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MOVING THE COMPACT SUBSETS 0F‘MAN!FOI’.DS .

Thesis for the Degree of Ph. D.
MICHEGAN STATE UNIVERSHY
JOHN KENNETH COOPER, it.

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ABSTRACT
MOVING THE COMPACT SUBSETS 0F MANIFOIDS
BY

John Kenneth Cooper, Jr.

We make the following definition. A tOpological space X
is said to be l-movable when for each proper compact subset, A, of
X there is a homeomorphism, h, of X onto itself with A 0 h(A) = ¢.
Some results are:

1. If an open connected Z-manifold, M, imbeds in a 2-sphere
with n-handles, N, then there is an imbedding of M in N that is
dense in. N.

2. If Mp is a closed connected n-manifold, n 2 2, C a
closed 0-dimensional subset of Mn, and Mn - C is l-movable, then

n

M is the n-sphere and C has an invert point.

3. A counterexample to the converse of 2 is presented for

4. If L is the (n-2)-skeleton of K (where K is a
simplicial complex, not necessarily locally finite, of dimension n)
then \K‘ - ‘L‘ is 1-movab1e.

5. Numerous examples restricting certain possible improve-
ments of the result of 4.

The notion of l-movable is generalized to that of k-movable

and we have results:

John Kenneth Cooper, Jr.

6. Every open connected triangulable n-manifold is n-movable.

7. For n 2 5, every contractible open n-manifold is 2-
movable.

8. If M is a k-movable manifold with boundary, then that
boundary is not compact and has either 0 or an infinite number of
compact components.

9. Let Mn be a closed n-manifold. If Mn is k-movable,
then Mn is the union of (k+1) open n-cells.

We consider the l-movability of certain open 3-manifolds
and one of the facts obtained is that all W—spaces (and hence all

contractible domains in E3) are l-movable.

MOVING THE COMPACT SUBSETS OF MANIFOLDS

BY

John Kenneth Cooper, Jr.

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1971

TO CINDY

ii

ACKNOWIEDGEMENTS

I want to thank Dr. Patrick Doyle for his patience, encourage-
ment and guidance. My admiration for him as a mathematician is
exceeded only by my respect for him as a person.

I wish to thank R.O. Hill and C.L. Seebeck for their
suggestions. I would like to thank J.G. Hocking and E.A. Nordhaus
for the interest they have taken in me over a number of years.

I want also to thank my father, who introduced me to mathematics.

iii

Chapter
I

II

III

IV

VI

TAB IE OF CDNTENTS

INTRODUCTION

2 -MANIFOLDS

1-MOVABILITY AND 0-DIMENSIONAL SETS IN
MANIFOLDS

SIMPLICIAL COMPIEXES AND l-MOVABILITY
PRODUCTS AND k-MOVABILITY

MANIFOLDS AND k-MOVABILITY

CERTAIN OPEN 3-MANIFOLDS AND l-MOVABILI'I'Y

BIB LIOGRA HIY

iv

Page

13
35
SO
53
58

64

ll’,|

 

 

I'lll‘l

 

INTRODUCTION

K. Borsuk made the following definition in 1934 (see
R.l” Wilder [23]); a Subset M of Euclidean n-Space is said to
be free, when for each s > 0 there exists an e-transformation
of M onto Mi and M n'M' = ¢. A topological Space x is said
to be invertible when for each proper closed subset A there exists
a homeomorphism h of X onto itself with A D h(A) = ¢. This
concept was created by P.H. Doyle and J.G. Hocking [6] and has
been studied extensively.

Thus the following definition seems a natural one to make.
Definition 0.1: A topological space X is k-movable (for some
positive integer k) if and only if for each proper compact sub-
set B of X, there exist compact subsets Bl’BZ"'°’Bk oi B and
1,h2,...,h of X onto itself with B = U B

k i=1
and B n hi(Bi) = ¢ for each 1 = 1,2,...,k.

homeomorphisms h

1

Remark 0.2: A compact Hausdorff space X is l-movable if and only

if it is invertible.

Example 0.3: Both ED and Sn are l-movable.
A l-movable compact manifold must be a Sphere, but no such

0 o . 1
restriction is p0531ble for non-compact manifolds as E x M

where M is any manifold is 1-movab1e (for example: E1 x Sn

and En+1).

Remark 0.4: A k-movable topological space is L-movable for each
integer L 2 k.

Although k-movable is defined in the topological category,
reSults are more interesting in the category of manifolds and much
of this work will concern manifolds.

The moving number of a space X is defined to be the
minimum {k‘X is k-movable}, and +m if ¢ = {k‘X is k-movable}.

Chapter I of this thesis contains the result that for an
open connected 2-manifold, M, that imbeds in a Z-Sphere with n-
handles, N, there is an imbedding of M in N that is dense in
N. The main theorem of Chapter II is: if Mn is a closed
connected n-manifold, n 2 2, C a closed O-dimensional subset of
Mn, and Mn - C is l-movable, then Mn is the n-Sphere and C
has an invert point. A counterexample to the converse of this
theorem is presented for n 2 3.

Chapter III is an investigation of the l-movability of
|K‘ - lL‘ where K is a simplicial complex and L is a sub-
complex of K. We have the result that if L is the (n-2)-
skeleton of K (where K is a simplicial complex, not necessarily
locally finite, of dimension n) then IKl - |L| is l-movable.
This chapter includes numerous examples restricting certain possible
improvements of the above result.

In Chapters IV and V results are obtained concerning n-
manifolds and k-movability in relation to product Spaces and manifolds
with boundary. Among the results are the following: Every open

connected triangulable n-manifold is n-movable. For n 2 5 every

contractible open n-manifold is 2-movab1e. If M is a k-movable

manifold with boundary, then that boundary is not compact and has
either 0 or an infinite number of compact components. If Mn is
a closed n-manifold, then Mn is k-movable implies Mn is the
union of (k+l) open n-cells. A connected Hausdorff Space that
is the union of k open n-cells is k-movable.

Chapter VI concerns the 1-movability of certain open 3-
manifolds and includes the fact that a11.W-Spaces (and hence all
contractible domains in E3) are l-movable.

The following notation will be used for certain sets and

topological Spaces
2 = {n‘n is an integer}
E = {X‘X = (x1,x2,...,xn) an n-tuple of real numbers}

where E0 is given the topology determined by the Euclidean

distance. A space is called a n-cell, an open n-cell, or an

 

(n-1)-sphere when it is homeomorphic respectively to

{x E En‘distance (0,x) S 1}, [x E En‘distance (0,x) < 1}, or
Sn”1 = {x E En‘distance (0,x) = l} with the subspace tOpology
induced by that on E“.

