Mn n-manifold, EdLn w ¢. and
Pn - Mn - Int Ln is a relative n-manifold.
2529;. If 1) holds, then EdLn c.2n and the result
follows from 1.3.13. If ii) holds, then EdLn - Fan =
FrPn = EdPn. Since EdLn - EdPn, the result follows by
applying 1.3.19 to Pn.

- -l
f : En 1 y R1 by setting f(x) = (1 - iglxfi)%. Then f is
continuous and f(x) - 0 if and only if x E Sn-Z. For each

t e [0.1] define ft : En"l ” R1 by ft(1) I tf(1)-

Definition 1.3.21 Let in? s {x e Rn|x1 g_0}, n 3,2. For
each t 6 [0,1] define BR(t) = P(0,ft;En'1). an(t) =
Bn(t) n SHE, and 33‘1(t) = éBn(t) n iii.

 

 

 

Figure 1.1

18

Theorem 1.3g22 Let s, t e [0,1] such that 0 < s < t < 1.
Then there is a homeomorphism h(s,t) Of Bn(l) onto itself
such that

i) h(s,t)anu) - id,

11) h(s,t) carries Bp(t) onto Bn(s),

iii) g(s,t) a h(s,t)|%Bn(l) is a homeomorphism Of
%3n(1) onto itself and g(s,t)l B§'1(l) is a
homeomorphism Of 83-1(l) onto itself,

iv) g(s,t) carries éBn(t) onto SBn(s).
2222;. This is an immediate consequence Of 1.3.11.
Remark. Note that for any 3, t 6 (0,1), P(fs,f;En-1) is
homeomorphic to P(0,ft;En-l) and that Bn(l) - En n éRn.

b. Piecewise linear manifolds

The terminology that will be used for simplicial com-
plexes is essentially that used by Zeeman in [15], but is
modified to agree with the terminology used by Hudson and
Zeeman in [11].

By an n-simplex tn, 0 5,n, is meant the convex hull of
n+1 linearly independent points (vertices) {VJ}J:O in RP,

n 5,p. By an r-face tr of tn, denoted by tr < tn, is meant
the convex hull Of r+l distinct points Of {vJ}J:o.

A simplicial complex K of RP, p g_l, is an at most
countable collection Of simplexes Of Rp such that i) if
t E K, then all faces of t are in K, ii) if s, t E K, then

s n t is a common face Of s and t, and iii) each vertex Of

19

K is the face Of at most a finite number Of elements Of K.
L is called a subcomplex Of X if L is a simplicial complex
and L c K. If tn is an n-simplex, n 2,0, let Eg a

{SIS < tn} and £n+1 = {sls < tn+l’ s # tn+l}‘ If s, t are
simplexes in Rp, then s and t are said to be Joinable if
the union Of their vertices forms a linearly independent
set Of points in Rp. If s and t are Joinable, then the
Jng Of s and t, denoted by st, is defined to be the sim-
plex spanned by the union Of their vertices. For t 6 K,
the set st(t,K) = [s 6 Klt < s} is called the gpgp_of t in
K; the subcomplex lk(t,K) = {s e KIS is Joinable to t and
st 6 K} is called the 1135 of t in K.

Let K be a simplicial complex in RP. The polyhedron
determined by K, denoted by IKI, is the set |K| = tht with
the weak tOpology determined by the simplexes Of K. A com-
plex K' is called a subdivision Of X if IK'I = IX) and each
simplex Of K' is contained in some simplex Of K.

Let K and L be simplicial complexes in Rn and RP
respectively. A continuous function f : IKI * ILI is
called simplicial if f(s) 6 L for all s 6 K; f is called
piecewise linear, denoted by PL, if there are subdivisions
K' Of K and L' of L such that f : IK'I w IL'I is simplicial.
If g is a homeomorphism Of ILI onto X and h is a homeomor-
phism of IX) onto I, then f : X r Y is called piecewise
linear if h-lfg is a PL map of [LI into )KI.

Henceforth, PL will be used for the term piecewise

20
linear.

Definition 1.4.1 Let tn be an n-simplex, n g_0, and K a
complex in Rp. IKI is called a combinatOpLgL_n-cell, if
there is a PL homeomorphism f Of lthl onto lxl. IKI is
called a combinatorial n-sphere, if there is a PI.homeomor-

phism f of |tn+1| onto IK|.

Definition 1.4.2 An n-manifold Mn, n 2.0, is called a 2L
n-manifold if there is a homeomorphism f Of IKI onto up
where K is a complex in RP, and such that if n 2_1, then
Ilk(v,K)| is a combinatorial (n-1)-ce11 or a combinatorial
(n-l)-sphere for all vertices v Of K. The pair 1§,gl is
called a gL_triangulation Of Mn.

It is a well known fact that if (K,f) is a PL triangu-
lation Of Mn, then any subdivision K' of K yields a PL tri-
angulation (K',f) Of Mn. Also if Mn is an Open subset Of
an or tan, then there is a complex K of an such that (K,id)
gives a PL triangulation of up where id is the identity map.

Definition 1.4.3 Let Mn be a PL n-manifold with PL triangu-
lation (K,f). An m-manifold Ln, 0 $_m45_n, which is embed-
ded as a subset of an is called 2;,gg_gfl if there is a sub-
division K' Of K and a subcomplex L' of K' such that
(L',fI|L'I) is a PL triangulation Of f(IL'I) - Lm. Note
that if Lm is PL in Mn, then Lm is necessarily closed in Mn.

A manifold Lm embedded in up may be a PL m-manifold,

21

but may not be PL in ME. For example, let (K,id) be a PL

triangulation Of R2, where K is a simplicial complex in R2

and id is the identity map. Then S1 is a PL l-manifold,

1 2

but S is not PL in R .

Theorem 1.4.4 Let Mn, n g_2, be a PL n-manifold and let Ln
0 o

be a PL n-manifold in MS. If i) Ln c Mn or ii) Ln n M“ -

Qn-l

Mn.

is a PL (n-1)-manifold in Mn, then Ln is collared in

25222. This result is easily established and the
method Of proof is only outlined. The first step is to use
the method employed in the proof of lemma 1? of chapter 3
of [15] to show that Mn - Int MnLn is a PL n-manifold in
up. If ii) holds, then i) applied to Qp-l c in and
Qn’l c-Ln shows that both Ln and up - IntMnLn are relative
Mn n—manifolds. Since Ln is closed in Mn, the fact that Ln

is collared in Mn follows from 1.3.20.

Lemma 1.4.5 Let Mn be a connected n-manifold, n 2.2, and
C CrMn a prOper compact subset. Then there exists a domain
D C’Mn such that C C D and C1D is a proper compact subset
Of up. Furthermore, if C C En, then D may be chosen so
that C1D C Mn.

22223. This is easily established. Let x 6 Mn - C.
Then Mp - x is a connected n-manifold and the result fol-
lows from the fact that Mn - x is connected, locally con-
nected, and locally compact. Note that the result is not

true for n a l.

22

If K is a simplicial complex, let Sd K, j 6 Z+ denote

J
the Jth barycentric subdivision of K.

Theorem 1.4.6 Let Mn, n 2_2,be a connected, PL n-manifold
with triangulation (K,f), D a domain Of Mn, and C a prOper
compact subset Of D. Then there is a compact, connected,
PL n-manifold Ln in Mn such that
i) C C-Intann c Ln c D, Ln 2 D,
ii) Ln = f(ILI) where L is a subcomplex of SdJ(K),
362*.
iii) Ln c in if C c fin, and
iv) Ln n Rn = Ln n D is a PL (n-l)-manifold in My
if C n in # fl.

2522;. Since K is locally finite, it may be assumed
that up is embedded in some Rp as a closed subset and that
Mn = lKl. It follows from 1.4.5 that there is a compact,
connected set Cl such that C C-IntMnC1
Since n g_2, it may be assumed that Dpfi Mn. Let

cc1 c.D, C1 2 D.

0 < e g d(Cl,Mn - D) and Q . {x e hP|d(x,C1) < e}, where
d is the Euclidean metric on RP restricted to up. Note
that Q C_D and that if C n M9 = 0, then 6 may be chosen so
that Q C En. The existence Of Ln is now established using
the terminology and results of [11].

If J is a simplicial complex, and X C IJI, let
N(X,J) e {s e Jls < t, t n x # fl}. Let L1 = N(C1,K). Since
C1 is compact, L1 is a finite simplicial complex with

C1 C IntMnILll. Therefore there is a q 6 Z+ such that if

23

L = N(Cl,quLl), then for all 3 Z,q

i) mesh SdJL1 < 6/4 and

ii) N(|L|,SdJLl) . N(|LI,SdJK) andIN(|LI,SdJL1)Ic.Q.
Suppose that C n Mn # fl. Then there is an n-simplex tn 6 L
such that tn has an (n-l)-face tn-l C Mn. Let b be the
barycenter of tn-l’ There exists an n-simplex t5 6 SdzL
such that b 6 t5, t5 n h“ = t£_l is an (n-1)—face of ti,
and |t£| C.IntmnIEhl. If C n fin a fi. let R -

Sd L - {t£,t£_l} and s = t' if C n Mn a 2, let a = Sd L

2 n-l’ 2
and S = a. In either case, |s| is link collapsible on |s|.
Furthermore, if C n hp ¢ ¢. then |s| n h“ is link collaps-
ible on |s| n in = |tg_1|. Let J = N(|RI - |s|,SdQ+¢K).
Then J is a subcomplex Of N(ILI,qu+uL1) and thus IJI C»Q.
From theorem 1 Of [11] it follows that IJIis a compact,
connected PL n-manifold in M? such that IJI n up = a if

c n up = a, and IJI n in is a PL (n-l)-manifold in up if

C n Mn # C. If C O Mn 2 a, let Ln = IJI; otherwise, let

P - J u Sd t; and set Ln = IPI. If it is the case that

2
C n Mn # C, then C CIntMnLn CLn C-Q and Ln is a compact,
connected PL n-manifold in up such that Ln n h? is a PL
(n-l)-manifold in MI. Since P and J are subcomplexes

Of qu+4x, Ln satisfies ii) through iv). Since n 2,2, in
the proof we have assumed that D # Mn. Thus since Mp is

connected and Ln is compact, Ln f D and i) is also satis-

fied.

CHAPTER II

CONNECTED MANIFOLDS
WHICH HAVE A GENERATING DOMAIN

In this chapter we will give a characterization Of
those connected manifolds which have domain rank 1. It is

clear that such a manifold must be without boundary.
1. Characterization

pefinition 2.1.1 An n-manifold Mn, n 2.1, is said to have
Euclidean compact subsets if for each proper compact set

C C Mn, there is a homeomorphism h of the pair (C, C n Mn)
into the pair (éRn,%Rn).

For n 2.1, let (T,id) be a fixed PL triangulation Of
Rn, where T is a simplicial complex in Rn such that ITI =
Rn and id denotes the identity map. Throughout the remain-
der of this work, it will be assumed that an has this fixed

PL triangulation.

pefinition 2.1.2 Let (T,id) be the given fixed PL triangu-
lation Of Rn, n 2.1. The set M(T) a {LIL is a subcomplex
of some SdkT and ILI is a compact, connected, PL n-manifold

in an} is called the set 2; regular submanifolds Of T.
If L 6 M(T), then L is a finite simplicial complex in

24

25

an. Therefore it is easily seen that M(T) is a countably

infinite set.

Theorem 2.1.3 Let Mn, n 2.2: be a connected, non-compact
n-manifold without boundary such that Mn has Euclidean
compact subsets. There exists a sequence {ME};;1 Of com-
pact, connected n-manifolds such that
i) N: is collared in HF, k 6 Z+,
ii) there is an Lk 6 M(T) and a relative homeomor-
phism hk Of ILkI onto Mg, k 6 2*,
iii) Mg C.IntM:+l, k 6 2+, and
iv) Mn = U .
k=l

£1223. Since Mn is a non-compact, connected n-mani-
fold, there exists a sequence {Pk};;1 Of non-empty compact
subsets Of Mn such that Fk is a proper subset Of Fk+1 for
all k 6 Z+ and M3 = ggle. We now Obtain the sequence
{Mi};;l by recursive construction.

a) Suppose that k = 2. Since MP has Euclidean compact
subsets and in a 2, it follows from 1.4.5 that there is a
prOper domain H2 of Mn with F2 c.R2 and a relative homeo-
morphism f2 Of H2 onto a domain f2(R2) Of Rn. As a conse-

quence Of 1.4.6, there is an L 6 M(T) such that f2(F2) C

2
-1
IntRnILZI c |L2| c f2(H2). Define h2 = f2 ||L2| and set
n
M2 2 h2(IL2I). Since IL2| is collared in RP and f2 is a
relative homeomorphism Of Hz-into an, it follows that n2 is
a relative homeomorphism Of ngl onto M2, and that M: is

collared in Mn. Now there exists an n-cell Cn C-IntBnILzl

26

such that Cn a |L1| where L1 6 M(T). Let h1 = h2||L1| and
set M? = h1(|L1|). Then hl is a relative homeomorphism Of
ILlI onto ME, M? is collared in ME, and ME C-IntmnMg.

b) Suppose that k > 2 and that a finite sequence
{M§}§;1 of compact, connected n-manifolds has been con-
structed such that

v) i) and ii) are satisfied, 1 §_J $_k-l,
vi) PJ c: IntMnMS‘, 1 < 3 5 k-l, and

vii) iii) is satisfied, 1 5.3 $_k-2.

Let Ck = Fk U M£_1; then Ck is compact and Ck # Mn. Again,
it follows from 1.4.5 that there is a proper domain Bk Of
M? with Ck C Hk and a relative homeomorphism fk Of Hk onto
a domain fk(Hk) Of Rn. By 1.4.6 there is a complex Lk 6
M(T) such that fk(Ck) c IntnnlLkl c |Lk| c: fk(I-Ik). Define
hk = fil IILkI and set ME = hk(|Lk|). Then hk is a relative
homeomorphism Of lLkI onto Mi, u: is collared in up, and
M111 C Intmnnfl. It is clear that the finite sequence of
compact, connected n-manifolds {M§}JE1 satisfies the recur-
sive hypothesis for k+l. Therefore it is possible to con-
struct a sequence {M£T;_l Of compact, connected n-manifolds

with prOperties 1) through iv).

Lemma 2.1.4 Let (T,id) be the given fixed PL triangulation
Of Rn, n sz. and let {L3}?;1 be an enumeration Of M(T).
There exists a domain D? Of Rn such that if G is an Open
set and ILJ| C-G for some J 6 Z+, then there is a homeomor-

n
phism h Of R onto itself which is the identity outside a

27

prOper compact set and ILJI C-h(D§) C'G.
Proof. Let (K,id) be a fixed PL triangulation Of
n n n
Q = {x 6 R le > 0}, where [Kl = Q and id is the identity
+
map. For each j E Z , let C(j) = {x 6 Rnlxl = l/j} and for

1,3 6 2+, i < 3, let Sl(j,i) = {x 6 RnIl/j 5.x 5_l/i}.

1

Set q a (1,0,...,0) and choose an (n-l)-simplex tn-l C C(1)
O

such that q 6 Itn_1|. Let tn be the n-simplex which is the

convex hull Of p U |Eh-l" where p = (0,...,0). For each

+ +
j E z , let E = tn n Sl(j+l,J). Then for all j 6 z , EJ

J
is a PL n-cell in Qn and in Rn.

We now begin the construction Of DE. For each j E Z+,
there exists a PL homeomorphism 1J of Rn onto itself which
is the identity outside a proper compact set such that
13(ILJ|) - M? C-IntRnEJ. Note that M? is a PL n—manifold
in both Rn and Qn. Let In be the unit n-cube, and set
A = {x E Inlxn = 0} and B = {x E Inlxn = 1}. By recursive
construction a sequence {fal:=1 Of embeddings Of In into
H“ can easily be constructed such that

a) [fJ(In)}”=1 is a disjoint collection Of PL n-cells

in an and on;

b) for all j 6 2+, fJ(In)C'IntBn(EJ U E3+1lt

c) for all j 6 2+, fJ(In) 0 EJ = E3 is a PL n-cell in

an and Q2, and fJ(In) n EJ+1 a P3 is a PL n-cell in
Rn and QT;
d) for all j E 2*, fJ(In) n C(j+l) is a PL (n-l)-cell

in Rn and Qn;

28

e) for all 3 6 2+, fJ(In) n M? = fJ(A) is a PL

(n-1)-cell in Rn and Qp, fJ(A) C~Mng and
+ n
f) for all j E z , fJ(I ) n M§+l . fJ(B) is a PL

(n-1)-cell in an and on, fJ(B) c ޤ+1.