A Cantor set is a homeomorph of the standard Cantor set
in [0,1]. An invertgpoint of a Space X is a point p such
that for each neighborhood U of p, there exists a homeomorphism,
h, of X onto itself with h(X - U) CZU. A topological Space X
is said to be the monotone union of a sequence of topological Spaces
xi, 1 = 1 2,..., when X1 is a subset of X1+1 for i = 1,2,...

9

co
and X = U xi' A pseudo-isotgpy of a topological Space X is a
i=1

homotopy that is a homeomorphism at every level except at possibly
the last. For a subset, A, of a tOpological Space, A. will denote
the closure of A and bdy A the boundary of A.

A n-manifold, Mn, is a separable metric Space each point
of which has a neighborhood whose closure is an n-cell. A manifold
is a topological Space that is an n-manifold for some n. All

manifolds in this theSis are connected. The interior of M is

 

the set of all points of the n-manifold Mn that have open n-cell

neighborhoods, and the boundary of Mn is My - (the interior of Mn).

 

A manifold with boundary is a manifold whose boundary is non-void.
A manifold without boundary is a manifold whose boundary is the
empty set. An open manifold is a manifold without boundary that is

non-compact, and a closed manifold is a compact manifold without

 

boundary.

In metric spaces, dist (x,y) will denote the distance between
x and y. In a metric space M, with (x,t) E M X E1, Sx,t =
{y 6 M‘dist (x,y) < t}. For tame and locally tame Cantor sets see

[19]. For locally tame in general see [18].

The end of a proof will be denoted by"[3".

CHAPTER I

2-MANIFOLDS

Ian Richards in [21] classified open 2-manifolds. Richards'
work implies that every domain in 32 has an imbedding which is
dense in 82, and this result is generalized by Corollary 1.5
to the 2-sphere with handles. We then prove that the Open M6bius
band is not l-movable and (using a recent result of R. Jones [12]
that each open 2-manifold is the union of two open 2-cells) that all
open 2-manifolds are 2-movable.

We begin by proving some lemmas. In this chapter Tn
will denote the closed orientable surface of genus n, namely

the 2-sphere with n handles.

2
Lemma 1.1: Let M be a compact 2-manifold without boundary,
2
C be a compact 0-dimensional set, C<: M , and U an open set

2
in M , U # ¢.

Then there is a homeomorphism h of M2 onto itself

with h(C) C U.

‘ggggf: Let {U0} be an open covering of M2 with U0 homeo-

morphic to E2 for all 0. Now let A > 0 be the Lebesgue

number of this covering. The proof of Lemma 1 in [1] ensures

a finite set of pairwise disjoint, closed 2-cells each of diameter
‘ t

< A, Say BI’BZ’°'°’Bt’ with CC: . 1int Bi' Choose pi 6 Bi for each
1:

1 = 1,2,...,t and qi e u for i = 1,2,...,c (with qi i qj
for i i j). There exists a homeomorphism f of M2 onto it-
self with f(Pi) = q1 for all i = 1,2,...,t. So f-1(U) is
an open neighborhood of {p1,p2,...,pt}. Since C is compact

and Cc: tJ int Bi’ there is a homeomorphism g of M2 onto

i=1 t
itself (g is the identity on M2 - U int Bi’ and in int Bi
i=1

shrinks Bi 0 C into f-1(U) 0 Bi) with g(C) CZf-1(U). Thus

h = fg is the desired homeomorphism.[3

Lemma 1.2: For each integer n > 0, let C be a compact 0-

dimensional subset of Tn'

Then Tn - C cannot be imbedded in Tt where t < n.

Iggggfiz For each integer n > 0, let a simplicial complex of
dimension 2, K, be a triangulation of Tn’ and C a compact
O-dimensional subset of Tn' Let U be the interior of one of
the 2-dimensional simplices of K. Then by Lemma 1.1, there is
a homeomorphism, h, of Tn onto itself with h(C) CLU. Thus
h(Tn - C) = Tn - h(C) z'rn - U23 \K(1)\.

Suppose there were an imbedding, f, of Tn - C onto

T with t < n. then fh-1\

t would be an imbedding of

MW

in Tt. But this contradicts the theorem of 5.5 in

MN

[24], which says that the l-skeleton of a triangulation of an
orientable Z-manifold is a minimal imbedding (i.e. the l-skeleton

cannot be imbedded in a surface of lower genus).

Hence the lemma.[3

Lemma 1.3: Let M2 be a surface formed, as in Theorem 3 of [21],
from a sphere S by first removing a closed, totally disconnected
set X from S, then removing the interiors of a finite or in-
finite sequence D1,D2,... of non-overlapping closed discs in

S - X, and suitably identifying the boundaries of these discs in

pairs, except perhaps the boundary of one disc may be identified with
itself to produce a "cross cap"; in the finite case there is Such a
"cross cap" and in the infinite case, there may or may not occur such

a "cross cap". The sequence {Di} has the property that for any open
set U in S, X<: U, all but a finite number of the D1 are contained

in U. P will denote this last property in the proof.

Then M2 cannot be imbedded in TK for any integer K 2 0.

Proof: Suppose M2 is as above and can be imbedded in TK for

some integer K 2 0. Let n be an integer greater than K. As

2
M2 imbeds as an open submanifold of orientable TK’ M is

2 .
orientable. Thus in the construction of M from S in the
hypothesis, no "cross cap" could have occurred. So we cannot have
the case of a finite number of Di' Hence we may assume that the

2
Di have been indexed so that in the identification to create M ,

the boundary of D is identified with the boundary of D

2j-l 2j

for j = 1,2,... .

Let g be the quotient map giving M2 from
co co 2
S - (X U U int Di)’ g: S - (X U U int Di) —+bd . Let
B = M - g( U boundary D.). Then B imbeds in T , as M
. 1 K
i=2n+l

does. But B is the quotient under the quotient map

g‘ m m . By property P of the hypotheses,
s-(m a int DiU U Di)
i=1 i=2n+1 m
for each i 2 2n+l, d1 = distance (Di’ X U U Dj ) > 0. let Vi
j=--1J
j¢i
be the Open di/3 neighborhood of Di in S. Let pi 6 int Di
a: m
for each i 2 2n+l. Then clearly S - (X U U int D. U U D.)
i 1
2n i=1 i=2n+1
is homeomorphic to S - (X U U int D. U U {pi }) by a homeo-
, 1
i=1 i=2n+1

morphisun, h, which is the identity outside (J ‘v 1, (shrink Di
i=2n+11

to the point pico inside Vi for each i 2 2n+1).