 

 

 

 

 

 

 

Figure 2.1

Let Mp = (JQIME) U (Jfilf3(ln)). It is easily seen
that up is a PL n-manifold in on. It follows from 1.4.4
that up is collared in Qn and hence from 1.3.17 that fin is
collared in Rn. Since Can(EdBnMp) = Ednnnn U p is compact,
we can choose a tapered collaring D? of up in Rn such that
the support of D: is compact. NOw suppose that j 6 Z+ and
that C is open in an, ILJ| c.G c.Rn. Let C1 s 13(6); then

G1 is Open in Rn and M? c Gl' Define
P1 {as J 3 1 }
j " n 3-1 3-1 .
fJ_1(I ) U (qglsq) =1F3_1 U (qglEq). J > 1
2 n °° co
PJ - fJ(I ) U 4q23+13q1 U p = 33 U (qu+lsq) U p, and
N? = an - Intnnug. Since |L1| = ILkl for some k > 1,

29

we may assume that j > 1. Then N? is a PL n—manifold in
i and P3 are PL n-cells in N? and Rn. Furthermore

Q? = P? - (boundary of EdNnPfi) is a collared relative N?
J

Rn and P

j =(boundary of
Edanj) = EdN fir n Nn= sn‘l , k = 1,2, where 33—1 =

NJ n-l
2

n-manifold and Cl Nn(EdNan) - EdNan

boundary of fj-1(B) and S = boundary of fJ(A).

   
  

 

e2(Bn(1))

Figure 2.2

Therefore it follows from 1.3.12 and the remark following
1.3.22 that there exist embeddings g1 and g2 Of Bn(1) into
N? such that

a) gloanun n 82(Bn(1)) = 9!;

b) ak(Bn(0)) = p}; n if; = p]; n iv“, 1. = 1,2,
o) sk(Bn(1)) n M? = gk(Bn(O)), k = 1,2; and

d) sk(Bn(%)) P?, k = 1,2.
Note that Mn C M? U P% U Pg. Now since G1 is Open and
sk(Bn(0)) C G1, k = 1,2, there is a t e (o,%) such that
sk(Bn(t)) c G1, k = 1,2. Define g : R“ w an by
{a x i1 81(Bn(l)) u gamma»

g(X) 1 n
Skh(t9%)81-{ (I), x E 8k(B (1)), k = 192

30

Then g is a homeomorphism Of Rn onto itself which is the

identity outside a prOper compact set; that is g E CH(Rn).
Furthermore g(Mn) C G1
is Open and Mn C G

— = -l o
and gIME — id. Let G2 8 (G1).

then G Since D? is a tapered collar-

2 2'
ing of Mn with compact support, there is a homeomorphism

hl e CH(Rn) such that h1|wn = id and h1(D§) C.G Let

2.
h = 13 ghl. Then h e CH(Rn) and ILJI c h(D?) CVG; and the

proof is complete.

Henceforth D? will always denote the tapered collar-

ing Of My defined in the proof Of the last lemma.

Theorem 2.1.5 Let Mn, n 2.2, be a connected, non-compact
n-manifold without boundary such that Mn has Euclidean
compact subsets. Then D? is a generator Of Mn.

2322:. As a consequence of 2.1.3 there is a sequence
{ME};.1 Of compact, connected n-manifolds such that i) M:
is collared in up, k e z+; ii) there is an Lk E M(T) and a
relative homeomorphism hk of ILkI ongo ME, k e z+; iii)

g: c.IntM:+l, k 6 Z+; and iv) Mn = kglhfl. It follows from
1.3.17 that there is a collaring CH: of M: in Inth:+1 and
a collaring CILkI of ILkI in Rn. From 2.1.4 it follows

that there is a homeomorphism f e CH(Rn) such that

k
ILkI C fk(D§) C-CILk|; and from 1.3.9 it follows that there
is a relative homeomorphism hi Of CILkl onto CHE which is
an extension of hk. Then for all k e z+,

n: C h§(fk(D$)) C-IntM:+1. Therefore D? generates Mn.

31

Theorem 2.1.6 Let Mn, n 2,2, be a connected n-manifold
without boundary. Then up has a generating domain if and
only if my has Euclidean compact subsets.

£3222, Suppose that MD has a generating domain D,
and let K c Mn be a prOper compact subset and x 6 Mn - K.
Since n 2'2, G1 = Mn - x is a domain Of Mn. Since D is a
generating domain for NI, there is a domain D1, with
K cDl c.G1, and a homeomorphism f1 Of D onto D1. Since
Mn is an n-manifold, there is a proper domain G2 of Mn and
a homeomorphism g Of ERn - éRn onto 62. Since D also gen-

erates G2, there is a domain D2 C.G and a homeomorphism

1 -1
2 f2f1 °

homeomorphism Of (K,¢) into (éRn,%Rn), and thus Mn has

2
f

Of D onto D2. Let f = g' Then fIK induces a

Euclidean compact subsets.
Now suppose that Mn has Euclidean compact subsets.

If D is a prOper domain of Mn, then D is a connected, non-

compact n-manifold without boundary which has Euclidean

compact subsets. It follows from 2.1.5 that D? generates

D. Since D was arbitrary, D?

Mn.

is a generating domain for

Corollary 2.1.2 Suppose that Mn, n 3.2, is a closed con-
nected n-manifold. Then Mn has a generating domain if and
only if MP is an n-sphere.

2392;. If Mn is an n-sphere, then clearly Mp has a
generating domain. Now suppose that up has a generating

domain. Let B“ be an n-cell in Mn such that E? is bi-col-

32

lared in Mn. Since Mn has Euclidean compact subsets, it
follows from 1.4.5 that there is an embedding h Of Ln =

Mn - IntBn into Sn such that h(Ln) = h(Bn) is a bi-collared
(n-l)-sphere. Since Frsnh(Ln) = h(sn), it follows from [2]
that h(Ln) and Ln are n-cells. Therefore Mn is an

n-sphere.

Although each Open connected subset Of Sn has D? for
a generating domain, it is not true that every connected
n-manifold with a generating domain is homeomorphic to a
domain of Sn. Examples Of such manifolds are considered in

[12].

2. Compact, connected n-manifolds with

in w 2 and DR(En) = 1

gheorem 2.2.1 Let Mn, n 3,2, be a compact, connected
n-manifold such that Rn # C and DR(En) = 1. Then my can be
embedded in an such that in is bi-collared in Rn.

£5933. Let M: = Mn x [0,1) Uc Mn be an abstract col-
laring Of Mn. It follows from 1.3.14 that M2 has a genera-
ting domain. Since En c.E: is a prOper compact subset and
Mn is bi-collared in M2, it follows from 2.1.6 and 1.4.5

that an can be embedded in an in the required manner.

Definition 2.2.2 Let {B:}k:1, q 6 z+, be a disjoint col-
lection Of n-cells contained in Sn, n 2,2, such that BE,

1 $_k 5.q, is bi-collared in Sn. A space X which is homeo-

33

q
morphic to Sn — (kngfl) is called a punctured n-sphere
(with q holes). A space X which is homeomorphic to
Sn - (killntBE) is called a compact punctured n-sphere

(with q holes).

The following result follows immediately from 2.2.1,
2.1.7, and the fact that a l-sphere is the only closed

l-manifold with domain rank 1.

Corollary 2.2;3 Let Mn, n 3.2, be a compact, connected

. O
n-manifold such that Mn # C and DR(Mn) = 1. If DR(C) a 1
for all components C Of Mn, then up is a compact, punctured

n-sphere.

CHAPTER III

CONNECTED MANIFOLDS WITH BOUNDARY
WHICH HAVE DOMAIN RANK 2

Because of invariance Of domain in manifolds, it is
clear that an n-manifold Mn with boundary has DR(Mn) g_2.
In this chapter we give a characterization Of those connec-

ted n-manifolds with boundary that have domain rank 2.
1. Characterization

For n g_l, let (Tl,id) be a fixed PL triangulation Of
SRn, where T1 is a simplicial complex Of Rn such that
|T1| . tan and id denotes the identity map. Throughout the
remainder Of this work, it will be assumed that as“ has

this fixed PL triangulation.

Lefinition 3.1.1 Let (Tl,id) be the given fixed PL trian-
gulation Of SRn, n g_1. The set R(T1) = {LIL is a subcom-
plex of some Sdle, |L| is a compact, connected PL n-mani-
fold in San which is a relative tan n-manifold, and |L| n
in“ is a PL (n-l)-manifold in is“) is called the set of

re lar, relative submanifolds Of T1.
Note that R(T1) is a countably infinite set.

Theorem 3.1.2 Let Mn,n Z,2, be a connected, non-compact

34

35

n-manifold with boundary such that Mn has Euclidean compact
subsets. There exists a sequence {Mi};=l of compact, con-
nected, relative Mn n—manifolds such that
i) Mi is collared in as, k e z+;
ii) there is an Lk 6 R(T1) and a relative homeomor-
phism hk of ILKI onto ME, k 6 Z+;
iii) h: c.1nth§+1, k 6 2+; and
iv) ”n = kglmi°

2322:. Since Mn is a non-compact, connected n-mani-
fold with boundary, there exists a sequence {kaigl Of non-
empty compact subsets Of M? such that Fk C'Ek+1 and Fk n
fin # C for all k 6 2+; and up = égle. We now Obtain the
sequence {Miffigl by recursive construction.

a) Suppose that k =:2. Since up has Euclidean compact
subsets and N? i C, it follows from 1.4.5 that there is a
proper domain R2 of Mn with F2 C H2 and a relative homeo-
morphism f2 of H2 onto a domain f2(H2) of an“; that is, f2
induces a homeomorphism Of the pair (R2, R2 0 En) into the
pair (ERn, %fin). As a consequence Of 1.4.6, there is an
tRn|L2| C ILZI c.f2(32).
= fglllel and set M; - h2(|L2I). Since |L2| is

L2 6 R(Tl) such that f2(F2) C Int
Define h2

collared in SRn and f2 is a relative homeomorphism Of H2
into SRn, it follows that h2 is a relative homeomorphism Of
|L2| onto M2, and that E2 is a relative Mn n-manifold which
is collared in MI. Now there exists an n-cell Cn C

Int%Rn|L2| such that 0n = |L1| and |L1| n as“ is a PL

36

(n-1)-cell in as“, where L e 3(T1). Let h1 = h2||L1| and

1
set M? a h1(|L1|)- Then h1 is a relative homeomorphism Of

ILll onto ME, M? is a relative Mp n-manifold which is col-
n n
lared in M , and M1 mn

b) Suppose that k > 2 and that a finite sequence

k-l
{Mj}j=l Of compact, connected, relative Mn n-manifolds has

C Int mg.

been constructed such that
v) i) and ii) are satisfied, 1 5_j §_k-1;
n
c _ .

vi) FJ IntMnMj’ l < j 5,k 1, and

vii) iii) is satisfied, 1 5.3 S_k-2.
Let Ck = Fk U Mfi-l; then Ck is compact, Ck O Mp # ¢9 and
Ck # Mn. Since up has Euclidean compact subsets, it fol-
lows from 1.4.5 that there is a proper domain Rk Of Mn with

Ck C'Ek and a relative homeomorphism f of Bk onto a domain

k
fk(Hk) Of ERn. By 1.4.6 there is a complex Lk 6 R(T1) such
thit fk(Ck) c.1ntéanlLkl c ILkI c.fk(Hk). Define hk =

f; IILkI and set ME = hk(|Lk|). Then hk is a relative
homeomorphism Of ILkI onto Mi, n: is a relative Mp n-mani-
fold which is collared in Mn, and ME_1 C-IntMnEE. It is

n

clear that the finite sequence {Mj}j:1
ted, relative Mn n-manifolds satisfies the recursive hypo-

Of compact, connec-

thesis for k+l. Therefore it is possible to construct a
n
sequence {Mk}§=l Of compact, connected, relative Mn n-mani-

folds with properties 1) through iv).

mema 3.1.3 Let T be the given fixed PL triangulation Of

1
%Rn’ n 2,2, and let {Lj}:;l be an enumeration Of R(Tl).

37

There exists a domain D2 of %Rn, D:

G is an Open set and ILJI C G for some j 6 Z+, then there

n as“ C C, such that if

is a homeomorphism h Of %Rn onto itself which is the iden-

tity outside a prOper compact set and ILJI C h(Dg) C-G.
£5222. Let (K,id) be a fixed PL triangulation Of

on e {x e aanlxl > 0} where IKI = on and id is the identity

map. For each j 6 Z+, let C(j) = {x 6 éRnIx1 = l/j} and

for i, j 6 2+, 1 < j, let sl(j,i) - {x e Skull/j 5_x1 5.1/1}.

Set q = (1,0,...,0) and choose an (n-l)-simplex tn-l C-c(1)

such phat tn-l O %Rn = tn—Z is an (n-2)-face Of tn-l and

q 6 |th_2|. Let tn be the n-simplex which is the convex

hull Of {p} U |th_l|, where p = (0,...,0). For each J 6 2+,

4.
let E - tn 0 sl(j+1.J). Then for all j e z , E is a PL

3
n-cell in %Rn and Qn; and E

J

j n SRn is a PL (n-l)-ce11 in

SRn and Q n.

We now begin the construction Of Dn For each j 6 Z+,

2.

there exists a PL homeomorphism 1 e ca(%an) such that

J
u C .

13(ILJ|) M3 IntéanEJ Note that M? is a PL, relative

%Rn n-manifold and also a PL, relative Qn n-manifold. Let

n

I be the unit n-cube, A a {x 6 Inlxn = 0}, B a {x 6 Inl

xn = 1}, and C = {x 6 Inlxn_1 = 0}. By recursive construc-

tion a sequence {f of embeddings Of In into ERn can

Q
J}J=1
easily be constructed such that

a) [fJ(In)}:;1 is a disjoint collection Of PL n-cells
in éRn and Qn;

b) for all j 6 2+, fJ(In) O %Rn = fJ(C) is a PL (n-l)-

d)

e)

f)

s)

h)

38

cell in éRn and Qn;
n
for all 3 6 2+, fJ(I ) C Int%Rn(Ej U EJ+1);
for all j 6 Z+, fJ(In) n E = E' is a PL n-cell in

J J

= F' is a PL n-cell

tan and on, and fJ(In) n E3+1 3

in an” and Q“;
for all a e z+, E' n in“ = Ea'is a PL (n-1)-cell in

J
in” and on, and F3 0 in“ = Fg'is a PL (n-1)-cell in
n n
23 and Q ;

for all j C 2+, fJ(In) n c(j+l) = E?‘1 is a PL
(n-l)-cell in tRn and Qn, and E3-1 0 éRn 3 E3'2 is
a PL (n-2)-cell in éRn and Qn.

for all a 6 2+, fJ(In) n M? - fJ(A) is a PL (n-1)-
cell in is“ and Q9, fJ(A) C-Edianhfi, and fJ(A) 0
tan is a PL (n-2)-cell in is“ and QR;

for all 3 6 2+, fJ(In) n up = fJ(B) is a PL

3+1

n n
(n-l)-cell in %R and Q , fJ(B) C Edianmg+lf and

fJ(B) 0 tan is a PL (n-2)-cell in in“ and on.