Now, U [pi ] is a O-dimensional set, and X
i=2n+1

is O-dimensional (since it is compact and totally disconnected,

see [11] Theorem D, p. 22), so D = X U U {pi } is O-dimensional.

i=2n+1
D is also closed.
_1 2n
So gh : S - (X U U [pi } U U int D 1) a B is actually
i=2n+1 i= =1
a quotient map, and identifies only boundary D2j with boundary
D2j-l for each 3 = 1,2,...,n.

This quotient, B, is homeomorphic to a Sphere with n
handles less a closed 0-dimensiona1 set.
But then K,< n and B imbeds in TK’ which contradictslemma

1.2. Hence the supposition was false, and so the lemma is proved.[]

Theorem 1.4: let M be an open connected 2-manifold which imbeds
in Tn’ k = minimum {t‘M can be imbedded in Tt}.
Then (1) If n = k, then M is homeomorphic to Tn - D

where D is a closed O-dimensional subset of Tn.

(2) If n > k, then M is homeomorphic to Tn - D,

where D is a l-dimensional set,either (a) a finite set plus

8—8— o.. —-8 or (b) a O-dimensional set plus the
1 2

... (n-k)

union of (n-k) pairwise disjoint homeomorphs of the 1-sphere.

Egggf: Let M, Tn, and k be as in the hypotheses of the theorem.
By Theorem 3 of [21], M is homeomorphic to either one of the
surfaces of the hypotheses of Lemma 1.3 or to the complement of
a closed O-dimensional set C in TL for some integer L 2 0.
The first of these cannot occur (else by Lemma 1.3, M could not
imbed in Tn, which contradicts the hypotheses of the theorem).
Thus M is homeomorphic to TL - C where C is a closed
0-dimensional subset of TL. By Lemma 1.2, M cannot imbed in
Tt for t < L. But M imbeds in Tk’ so k 2 L. By the minimality
of k, k s L. Hence k = L.
(1) If n = k = L, then letting D = C we have con-
clusion (l).
(2) If n > k = L, then ‘C‘ = the cardinality of C is
either < 2(n-k) or 2 2(n-k).
a) In the case that ‘C‘ < 2(n-k) 01 is not compact, so
‘C‘ 2 1). let p 6 C, and U be an open neighborhood
of p, which is homeomorphic to E2 by homeomorphism g: U a E2
and C D U = {p}. In U, let A = g-1(B), where B C E2 is
the union of 2(n-k) pairwise disjoint closed 2-discs,

Dl’D2’°"’D2(n-k)’ and 2(n-k)-1 arcs, connected in the manner

10

of the sketch 0 O ....D .

D1 I’2 I’3 D2(n-k)

Since B is cellular in E2, A is cellular in TL, and

so there is a homeomorphism, f, from IL - {p} onto TL - A.

 

Thus I = f|T -C is ahomeomorphism from T - C onto

2(n-k)
(T - A) - f(C - {p}). Let F = 0 pint D., and in T - F
L 1 L
i=1
make the identification of the boundary D21_1 with the boundary
of D2i for i = 1,2,...,n-k. This gives Tn as the quotient

Space of Tg - F (as L = k). Call the quotient map h,

h: TL - F * Tn. So h‘IL'A: T - A,» Tn - h(A - F) is a homeo-

L

morphism and h'= h‘ is a homeomorphism onto

(TL-A)-f(C-[p})

Tn - [h(A—F) U hf(C - [p})]. Thus f is a homeomorphism from

TL - C onto Tn - [h(A-F) U hf(C - {p])]. But M is homeomorphic

to TL - C and hf(C - (pl) if finite and h(A-F) is clearly

homeomorphic to 8—8— ... —_8 ,
1 2

(n-k)

b) In the case that 101 2 2(n-k), let be

p13P2:°'-9P2(n_k)
2(n-k) distinct elements of C. So clearly there exist pair-
wise d13301nt open sets Ul’U2"°"U2(n-k) of TL Wlth pi 6 U1
for i = l,2,...,2(n-k) and U1 homeomorphic to E2 for

i = 1,2,...,2(n-k). For each i = 1,2,...,2(n-k) let Di be

a closed disk, cellular in Ui’ Di<: Ui' Then there exists a
homeomorphism hi from Ui - {pi} onto Ui - Di’ with hi

equal to the identity outside a compact set which is a neighbor-

hood of D1 in Ui' Define a homeomorphism h from

11

2(n-k)
.. _ =11
TL {p1,p2,...,p2(n_k)} onto TL :11 Di by h‘Ui i for
1 = 1,2,...,2(n-k) and h| 2(n_k) = the identity. Let
TL- 121 Ui
2(n-k)
B = U int Di’ and in TL - B make the identification of the
i=1
boundary of D21.”1 with the boundary of D21, for i = 1,2,...,n-k.
This gives Tn as the quotient space Of TL - B (as L = k).
Call the quotient map g, g: TL - B'» Tn' g = g‘ 2(n-k) is
T ' U D.
" i=1 1

clearly a homeomorphism onto Tn - X where X is the pairwise
disjoint union of (n-k) homeomorphs of the l-sphere. Let

hI= h|T -C’ then E'h' is a homeomorphism from TL - C onto

Tn - [X U gh(C - [p1,p2,...,p2(n_k)})]. But M is homeomorphic
to TL - C and gh(C - {p1,p2,...,p2(n_k)]) is a 0—dimensional

set with X as above, so we have (2) (b). E]

Corollary 1.5: Let M be an Open connected 2-manifold which
imbeds in Tn' Then there exists N CITn such that N is

homeomorphic to M and N = Tn'

Proof: let N = Tn - D of Theorem 1.4. D clearly has void
interior, so N.= T - D = T .[3
n n
Theorem 1.6: Let M2 be a compact 2-manifold without boundary
2

2
and C a 0-dimensional compact subset of M . If M - C is

l-movable, then M2 is a 2-Sphere.