_——————“*—“""'7_—.
L_——————j E

 

 

 

 

 

 

 

 

 

 

39

Let Mn = (301M§)U ( U1 f (In)). Then it is easily
seen that up is a PL n-manifold in Qn and that Mn n C“
Mn n as“ is a PL (n-l)-manifold in Qn . Therefore it fol-
lows from 1.4.4 that Mn is collared in Qn and hence from
1.3.17 that Mn is collared in SRn. Since C1%Rn(Ed%RnMn) =
Ed%BnMn U p is compact, we can choose a tapered collaring
D2 Of Mn in tRn such that the support of D3 is compact.
Now suppose that G is Open in éRn, ILji C-G C SRn. Let

CI = 13(6); then C1 is open in an“ and h? c.C1. Define

2,3 -
Pi = ‘{ i- 1 -1 -}
fJ_1(In ) U (U E ) - F' U ( U E ). i > 1

q= -1 q j-1q=1 q
2 - E.
PJ f (In ) U (qUJ+1Eq ) U p - EJ U ( qUJ+lqu U p. and
n = -
Since |L1I = |Lk| for some k > 1, we may assume that j > 1.

Then N? is a PL n-manifold in sRn and N? is a relative iRn

n-manifold. Furthermore, Q? = P? - (EdNnPk

j j n Ed%RnN?) is a

collared, relative N: n-manifold and

k

Cl Ed = EdN P
Ni( Niqi NJ Q3 N3 3
fore it follows from 1.3.12 and the remark following 1.3.22

) - EdNn n Ed%RnNn, k . 1,2. There-
that there exist embeddings g1 and g2 Of (éBn(l) ,Bi-l(1))
into (N?,N? n tan) such that

a) s1<tan<1)) n e2<tan<1ii - a;

n k _ k _ .

b) sk(23 (0)) . PJ 0 M? — P1 0 Ed%RnN?, k _ 1,2,

c) ek(ttn<1>) n n? = gk<ian(o)>, k = 1,2; and

d>qefien-P§k=in.

#0

   
 

82(%Bn<l))

 

p /q
Figure 3.2
Note that Mn c M? U P: U P3. Since 61 is open in tan and

gk(%Bn(0)) c c1, k = 1,2, there is a t e (o,%) such that
gk(%Bn(t)) c cl, k = 1,2. Define f : in“ ~ in“ by
x. x ¢ s1<tsn(1)) U s2<ssn(1))
f(x) = als(t.s)s;1<x). x e 81(%Bn(l)) .
szs<t,t)sgl(x), x e 32(san<1))
Then f e CH(%Bn) such that f(Mn) c cl and f|h§ = id. Let
G2 = f‘1(cl); then c2 is open and Mn c 62. Since D2 is a
tapered collaring of up with compact support, there is a
homeomorphism h1 e CH(%Rn) such that h1|Mn = id and
h1(D§) c oz. Let h - lslfhl. Then h is a homeomorphism
of %Rn onto itself which is the identity outside a proper

compact set and [La] C h(Dg) C G.

Henceforth D3 will always denote that tapered collaring

of Mn defined in the proof of the last lemma.

Theorem 3,1.4 Let Mn, n g 2, be a connected, non-compact

n-manifold with boundary such that up has Euclidean compact

subsets. Then D2 is a generator of Mn.

41

25292. AS a consequence of 3.1.2, there is a sequence
{M§T;=l of compact, connected, relative Mn n-manifolds such
that i) Mfi is collared in Mn, k 6 2*; ii) there is an
Lk 6 R(Tl) and a relative homeomorphism hk of ILkI onto
ME, k e 2*; iii) h: c IntME+l, k 6 2+; and iv) up = £21 .
It follows from 1.3.17 that there is a collaring on: of ME
in IntMil+1 and a collaring CILkI of ILkI in fifln. From
3.1.3 it follows that there is a homeomorphism f, e CH(%Rn)
such that ILkI c.fk(D2) c CILkI, and from 1.3.9 it follows
that there is a relative homeomorphism hé of CILkI onto
cm; which is an extension of hk' Then for all k E Z+,

ME C'hk(fk(Dg)) C Intufi+1. Therefore D2 generates up since

Mn . kEAhfi(fk(Dg)) and hi(fk(Dg)) is open in Mn, k 6 2*.

Theorem 3.1,5 Let Mn, n 2.2, be a connected n-manifold
with boundary. Then DR(Mn) = 2 if and only if Mn has
Euclidean compact subsets.

2323;. Suppose that DR(Mp) = 2 and that {01,92} is a
set of generating domains for M3. Since DR(MF) - 2 and
in fi fi, we may assume that D1 is an n-manifold without
boundary and that D2 is an n-manifold with boundary. iLet
K C Kn be a prOper compact set. If x 6 in, then K U x - K1
is still a prOper compact set. Let y 6 Mn - K1 and set
G1 = Mn - y. Since n 2,2, G1 is a domain of Mn such that
G1 n hp i H. Therefore D2 must generate Cl and thus there

is a domain H1, K1 C Hl C G1, and a homeomorphism f1 of the

pair (D2,D2) onto (111,11l n Mn) = (31,111 0 cl). Since

42

in f 2, there is a homeomorphism g of the pair (éfin,%én)
onto the pair (G2,G2 0 Nn) where 62 is a prOper domain of
Mn. Since D2 must also generate 62, there is a domain
H2 c.62 and a homeomorphism f2 of (D2,EZ) onto (H2,Hz n fin).
Let f = g‘lfzfil. Then fI(K,K 0 Nn) gives an embedding of
(K,K 0 E“) into (%Rn’%fin). Thus up has Euclidean compact
subsets.

Now suppose that Mn has Euclidean compact subsets. If
D is a prOper domain of Mn, then D is a connected, non-
compact, n-manifold which has Euclidean compact subsets.
If 5 = fl, then it follows from 2.1.5 that D? generates D.
If S i g, it follows from 3.1.n that Dn generates D. There-

2
fore DR(Mn) = 2.

Remark. It follows easily from the last theorem that if an
n-manifold Mn, n g_2, with boundary has DR(Mn) = 2, then

each component of Mn is an cpen (n-1)-manifold.
2. Some Special manifolds of domain rank 2

perinition 3.2.1 Let n9, n 2,2, be an n-manifold such that
fin 2 an and in I an'l. Then up is called a §;§_manifold.
Theorem 3.2.2 If Mn is a K-R manifold, then DR(Mn) = 2.

23222. Suppose that up is a K-R manifold, n i 3. Then
it follows from [7] that Mn 2 %Bn and thus DR(Hn) = 2. Now
suppose that n = 3 and let (L',f') be a PL triangulation of
En (the existence of such a triangulation follows from

[1]). As remarked in [8], it is possible to extend (L',f')

43

to a PL triangulation (L,f) of an abstract collaring
M2 . M3 x [0,1) Uc M3 of M3. Let M3, M3 x [0,1), and M3 be

identified with their respective embeddings under the iden-

tification p : M3 x [0.1) + M3 s M2. Suppose that K is a

compact set, K C M3. Since (L,f)|f.1(r.13 x [0,1)) gives a
PL triangulation of N3 x [0,1) 2 %RB, there exists a PL 3-

cell D3 in M2 such that D3 C N3 x [0,1), D3 n N3 is a PL

0 O
2-ce11 D2 in M2, and K n M3 C-DZ. Let 8% denote the
3

c,
in the 3-sphere Si. Since D3 is PL in M2, D3 is a bi-col-
lared 2-sphere in 8%. Therefore Si - Intsgn3 is thus a 3-

1-point compactification of M and consider mg as embedded

cell B3 and E3 = EB. Furthermore, K C B3 and K n N3 =

K 0 E3. Since K n E3 C’BZ C E3, there is a point x 6 E3 - K
such that K C B3 - x. Therefore, there is an embedding h

of (x.x n 23) - (x,x n 23) into (B3 - x, boundaryua3 - x)).
Since B3 - x 2 $33, M3 has Euclidean compact subsets and it
now follows from 3.1.5 that DR(MB) a 2.

n
refinition 3.2.3 Let M , n 2_2, be an n-manifold such that

o
T 0
Mn a an and My has two components, both of which are homeo-

morphic to Rn-l, then up is called a pseudo n-slab. If Mn

1

is homeomorphic to Rn' x [0,1], then Mn is called an

Theorem 3.2.u If M3 is a pseudo 3-Slab, then DR(MB) = 2.

Proof. Suppose that M3 is a pseudo 3-Slab and that Bi

and 2% are the components of M3. Let (L',f') be a PL tri-

#4
angulation of M3. It is possible to extend (L',f') to a PL

triangulation (L,f) of an abstract collaring M3 =

fiB 3 3 3 2 2 2
x [0,1) Uc M of M . Consider M ’Bl’BZ’Rl x [0,1), and
R: x [0,1) as embedded in H2 under the identification

p 3 N x [0,1) + M3 * M2. Note that (L,f)If-1(B: x [0,1))

2 T 3
gives a PL triangulation of Bk x [0,1) = %R , k s 1,2. Let

K c.M3 be a compact set. There exist PL 3-cells D2 in M2,
such that Di C-Biox [0,1), D? n R: I D: is 3 PL 2-cell in
M2, and K n R: C1D: for i = 1,2. Let Si be the 1-point
compactification of Mo and consider M2 as embedded in the
3-sphere sf - M2 U p. Since K1 = K U Di U D: is compact in
M2 2 R3, there is a PL 3-cell B3 in M2 such that K1 C-

IntMgBB. Therefore we may assume that Si has a PL triangu-

2 2
lation (J,s) obtained by extending (L ,f||L I) where ILZI =

- 2
f 1(B3) and L is a subcomplex of some subdivision of L.

3 3 3 3 3 2 2
Let F% a K U D2, F2 = K U D1, D3 = S1 - IntB , 81 = Bl U p,
and $2 = B: U p. For i a 1,2, A? a B3 - IntDi is a PL 3-

3 ‘3' 02 2 3 2

annulus in 81' Singe F1 n D1 CD1 CD1 2 D1 0 S1 and
SE n B3 i 2, A: - D1 lies in some component of A; - F1,
1 a 1,2. Therefore there exist PL 3-cells B2 in Si,
B; c.ai - P1 such that B? n 52 and B? n B3 are PL 2-cells

in Si, 1 a 1,2. Furthermore, we can construct B? and B3
so that B; n 33 - H. Therefore it follows from the results
of chapter 3 of [15], that E; = of U B; U 03 U 33 U D3 is

a PL 3-cell in 83. Let E3 = Si - IntEgg then E% is a PL

2

3 such that K c.E3 K n M3 = K n 23, and

3-ce11 in S1 29

45

K n 23 # 23. Let x 6 S3 - K and set L3 = E3 - x. Then
there is an embedding h of (K,K n E3) = (K,K n 22) into
(L3,L3). Since L3 2 $33, M3 has Euclidean compact subsets
and it follows from 3.1.5 that DR(MB) = 2.

In contrast to 3.2.2 we have the following theorem about

pseudo n-Slabs.

Theorem 3.2.5 Let Mn, n 2'4, be a pseudo n-slab. Then Mn
is an n-slab if and only if DR(Mn) - 2.

23222. If Mn is an n-slab, then there is an embedding
h of (Mp,hp) into (%Rn,%En) and thus DR(Mp) a 2.

Now suppose that DR(Mn) = 2. Let M: -

Mn x [0.1) UG Mn be an abstract collaring of up. Since

M: E Rn, we may assume that M2 = Rn, Mn C Rn and that the
components RE and R; of Mn are bi-collared, closed subsets

' n+1
of En. For n 2.0, let 83 a {x 6 Bn+1|k21|xk| = 1}. Then

83 is a combinatorial n-sphere in Rn+l

of SS in Bn+2

technique which Greathouse used to prove the theorem of

, and the suspension

may be taken to be 83+1. We now employ the

[10]. Let S? - an U {p} be the l-point compactification of

Rn and let S?‘1 a a?‘1 U {p}, i - 1,2. It follows from the

corollary to theorem 2 of [6] that S?’1 is bi-collared in
S? for i a 1,2. Therefore we may assume that S? = 83,

P = (0,...,O,l,0), S?"1 = 88‘1, and that 32-1 lies in the

northern hemisphere of 33 with SE'1 0 32-1 8 {p}, Let Bn-l

be a combinatorial (n-1)-cell such that BF‘l c.33‘1 and

#6

° 1

p 6 En. . Let r be the south pole of Sn

0’
the line segment Joining p to r in SS, L the line segment

q the midpoint of

n

n, B: the cones (n-cells) in So

Joining p to q in 83, and Eq

with base Bn'1 and cone points q, r respectively. Let

s?‘1 = (35"1 U SE) - gn‘l. Then 33-1 is a bi-collared

(n-l)-sphere in $3 and 32-1 n 83-1 = fl. Let u 6 32-1 and
n-1 n-1

v 6 Bl - B . Then there exists an embedding f : I * up
such that f(O) - u, f(1) = v, and f(0,1) c.nn. Since
DR(Mn) - 2. it follows from 3.1.5 and 1.u.5 that there is a
prOper domain G C Mn, f(I) C G, and an embedding h of

(G,G n in) into (tan,éfin). Therefore there exists and n-
cell Fn CMn such that Fn 0 Sn"1 is an (n-1)-ce11, i a 2,3.

1
Thus An 2 Mn U B2 is an n-annulus. From this point the
proof proceeds exactly as the proof of the theorem of [10]
to show that up is an n-slab; and thus the remainder of the

proof will be omitted.

CHAPTER IV

MANIFOLDS WITH COMPACT BOUNDARY
WHICH HAVE DOMAIN RANK 3

It is clear that if Mn is an n-manifold with boundary,
n g,2, such that some component of Mn is compact, then
DR(Mn) 3,3. In this chapter we will characterize those n-

manifolds with compact boundary which have domain rank 3.

l. A generator for a certain dominion

of a compact, punctured n-sphere

Qefinition h.1.1 A homeomorphism h of Sn onto itself is
called strictly stable if h is the identity on a non-empty
open set. A homeomorphism h of Sn onto itself is called
stable if h is the product of a finite number of strictly
stable homeomorphisms of Sn. SH(Sn) will denote the group

of stable homeomorphisms of Sn.

Lemma h.1.2 Let {BE}§=1 be a finite disjoint collection of

n-cells contained in Sn, n g_2, such that B: is bi-collared
n

in S , 1 S.k $_q. Suppose that Mn is a connected n-mani-

fold, Mn C Sn, and that h and h2 are stable homeomorphisms

l

o
n n

of S onto itself such that kglfik C-Inth1(Mp) = h1(Mp),

- T n n

i - 1,2. Then hl(Mn) - kflIntBk = h2(M ) - k§llntBk.

Proof. Let h a h h"

n n n
2 1 and L = hl(M ). Then h E SH(S )

47

#8

and k=lB§ C-Inth n Inth(Ln). It suffices to show that
h(Ln) - k§lInth(B§) is homeomorphic to h(Ln) - kQIIntBE.
For each k, 1 5_k 5_q, let ok 6 IntBE. Since n g_2 and h
is a homeomorphism, there exists a disjoint collection
{D:}k:l of n-cells contained in Inth(Ln) such that E: is
bi-collared in Sn and {ck} U {h(ckn c Int 1):, 1 g k 5 q.
Therefore there exists a homeomorphism g1 of Sn onto itself

q n
- kglIntDk) = id and g1 (h(ck)) = ck,
1 g_k g_q. For 1 5,k 5_q, let Uk be a neighborhood of ck,

ok 6 Uk C IntB: and let Vk be a neighborhood of h(ck),

n
h(ck) 6 Vk C-Inth(Bk) such that g1(Vk) C Ck. Since

{h(B:)}k:l is a disjoint collection of n-cells with bi-

such that g1|(sn

collared boundaries, there is a disjoint collection {0:}kgl

of n-cells conunned in Inth(Ln) and-a collection of embed-

n

k!

fk(En(%)) = h(BE), and f (O) = h(c ), l < k < q, where
k k -' '-

q n n n
dings {fk}k=l of E into S such that fk(E ) — C

En(%) ='{x 6 Enld(x,0) 5 3}. Therefore there exists a
homeomorphism g2 of Sn onto itself such that g2 restricted
to (Sn - kglIntCE) = id, and such that for 1 $,k 5.q,
g2(h(ok)) - h(ck) and g2(h(B:)) C Yk‘ Then g = glg2 is a
stable homeomorphism of Sn, g|h(Ln) is a homeomorphism of
h(Ln) onto itself, and g(h(B§)) c.IntB: for l g_k §_q.
Since a: is a bi-collared (n-1)-sphere in Sn, it follows
from 9.1 of [5] that B; - Intg(h(B:)) is an n-annulus,

l 5_k S,q. Since {3:3kil is a disjoint collection of n-

cells with bi-collared boundaries contained in Inth(Ln),

49

there is a homeomorphism f e SH(Sn) which carries
h(Ln) - kgllnt(B:) homeomorphically onto
n n n n g
h(L ) - kgllntg(h(Bk)). Since h(L ) - kQlIntg(h(Bk)) —
h(Ln) - kéllnt(Bfl) and g|h(Ln) is a homeomorphism of h(Ln)
n n I n n
onto itself, h(L ) - k§11ntBk — h(L ) - kQllntMBk) and the

desired result is established.