Proof: Let M2 - C be 1-movable, and U the interior of D,
a closed 2-cell in M2. By Lemma 1.1, there exists a homeomorphism,

h, of M2 onto itself with h(C) CLU. Then using D as the

12

initial 2-cell in the proof of Theorem 1 of [7], there is a
standard decomposition of M2, M2 = leJ A (with P2 homeo-
morphic to E2 and U<: P2). But A is compact in M2, so
h-1(A) is compact in M2 - C. M2 - C is l-movable, so there
exists a homeomorphism g of M2 - C onto itself with
g(h-1(A)) fl h-1(A) = ¢. Thus h(gh-1(A)) fl hh-1(A) = ¢. By

Lemma 2.1, there exists a homeomorphism g of M2 onto itself

with §| 2 = g. h’gh-1 is a homeomorphism of M2 onto itself.
Ml"3 -1 -1 -1 2
But hgh' (A) n A = hgh (A) 0 hh (A) = ¢. Thus (hgh )(A)<: P .

So by Corollary 1 of [7], M2 is a 2-Sphere.[J

Theorem 1.7: The Open MObius band is not l-movable.

 

2122:: Let the Open MObius band M be given by the quotient map

g: [0,1] X (0,1) a M by the identification of (0,t) with (1,1-t)
for each t E (0,1). A = g([0,1] X [%', §]) is a compact subset

of M and A23 U = g([0,1] X (%-, 2)) which is an open Mfibius band
and so not orientable. If M were l-movable there would exist a
homeomorphism h of M onto itself with h(A)<: M - A. But M - A
is homeomorphic to S1 X (0,1) and so is orientable, while

h(U)<: M - A and so h(U) is orientable. Hence U is orientable.

This is a contradiction.[3
Theorem 1.8: All open 2-manifolds are 2-movable.

Proof: By a theorem of Jones [12], an open 2-manifold, M, is the

union of two open 2-cells, and so M is 2-movable by Theorem 5.5.[3

CHAPTER II

l-MOVABILITY AND O-DIMENSIONAL SETS IN MANIFOIDS

Theorem 1.6 tells us that if the complement in a closed 2-mani-
fold of a closed O-dimensional set is 1-movable, then that manifold
is a 2—sphere. In this chapter we consider the complement of closed
0-dimensional subsets of closed n-manifolds. The main result of
this chapter is Corollary 2.5, if M“ is a closed n-manifold, C
a closed 0-dimensional subset of Mn, and Mn - C is l-movable,
then MP is the n-Sphere and C has an invert point. We end
this chapter with some examples. For each integer n 2 3 there
are two Cantor sets in Sn such that the complement of one of
them is l-movable while that of the other is not l-movable. But
a Cantor set clearly has an invert point and so the converse of

Corollary 2.5 is not true for n 2 3. We begin with a lemma.

Lemma 2.1: Let Mn be a compact n-dimensional manifold without
boundary (n 2 2) and C a closed 0-dimensional subset of Mn.
Then for each homeomorphism h of Mn - C, there exists

a homeomorphism h of Mn such that h| n = h.
M -C

Proof: Given the homeomorphism h of Mn - C as above, define
h as follows in i) and ii).

i) h(x) = h(x) ‘v x E Mn - C, so h] n = h.
M -C

13

 

 

14

ii) Given e 6 C c M“, there is an open neighborhood U
of e with a homeomorphism g: En mu with g(0,...,0) = e.

Let U for i = 1,2,..., then U1 is

i ‘ g(3(o,...,0), l/i:

homeomorphic to ED and fl vi = e. By Theorem IV 4. of [11],
i=1
p. 48, En cannot be disconnected by a subset of dimension 3 n - 2,

o 5 n - 2, so Ui n (Mn - C) # ¢ is connected. Thus h(Ui 0 (Mn - C))

 

¥'¢ is connected and so h(Ui n (Mn - C)) is connected and closed,
n n -l n
thus compact in M . Let x E M - C. Then h (x) E M - C

and h-1(x) f e. Clearly there exists in Mn - C an open neighbor-

hood V of h-1(x) and integer jO so that V H Uj = ¢, and

0
so V n Ui = ¢ for all i 2 jo. Thus h(V) is an open neighbor-

hood of h(h-1(x)) = x which is disjoint from h(Ui 0 (Mn - C))

 

as h is a homeomorphism. Thus x 4 h(ui 0 (Mn - C)) for all

 

i2jo and so xi 0 h(Uifl (Mn-C)). Thus
i=1

m
H h(Ui 0 (Mn - C)) C:C, and is a compact non-void connected
i=1

 

set. The components of C are points. Hence

co
0 h(Ui 0 (Mn - C)) = {t} for some t E C.
i=1

 

Now define h(e) = t and denote Ui by eUi i = 1,2,... .

iii) To show that h is continuous.
h is clearly continuous at each point of Mn - C, open in M“.
Let a E C and h(a) E 9 where e is open in M“. Let Vi = an

of ii) for i = 1,2,... . Then {8(a)} is the intersection of a

 

decreasing family of non-empty compact sets, h(Vi 0 (Mn - C))

Since {h(a)] c:e, Theorem 1.6 b) of [8], p. 226 implies there is

 

a positive integer 10 so that h(Vi 0 (Mn - C)) c:e.
O

15

Consider Vi (an open neighborhood of a). Then

0
h(vi 0 (Mn - C)) - h(vi 0 (Mn - C)) c: 9. let b e v].L n c.
O O 0

Now let W1 8 bUi of ii) for i = 1,2,..., so the W1, i = 1,2,...,

form a neighborhood basis for b in, Mn' Thus 3 jo so that
wj cvi. 30 w 0(Mn-C)cvi moan-C) and
o o 3o o

h(b)€h(W moan-C))cthwi nmn-cnce.

O O

 

1

Thus h(vi ) C19 and so h is continuous at a. But a
was an arbitrary element of C, hence h is continuous.

iv) To Show that h: Mn «‘Mn is surjective, we observe
that as h is continuous and Mn is compact, h(Mn) is compact,
hence closed in the T2 Space Mn. But Mn is connected and C
is O-dimensional thus Mn - C = Mn. Now h(Mp)ZD h(Mn - C)
= h(Mn - c) = M“ - c with h(M“) closed, so 804“) :> M“ - c = M“.

v) h is injective by the following.

As h is a homeomorphism of Mn - C, SO is h-l, and so by i),

/,
ii) and iii) there is a continuous map ,(h- ) from Mn to Mn
/’ /’\
so that (hi)! n = h 1. Note that (h'1)h| n = h'lh = 1 n ,
M -C M -C M -C
the identity map on Mn - C. But 1 | = 1 also,

n n n
M M -C M -C
and as Mn is a T2 Space with Mn - C dense in Mn we have
”R n n
(by 1.5(2), p. 140 of [8]) (h )h: M ..M is the identity map
on Mn. Thus h is injective.
vi) Since h is a continuous bijective map from the
compact Space Mn to the T

2 Space M“, h is a homeomorphism

(by Theorem 8, p. 141 [14]).[3

16

Theorem 2.2: Let MP be a closed n-manifold which is the carrier
of a finite simplicial complex, T, c<: Mp,'C is 0-dimensional and
closed, n 2 2 and Mn - C is 1-movable. Then either C has an
invert point or there exists p, c E C, p i c, where each homeo-
morphism, h of C has h(c) = c and h(p) = p, and given

6 > 0, there exist (Open in C) neighborhoods 9p of p and 9C

of c with 9pc: Sp , ec<: Sc and a homeomorphism h of C

:3 Se

such that h(C - 9p) C 9c (thus p and c are the only fixed
points of C). This h is the restriction to C of a homeomorphism
of Mn onto itself with the above property for 9p and 9C open

. n
in M .

Proof: i) There exists a sequence Ki’ i = 1,2,... of compact connected
sets CMn - C such that Mn - K.i c:{x E Mn‘dist (x,C) 5 xi]
and xi 4 0. To construct the [Ki], let T(O)=T, let T(1) be

the barycentric subdivision of T(i-1)

for i = 1,2,..., and xi
be the mesh of T(i). Then xi 4 0 (Theorem 5-20, [10]). Let
L1 = [s E T(1)‘ ‘3‘ n C = ¢] and thus the number of components

. . . n . .
of ‘Li‘ is finite, say L1’L2""’Lt; ‘13] CZM - C which is
path connected (connected by Corollary 1, p. 48 of [11], and

each point has a path connected neighborhood so Theorem 5.5, p.

116 of [8] gives path connected), thus there exists arcs pj

n

in S - C from a point oft L1 to a point of Lj for
i i
j = 2,...,t.. Let K. = ( U L ) U ( U p ) which is clearly
i i j=1 j j=2 j

connected, C Mn - C and compact and K1 2 ‘Lil' If s 6 Mn - Ki
then S is a point of simplex of T(1) which also contains a

point of C and has diameter 5 xi; so dist (S,C) 5 xi.

17

ii). For each positive integer t there exists a compact
connected set Mt such that Mt<:.Mn - C and the diameter of each
component of Mn - Mt is < %3 and Mn - Mt C:{x 6 M3‘ dist (x,C)
< %]. Construct Mt as follows for each positive integer t.

As C is O-dimensional, for each b 6 C, there is an
open neighborhood, 9P, of b in Mn with (boundary 9b) 0 C = ¢

and 9b C {x 6 Mn‘dist (x,b) < %E . C is compact so a finite

number of these suffice, say 9b ,...,eb , to cover C. Let

1 L
n L L n
N=(M ~Ueb)U(Uboundary 9b). Then NCM -C,N is
i=1 3 i=1 1'
clearly compact, and each component of MD - N is a subset of
one of the 9b and thus has diameter < the diameter of 9b '< %3
J' 1

As N 0 C = ¢, we have d

distance (N,C) > 0, SO there is a
xi from i) so that xi < minimum (d,%), Thus N czxi and
each component of Mn - K1 is thus a Subset of one in Mn - N

and has diameter < %u Let Mt = Ki’ then clearly Mn - Mt C

{x 6 Mn‘ dist (x,C) 5 xi} C {x E Mn‘dist (x,C) < %].
iii) As Mn - C is l-movable, there exist homeomorphisms

n . _
ht of M - C onto itself such that ht(Mt) 0 Mt - ¢. But

ht(Mt) is connected and so a subset of one of the connected com-
ponents of (Mp - Mt) - C, which must be a subset of one of the

components of Mn - Mt’ all of which have diameter < %u So

ht(Mt) has diameter < %u Now, for all t, Mt # ¢, so there

co

. a w' a . . .
ex1sts a sequence < t>t=l ith t E ht(Mt) But this sequence is

. n .
in a compact space, M,, and so has a Subsequence which converges

in M“, say <a a c 6 Mn.

>00
tj j=l
Thus the sequence of compact connected sets [ht (Mt )}:=1

converges to c.

18

We now show that c E C. Assume c é C. So d = distance

(c,C) > 0. Then there exists j1,j2,positive integers,so that

dist (c,a.) < 2, for all j 2 j , and -l;-< Q3 and diameter
t 4 l tj 4
2

d . . _ . . .
htj(Mtj) < 4 for all j 2 32. Thus for all j 2 J0 max1mum (11,32),

n. 9.-
ht amt )<: {x E M ‘dist (x,c) < 2} - Sc So for each

,d/2'

x E ht (Mt ) we have dist (x,C) 2 d/2 (else dist (c,C) < d,
1o jo d

a contradiction) and so dist (x,C) 2 §'>»%-> fl— > El— . By ii)

j2 jo
we have x é Mn - Mt (if not, dist (x,C) < fl—) thus
J0 jo

x 6 Mt . But then ht (Mt ) 9 Mt which contradicts the first
jo jo jo jo

sentence of iii), i.e. ht(Mt) 0 Mt = ¢, as Mtj # ¢. Thus the

0

assumption c d C is false and so c 6 C.
iv) Note Lemma 2.1 says that any homeomorphism of Mn - C
extends to a homeomorphism of Mn. Thus we have for each j a

homeomorphism gt which is the extension to Mn of ht , and
J J
so [g (M )}? converges to c.
t, t, j=l
J J
v) Assume C has no invert point (x is an invert point of
the topological Space X means for each open neighborhood 9 of x,

there exists a homeomorphism h of X so that h(X-e) : e). For

each j = 1,2,..., let Uj be the component of Mn - Mt. that has

J
. ' C o
c E Uj There is a k so that gtk(Mtk)<: Uj and Mtj M.tk So
g (M )<: U ; denote this g by h . Let a be that component
tk tj j Ck tj j
- d -
of M9 Mt such that fit,(9j) 2 Mt. an hence ht (ej) 2 Mp Uj

J
which is connected and contains Mt . C has more than one element

(or else it would have an invert point) so let 2 E C, 2 # c.