Theorem “.1.3 Let Mp be a compact, punctured n-sphere with
q holes, n 2.2. There is a prOper domain D C Mn, fin C.D,
such that if G is a prOper domain of Mn, fin C G, then D
generates G.

25222. Let p be the north pole of Sn. Without loss

0
of generality we may assume that Mn = Sn - kalBfl’ where

{B£}k:l is a finite disjoint collection of n-cells with
o
bi-collared boundaries, and that p 6 Mn = IntsnMn. We will

consider Rn = ITI as embedded in Sn as the subspace

n

S - {p}under an embedding e.

Let G be a proper domain of Mn, Mn C.G. Since G is a
prOper domain, there exists a g 6 SH(Sn) and an x 6 Mn - G

q n

such that g(x) = p and g|(kglsk) - id. Let cl - 3(6) and
qn n

set G2 = G1 U (kngk). Then G2 is a domain of B . It fol-

lows from 1.4.6 and 2.1.“ that there is an h 6 SH(Sn) which
is the identity in a neighborhood of p, such that

q n n n 3 n
kngk C h(Dl). Let D - h(Dl) - k=1IntBk; then D is a pro-
per domain of fin, Mn c.D. Since G2 is a domain of En, it
follows from 1.u.6, 2.1.n, and the fact that an has a PL

triangulation (T,e), that there is a sequence {f3}:=l of

50

elements of SH(Sn) such that

i) for all j e z+, f is the identity in a neighbor-

J
hood UJ of p;

+ n n n
n
111) G2 — ngfj(D1)'
For all j 6 2+, define gJ a g-lth-l. Then {gJ ?;1 is a

sequence of stable homeomorphisms such that for all

+ n n n
j 6 Z , kngk c.gJ(h(D1)) C 3j+1(h(Dl))’ and such that

w n n

= - . “.1. that
G 331(51(h(31)) k§11ntBk) It foilowsnfrom 2
n n n

for all j 6 Z , gj(h(D1)) - k§11ntBk a h(Dl) - kQIIntBk = D

and thus D generates G.
2. Characterization

Lemma #.2.1 Let Mn, n 2.2, be a connected n-manifold with
boundary such that DR(Mn) = 3. If Np has a compact compo-
nent, then N9 is an (n-1)-sphere.

£3232. Suppose that C is a compact component of Np.
Let {D1,D2,D3} be a set of generating domains for Mp. We

may assume that D is an n-manifold without boundary, that

1
D2 is an n-manifold with boundary such that all components
of D2 are cpen (n-l)-manifolds, and that D3 is an n-mani-
fold with boundary such that c c 53. Suppose that h? s c
and let x 6 hp - C. Since Hg is a connected n-manifold,
there exists an embedding f : I r M? such that f(O) e c,
f(1) = x, and f(0,1) C fin. Since n 2.2, it follows from
1.h.5 that there is a proper domain H1 such that

f(I) U C C-Hl. Also it follows from 1.3.13 that there is a

51

proper domain H2 of Mn such that H2 n in = C. Since

C C Hl n H2, D3 must generate both H1 and H2, which is im-
possible. Therefore C = Np. Suppose that x 6 C, then
DR(Mn - x) = 2 and if n > 2 it follows that DR(C) = 1.
Since C is a closed (n-l)-manifold, 2.1.7 shows that

C Q Sn"l if n > 2. If n = 2, then C 2 51, since S1 is the

only closed, connected l-manifold.

n
Definition h.2.2 Let J = [-1,1] and set SA a
Jn - (0,...,0), n 2.2. A space X which is homeomorphic to

SA? is called an n-semi-annulus.

pefinition u.2.3 Let Mn, n 2,2, be an n-manifold such that
ND is an (n-1)-sphere. up is said to have semi-annular
compact subsets if for each proper compact subset K C Mn,
there is an embedding h of (K, K 0 in) into (SAP,SA9).

Let (T2,id) be a fixed PL triangulation of SA? where
|T2| = SA? and id is the identity map.

Definition h.2.h Let T be the given fixed PL triangula-

2
tion of SAp, n g_2. The set S(T2) = {LIL is a subcomplex
of some SdkT2 and |L| is a compact, connected, PL n-mani-
fold in SA? with SAP c.|L|} is called the set of re lar,

boundary submanifolds of T2.

n
Theorem #.2.5 Let M , n g_2, be a connected, non-compact
n-manifold such that up has semi-annular compact subsets.

There exists a sequence {ME};;1 of compact, connected n-

52

manifolds such that
i) M: is collared in Mn, in C-ME C Mn, k 6 Z+3
ii) there is an Lk 6 S(T2) and a relative homeomor-
phism hk of ILkI onto ME;
iii) ME c.1ntM§+l, k 6 2+; and
w
1v) Mn = kglM§°
23222. The method of proof is similar to that of
2.1.3 and 3.1.2 and will be omitted.

Lemma 4.2.6 Let (T2,id) be the given fixed PL triangula-

tion of SAn, n 2_2, and let {L be an enumeration of

}@
JJ=1 ,
S(T2). There exists a domain D3 of SAP, SAP c.D§, such that
if G is an cpen set and ILJI CAG for some j 6 Z+, then there
is a homeomorphism h of (Dn,D§) into (SAn,SAp) such that

n

L C»h D c c.

I JI (3)

Proof. This is an immediate consequence of 4.1.3.

Henceforth D3 will always denote the domain of SAP

referred to in the last lemma.

Theorem 4.2;2. Let Mn, n g_2, be a connected, non-compact
n-manifold such that up has semi-annular compact subsets.
Then D? is a generator of NP.

2322;. The result follows directly from 4.2.5 and
4.2.6. The method of proof is similar to that used in the
proof of 3.1.4 and the details of the proof will be omitted.

Theorem 4.2.8 Let Mn, n g_2, be a connected n-manifold

with up 3 Sn'l. Then DR(Mn) = 3 if and only if an has

53

semi-annular compact subsets.

2322;. Suppose that DR(Mn) - 3 and that {D1,D2,D3} is
a set of generating domains for M3. Since DR(Mn) - 3 and
En is an (n-1)-sphere, we may assume that D1 is an n-mani-

fold without boundary, D is an n-manifold with boundary

2
such that each component of D2 is an cpen (n-1)-manifold,

and D is an n-manifold with D3 2 Sn-l. Now suppose that

3

K C Mn is a prOper compact subset; then K U M? = K1 is also

a proper compact subset. Let x 6 up - K1 and set

G = M3 - x. Since n g_2, G is a prOper domain of up with

1
Mn C-Gl. Since D

l

3 must generate Gl there is a.domain H1,

Kl C’Hl C-G and a homeomorphism f of (D3,D3) onto

1
(31,31 n Mn) = (Bl,hn). It follows from 1.3.13 that there

is a domain c2 of Mn and a homeomorphism g of (SAp,SAp) 3

(sn“1 x [o,1),sn'l x {0}) onto (c2,o2 n h“) - (c2,fin).

Since D3 must also generate G2 there is a domain HZ,

of ($3,D3) onto

H2 C-G2 and a homeomorphism f

(32.3

2

n fin) = (H2,Nn). Let f = g- f f"1 Then

2 . 2 1 °.
fI(K,K n Mn) gives an embedding of (K,K n up) into
(SAF,SAP) and thus up has semi-annular compact subsets.

Now suppose that My has semi-annular compact subsets,
and that G is a prOper domain of HF. If Np C-G, then it
follows from 4.2.7 that D; generates G. If G n fin-fi 2.