let x = distance (z,c). Then there is a positive integer jO

19

ga But Uj has diameter
s <£ and c€U.,so 245m. Thus zEMn-U ch (9,)
j 2 J J J t. J
J

and hence h-1(z) E 9.. But h ‘ is a homeomorphism of C

t j t C

J 1 3
onto itself, so h- (z) E C H e . Let b = h-1 (2). Then

t j j t
1 (Mo)

on
<bj>j=1 is a sequence in compact C, and SO has a Subsequence

so that for each integer j 2 jo, l/j <

<

"IH
L—bIH

<bj>:=1 (with j1 2 jo) which converges to a point, say p, of
L

C. Thus the sequence <9. 2:=1 converges to p.
JL
So, given 6 > 0, there exists an L such that 9. C13
J’L p,€
and —1- s L s l< s, so U, C S (as c E U, which has
t .1 L J CSS J
i, L L L
diameter < -l"< e) and h (e ) 2 Mn - U, . Denote h ‘
tj tj jL JL tj C
L L L
by h and 0. U C by e and U, n C by U . Then.for h = h ,
6 JL 6 J, c e

6p = 96, 9C = ue,we have h(ep) :2 C - 9C and so h(C - 9p) 6; 9C. Note

9C3 ,UCS
e c

-l
6 P’s , h (C - 9C)<: 9p. It remains to Show that

:6

P € 0p = 9 V's, p * c, and that p and c are fixed points
a
under homeomorphisms of C.
vi) p # c. Suppose p = c. Let 0 be any open neighbor-

hood of p. Then for some positive number a, S = S C10-
P:€ C93

Now by v) there exist open sets 9 and Us neighborhoods of p
s
and c respectively and a homeomorphism h of C so that

h(C - 9 ) G U and e c S and U C S . But C - e
e 6 6 6

C38

: - S c C - s h C - c h C - c U c S c .
Cm 9e°(°)‘98ec,e9

,6

Thus p = c is an invert point of C. This is a contradiction.
vii) There does not exist a homeomorphism h of C onto
itself such that h(p) = c. Suppose there were a homeomorphism

h of C such that h(p) = c. Let 9 be an open neighborhood

20

of c. Then for some 8 > 0, Sc BC: 9. let a = 1/2 min (B, dist
3

. S =adSchi
(c,p)) > 0 So Sc,a fl p,a ¢ n C,a e, 8 continuous

so there exists 5 > 0 such that h(S ) c:S and 5 < a-
P,O C d

3

AB C is O-dimensional there exists a clopen (closed and open)

neighborhood B of p, B c:S So h(B) CSC a is a clopen

9

p.6'
neighborhood of c. Since B is a neighborhood of p, there

exists a A > 0 such that Sp xC B and T«< a. By v) there

exist open sets 9x and UK and ahomeomorphism f of C so

that CS U CS
9). m.’ i c

homeomorphism u of C by letting “‘3 = h

a d f C - CZU . Def' e a
n ( 9x) A in

_ -l
‘3’ LL1MB) ' h lhgs)
and ”‘C-(B U h(B)) = the identity on C - (B U h(B)) (this is

a homeomorphism as B and h(B) are clopen, so C - (B U h(B))
also is clOpen, and h and h.1 are homeomorphisms). Consider

the homeomorphism fu on C. u(C - Sc ) Clp(C - h(B))<:
9

B
C - h B = C - B CZC - S . So f C - 9 (I f C - S C:
u. ( ) psi M ) M C:S)
f C - S <: f C - CZU C:S C:S c: . Th S 'S a
( p,x) ( 9x) A Csh C,a 9 u c i n

invert point of C. But this is a contradiction.

viii) p[c] is a fixed point of C; i.e. both are fixed.
Suppose not, then there exists a homeomorphism f of C so that
f(p) # p [f(c) ¥ c], then f(p) é {p,c} [f(c) é {p,c]], as no
homeomorphism of C can send p to c or c to p by vii).
Given any neighborhood 9 of c[p] there is a 5 > 0 so that
C 9]. Let a = l-min(B, dist (p,c), dist (f(p),c),

SCSB ’B 2
dist (f(p),p)) [a = 1/2 min (B: diSt (p,c), dist (f(c),c),

C19 S
[ P

dist (f(c),p))]. As C is O-dimensional, there exists a clopen
neighborhood B of p[c] with B c:S [B C:S ] where
P,5 CS6

6 > 0 is that given by the continuity of f so that

21

f(S ) , 6<a [f(Sc O<a].

C Same ,5) C Sf<c>.a’

Thswehavef CS andso fBCC-S
U (B) f(p),a ( ) p30

[f(B) C Sf(¢),cr and so f(B) C C - Sc a].

Psé

But B is a clOpen neighborhood of p[c] so there iS a

)(>0 with h<a so that SPACB[SCKCB]. Byv) there

9 9

exist open sets 9A, UK and a homeomorphism hh so that

-1

h C-e CU d h - ad 8 ,U CS .
if i) i a" i“ Ui)cei “ 9).: m. i c.>.
h hf h -8 ch C-S
T us K (B)<: h(C p,a) X( p

h - C:S
QC h(C 9).)CU C

A

-1 -1 -1 -1
- - h -
Eh). f(B) C h). (C Sew) C h). (C Sc») C x ((3 UK) C 9). C SPA]

-1 , -
and hhf(B)[hk f(B)] is clOpenm 13 n hA£(B) = @[B n hllf(B) = ¢].

:1.

Define ahomeomorphism u of c by “‘13 =hxf|B

and #1 = identity on

-1
p‘ = (h f) l -
hxf(B) A hxf(B) c (B U hxf(B))

C - (B U hkaBD

-1 _ -1
MB hi “3’ ”‘hilfm — (h).

identity on C - (B U hklf(B))].

f)-1‘ 1 , and p,‘ =

- -1
hx f(B) C-(B U h)\ f(B))

Consider the homeomorphism bl” of C [h;1p of C].

u(C - e) Cp,(C - hxf(B)) = C - u(h)\£(B)) = C - B C C - SPAC C - 9)\

so hAu“: - 9) C h)‘(C - 9A) C Uh C Sc,), C SC“), C SC:B 7.: 9. Hence

c is an invert point of C, a contradiction.

- -1
[MC - e) c h(c - hx1f(B)) = c - p,(h)\ f(B)) = c - B c: c - SCAG
c - U so hilum - 9) Ch-

l (C-Ux)CeCs CS C

S C
A A PSA pad PaB

Hence p is an invert point of C, a contradiction.]

e 0

ix) If p i 9e for some , then let a = 1/2 minimum

0 so
(so, dist (p,c)). So there exist open sets 9 and Ua and a
a

homeomorphism h of C such that h (C - e ) CCU , c E U ,
a a a a a

gas ,ch
a c

. f la a d
a p,a I p E 90’ rep ce 96, US’ n h, by

22

a d h t' l . f . h a U =
ea, Ua’ n 0 respec ive y I p i en T en 3 an n a ¢
C - 0
we have ha(p) ha(C 9a)<: Ua and p 4 no Thus p is not a
fixed point after all,aicontradiction. Hence p E ed as above.
And for the new choice of e = e and e = U , and h = h we
P a C a

CY

have p 6 9p. Making this change for all 60 such that p é 9
e
0
gives us the theorem.l:J

Theorem 2.3: The non-invert point case in the conclusion of

Theorem 2.2 cannot occur.

‘nggfz Assume as in the non-invert point case in the conclusion

of Theorem 2.2, that there exist p, c E C, C a compact 0-dimensional
metric space with h(p) = p and h(c) = c for all homeomorphisms

h of C onto itself. Also given any 3 > 0, there exist open

neighborhoods 9p of p and 9c of c, With apt: Sp , 6C C:S

96 C36

and a homeomorphism h of C onto itself such that h(C - 9p) 3 6C

(note that h"1 is thus a homeomorphism of C onto itself Such

-1
that h (c - 9C) c 9p).

He eweha that hC-S ChC- C CS
nc ve ( p,6) ( 9p) 9C C’s

'1 -l
a d h C - S CZh - C2 C23 .
n ( Cae) (C 9C) GP Pss

Now let d = 1/3 distance (p,c), so d > 0 and Since C

is 0-dimensiona1 there exist clopen sets D0 and E0 With

p 6 D0 C S and c E E Note D0 n E0 = o. As D0

p,d C Sc

0 d'
and E0 are open, there exists an do > 0 so that Sp,aO<: D0

and S CZE , so a s d . So by the hypotheses there exists
c,a0 0 0

of C onto itself such that hO(C - S )c:
a

a homeomorphism. h0

C:
S . S h - C - ‘ . t
p’ao 0 0(C E0) ho(C chdo) C Sp,ao L.D0 LB GO

C-(DOUEO)CC-E

O

0’ then h0(GO) CZDO. Note GO and hO(GO)

23

a
. 0 .
are clOpen sets. Let d1 = minimum {3,‘§-,‘5 distance ({p,c},
G0 U h0(GO))}. Then there exist clopen sets D1 and E1 with
p 6 D1 C Sp’dl and c E E1 C Sc,d1' As D1 and E1 are open,
there exists an a > 0 so that S c:D and S c:E ,
1 p,a1 l c,a1 1

so a s d . By the hypotheses there exists a homeomorphism

h

h

(homo) U D1) C C - D1, then h1(Gl) C E

l

1

1 l

of C onto itself Such that h1(C - S

(C -Dl) ch1(C - S

) c s

P 01

C,a1

P301

CE. Let

1

1.

GI

) C S . So

C,dl

= D -
0

Define inductively di’ Di’ E1, ai’ hi’ and G1 as follows

 

a.
for each integer i 2 2. Let d, = minimum [973 1.1, l-distance
i-l i 21 2 2
({p,c}, U (Gj U hj(Gj)))], then there exist clopen sets D,
j=0 1
and E1 Wlth p E Di C Sp’di and c E Ei C Sc,di' As Di
and E. are open, there exists an a. > 0 so that S C1D.
1 1 p,ai 1
and 8 CE,. Note that D,CD, and E,CE .
c,a i 1 1-1 1 i-l

homeomorphism hi of C onto itself such that hi(C - S

1

Now if i is even, by the hypotheses there exists a

So hi(C - E1) C hi(C - Sc )

’ai

CZD,. LEt
930- 1

(hi-1(Gi-1)U Ei) C C - E1, then hi(Gi) C Di'

h.
i

If i is odd, by the hypotheses there exists a homeomorphism

of C onto itself Such that hi(C - S

hi(C - Di) C hi(C - Sp

3

i

p,ai

CE.’ mt
1

(hi_1(Gi_1) U Di)<: c - Di, then hi(Gi)<: Ei.

G.
1

’ i
Gi ' Ei-z '

) C:S . So

Define the function f of C onto itself by f‘G = h

h-l

flhimi) = i ‘hi(Gi)’

D.
i

are clopen for all

f(P)

i, the

G.
i

= c, and f(c) = p.

and hi(Gi)

1

Since E1 and

are clopen for

) c:S

i G.
1

P.a.°

1

24

all i (inductively as CO is clopen and all the hi are homeo-

morphisms), and the family of sets {H‘H = C1 or hi(Gi) for some
non-negative integer i} is easily seen to be pairwise disjoint,
we have that f is a bijective function of C onto itself which

co

is continuous on U (G. U h,(G,)) = C - [p,c]. As d s 2:3 the
. i 1 i i 1
i=0 2

l

Di's converge to p and the Ei s converge to c, thus for i

= 2k+l, D. = {p} U U G U U h (C ), while for i = 2k,
1 a j=k+l 2j+1 j=k+l 2j 2j

a)
E. = {c} U U G . U U h (C , ). So for i even f(E.) CiD,
i j=k+1 23 j=k 2j+l 2j+l 1 1-1

and for i odd f(Di) CZE . By this, f is easily seen to be con-

i-l

tinuous also at p and c. Hence f, a bijective continuous func-

tion from a compact space to a Hausdorff Space, is a homeomorphism.

But f(p) i p (and f(c) # c) which contradicts our
hypotheses. Hence our assumption of the existence of a Space
satisfying the properties of the non-invert point case in the

conclusion of Theorem 2.2 was false.EJ

Theorem 2.4: If Mn is a closed n-manifold, C is a compact
n n n
0-dimensional subset of M , and M - C is l-movable, then M

is an n-Sphere.