G n En ¥ En, then since G has Euclidean compact subsets, it
follows from 3.1.4 that D:

G n Mn - fl, then since G has Euclidean compact subsets, it

generates G. Finally if

54

follows from 2.1.5 that D? generates G. Therefore
{D?,D2,D§} is a minimal set of generating domains for Mn

and thus DR(Mn) = 3.

Corollaryg4.2.9 Let Mn, n g_2, be a compact, connected
n-manifold with boundary. Then DR(Mn) = 3 if and only if
Mn is an n-cell.

23332. If Mn is an n-cell, then Mn has semi-annular
compact subsets and by 4.2.8 DR(Mn) 3 3. If DR(Mn) B 39
then it follows from 4.2.1 that Kn 3 Sn'l. Since DR(Mn) = 3
and in 2 sn‘l, DR(gn) = 1. Therefore it follows from 2.2.3
that Mn is a compact punctured n-sphere with 1 hole and

consequently theorem 5 of [2] shows that up is an n—cell.

Corollary 4.2.10 Let Mn, n g_2, be a connected n-manifold
such that NH 3 Sn'l. If DR(gn) = 1, then DR(Mn) 8 3.

2322;. Let M: = K“ x [0.1) UG Mn be an abstract col-
laring of Mn and consider Mn and En as embedded in ME. Let
K be a proper compact subset of Mn. Then Kl = K U in is
also a proper compact subset of Mn. Since DR(ME) a
DR(gn) = 1, there is an embedding h of K1 into R” such that
h(Nn) is a bi-collared (n-1)-sphere in Rn. Since Mn is con-

nected, K is not an n-cell and thus there is an embedding

1
8 Of (h(Kl),h(Mn)) into (SAn,SAn). Therefore MD has semi-

annular compact subsets and so DR(Mn) a 3.

CHAPTER V
MONOTONE UNIONS AND PRODUCTS

Definition 5.1.1 Let C* be a collection of non-empty
tOpological spaces. A tOpological space X is said to be
w

an open monotone union of C* if X gkglxk where

i) for all k 6 2+, xk is open in x and xk is homeo-

morphic to some element Ck 6 0*, and
+
ii) for all k 6 Z , Xk C-Kk+l.

X is said to be an open, homogeneous monotone union of C*

if X is an open monotone union of C* and for all k 6 Z+,
agh-
Theorem 5.1.2 Let C* be a collection of connected n-mani-
folds, n 2,2, such that either
i) for all C 6 0*, C = fl and DR(C) = 1;
ii) for all C 6 0*, C f D and DR(C) = 2; or
iii) for all C 6 C*, C 2 Sn'1 and DR(C) a 3.
Let X be an cpen monotone union of C*. Then X is a connec-
ted n-manifold such that
iv) if i) holds, then i = 2 and DR(X) e l;
v) if 11) holds, then K s a and DR(X) = 2; and
vi) if iii) holds, then K 2 Sn"1 and DR(X) = 3.

Proof. Let X be an open monotone union of C*. It

is clear that X is a connected n-manifold such that if i)

55

56

holds, then x = 2; if ii) holds, then x i U; and if iii)

holds, then X g Sn'l. The result now follows easily from

the characterizations given in 2.1.6, 3.1.5, and 4.2.8.

Throughout the rest of this chapter the boundary of an
n-manifold Mn will be denoted either by bd(Mn) or Mn.

Theorem 5.1.3. Let Mn, Mk be connected n and k manifolds
respectively such that Mn, Mk have Euclidean compact sub_
sets, and Mk is not compact. If either k 2_2 or Mn is not
compact, then Mn x Mk has Euclidean compact subsets.
£3292. Let C be a prOper compact set in Mn x Mk and
let p1 and p2 be the projections onto Mn and Mk respec-

tively. Define C1 = p1(C), i = 1,2. Then

C C Pl(C) x p2(C).
a) Suppose that k Z_2. If Mn is compact, then it fol-

lows from the remark after 3.1.5 that Mn - D and thus
Mn 2 8“. Therefore either cl 3 Sn or there is an embedding

hl of (Cl,C1 0 Mn) into (san,%Rn). Since Mk is not compact,
2 2 n Mk) into (Sak,%nk).
T

there is an embedding h
1) Suppose that C1 = Sn and that h1 is a homeomorphism

of c1 onto Sn. Then f : c1 x c2 ~ Sn x 23k defined by

f(x1,x2) = (hl(xl),h2(x2)) induces an embedding h of the

of (C2,C

pair (01 x c2,c1 x 02 n bd(M9 x Mk)) into the pair
(Sn x %Rk,bd(Sn x iak)). Since k g_2, Sn x Ek'l can be
embedded in Rn+k‘1, and hence there is an embedding g of

(Sn x enk,bd(sn x sz)) into (snn+k,ssn+k). Therefore

57

gf|(c,c n bd(Mn x Mk)) gives an embedding of

(c,c n bd(Mn x Mk)) into (enn+k,%Rn+k

) and thus Mn x Mk has
Euclidean compact subsets.
ii) Suppose that C1 is not an n-sphere. Then C1 # Mn,

C2 # Mk and there are embeddings h1 of (C1,C n Mn) into

1
(tan,sfin) and b2 of (c2,c2 n wk) into (sak,ték). Define
h : 01 x (32 r in“ x 23k by h(xl.12) = (h1(x1).h2(xz)).
Since tan x 23k 3 San+k, h induces an embedding g of
(C,C n bd(Mn x Mk)) into (éRn+k,%Rn+k). Therefore Mn x Mk
has Euclidean compact subsets.

b) If Mn is not compact, then an argument similar to
that given in ii) above shows that Mn x Nk has Euclidean

compact subsets.

Corollary 5.1.4 Let {Mfl(k)}§=l be a finite collection of
connected h(k)-manifolds such that for l g.k S_q,
h(k)
com t th % - .
pac , en DR(k21Mk ) 1

If for some j, 1 g,j 5,q, M§(J) is not

Proof. This follows immediately from 5.1.3.

ll.

12.

l3.

14.
15.

BIBLIOGRAPHY

R. H. Bing, ”An alternative proof that 3-manifolds can
be triangulated”, Ann. of Math. (2) 69 (1959) 37-65.

M. Brown, "A proof of the generalized Schoenflies
theorem”, Bull. Amer. Math. Soc. 66 (1960) 74-76.

M. Brown, "The monotone union of cpen n-cells is an cpen
n-cell", Proc. Amer. Math. Soc. 12 (1961) 812-814.

M. Brown, ”Locally flat embeddings of tOpological mani-

folds”, TOEOlogy of -manifolds and Related To ics,
arentice-Hall, l982)2832917"" “"“““‘

M. Brown and H. Gluck, ”Stable structures on manifolds,
I. Homeomorphisms of Sn”, Ann. of Math. (2) 79 (1964)
l-17.

J. c. Cantrell, "Almost locally flat embeddings of Sn-1
in Sn”. Bull. Amer. Math. Soc. 69 (1963) 716-718.

P. H. Doyle, "Certain manifolds with boundary that are
products", Michigan Math. J. 11 (1964) 177-181.

P. H. Doyle and J. G. Hocking, ”Special n-manifolds with
boundary", Proc. Amer. Math. Soc. 16 (1965) 133-135.

J. Dugundji, To 010 , (Allyn and Bacon, 1966).

C. Greathouse, ”The equivalence of the annulus conjec-
ture and the slab conjecture", Bull. Amer. Math. Soc.
70 (1964) 716-717.

J. F. P. Hudson and E. C. Zeeman, ”On regular neighbor-
hoods”, Proc. London Math. Soc. (3) 14 (1964) 719-745.

D. R. McMillan, Jr., ”Summary of results on contractible
open manifolds”, Tapologz of z-manifolds and Related
Topics, (Prentice-hall, 1982) 100-102.

M. H. A. Newman, ”The Engulfing Theorem for topological
manifolds", Ann. of math. (2) 84 (1966) 555-571.

E. H. Spanier, Algebraic To 010 , (McGraw-Hill, 1966).
E. C. Zeeman, Seminar on Combinatorial To 010 ,
(mimeographed notes, Inst. Hautes Etudes Sci., Paris,
19 3 .

58

11111111)“

1368

     

Mk)ifi\))\\)fl[)[(\))))[|flfl)i