Proof: Let Mn and C be as in the hypothesis and Suppose Mn
were not an n-sphere. A result of D. Galewski, Corollary 1.14

of [9] is that C has an open neighborhood, U, which imbeds in

B“. So Mn - U is a compact subset of Mn - C. d = distance

n

(c, M - U) > 0. Let c, {Mt }, gt be as in the proof of

J J

Theorem 2.2. c E U, which is open, and Mn is a n-manifold with-

out boundary, so there is an Open set V which is homeomorphic

n

to E and c E V CZU. Since the g (M.t ) converge to c, there

tJj

25

is a positive integer y Such that gt (M.t ) c:V for all integers

J J
l
j > y. There is an integer b > y so that g’< d. Now, tb 2 b
l 1 n n
so —Sh<d' Hence M 2(M -U), so gt (M -U)Cgt (Mt)CV.
th th b b b
So gt (Mn - U) is contained in the interior of a closed n-cell,

b
D, with bicollared boundary in V. Hence Mn - U is contained

in a closed n-cell (gtb)-1(D) with bicollared boundary, which

shall be taken as the initial n-cell in the proof of Theorem 1 of

[7]. Thus the "standard decomposition" of [7] which is obtained, Pn U

R = M“, has Mn - U<: Pn so R CZU. This says R has a neighborhood, U,

which can be imbedded in E“. But this contradicts Theorem 4 of [7].[3
Theorems 2.2, 2.3, and 2.4 together imply immediately the

following corollary.

Corollary 2.5: If Mn is a closed n-manifold, C a closed 0-

dimensional subset of Mn, Mn - C is l-movable and n 2 2, then Mn

is the n-sphere and C has an invert point.

Corollary 2.6: Let C be any Cantor set in E“, n 2 2. Then

En - C is not l-movable.

2522;: Let h be the inverse of the Stereographic projection homeo-
morphism, h: En a Sn - [p] C Sn, p 6 8“. So h] n is a homeomorphism
of En - C onto Sn - ({p] U h(C)) and H = [p] UCh(C) is clearly
O-dimensional and closed. Suppose En - C were l-movable, then by

Corollary 2.5, H has an invert point. But {p} is the only isolated

point of H and so H has no invert point. This is a contradiction.[]

Because of Corollary 2.5, let us now look at the comple-

. n
ments of some O-dimen81onal subsets of S .

26

Notation for lemma 2.7 is as follows: let "Cantor set"

mean a O-dimensional compact, perfect metric space, let

n+2
x - (x1,x2,...,xn+2) E E

Sn+l = {x E En+2‘ ‘x‘ = 1}

n

S n+1‘

= 0}

[x E S

xn+2

SI = {X E sn+1‘

p = (1,0.....0) e an“ .

Lemma 2.7: Let C be a 0-dimensional compact set, C C Sn - [p],

n 2 1, A<: 81 - [p], and a a homeomorphism from C onto A. Then

Sn+1 - C is homeomorphic to Sn+1 - A by the restriction to

+ +
8n 1 - C of an extension of a to Sn 1. There exists such an

A and a for each such C.

Proof: 1) The stereographic projection, g from Sn+1 - [p] to

En+1 = En X E1 can be taken so that g(Sn - {p]) = En x {0] and

8(81 ' [P}) = [(03'0-:0)} X E1°

ii) Thus C and A are compact and so g(C) and g(A)

are homeomorphic by g a g and compact,and so closed.

-1‘
8(0)
Also g(C) : En X {0] and g(A) C {(0,...,0)] x E1. So by Klee's

n+1 = n

Theorem 3.3 [15], 3 a homeomorphism h(3 E E X B1 such

that h(g(C)) = g(A) and h]
+1

g(C) = gag-1‘g(c). Extend g-1hg

to h, a homeomorphism of Sn onto itself, by h(p) = p. Then

1

h is a homeomorphism of Sn+ and h(C) = A. Note:

- - -1 -1 -1 -1 -1
fi‘c = 8 1hglc * s 1h\g(c) = s (h|g(c)) = 8 gas |g(c) = g gag s\C =

a‘c

= a. So h] n-l is the desired homeomorphism.
S

27

iii) C C Q_ (a "Cantor set") 5 S“. C has at most a

d
countable number of isolated points (as Sn is 22- countable)
say {ai]:=1, For each i, there clearly exists a "Cantor set",

T. C:Sn with a. E T. and diameter T < 1.,
i 1 1 i i

Q
Let g_= C U ( U Ti) C 5“. Now, the T1 are compact,
i=1
thus closed, and 0-dimensional in ‘Q, so that Theorem II 2. of

[11] applies to give 9_ is a O-dimensional metric space. Q
clearly has no isolated points and so is a perfect Space. Let

c be a point of :§ . If c E C, then c E Q, If c E C, then
d = distance (c,C) > 0. So there is a positive integer n with
%-< %u Thus the neighborhood N6 = {x E Sn‘distance (c,x) < e]

of c for s = %' is disjoint from TK for all K 2 n (else
d d

 

d 5 distance (c,aK) < §'+'§'= d). Thus N6 0 (ginTK) = ¢ for
all e S %3 But N 0'9 f ¢ for any 6 > 0, so for each
d e n-1 n-1
0 < e < ‘3 N C (C U U T ) # ¢. Thus c E (C U U T ) =
2 e _ K __ K
K—l K—l
n-l ‘_
C U U TK G C, Thus g. C Q and so g_ is closed, therefore
K'l

compact. Hence .9 is a "Cantor set".

iv) It is well known that any 2 "Cantor sets" are
homeomorphic (reference Corollary 2-98 of [10]). Since there
is clearly a homeomorphic image B of the standard Cantor set
in S1 - {p}, there is a homeomorphism f from g. onto B,
and so, f(C) G B C S1 - [p] is the desired set A.[:l

Lemma 2.8: Let C be a Cantor set C Sn - [p] (with notation

as in.Iemma 2.7) n 2 1. Then Sn+1 - C is l-movable.

28

Proof: Using the notation of Lemma 2.7 and of i) in its proof,

let D be the Standard Cantor set in [0,1] C E1 and

{x E E1\x +-l'+-t E D for some integer t] and consider

2
{(0,...,0)} x F C {(O,...,0)} x EICEn x E1. Then

F

2
ll

En X E1 - H is l-movable. Since if M is a compact set in

T=Ean1

t-l
ll

1 1
H, then S={yEE1|(x,y)EMcEan -HcEan

n . .
for some x E E ] is compact. So there eXiStS (an integer L > O

with ‘y‘ < L for all y E 8. Define h: T ~ T by h(x,y) = (x,y

+ 2L). h is clearly a homeomorphism of T onto itself with M H h(M) = ¢.

n+1

Now g-1|T is a homeomorphism from T onto S -
-l n+1 n+1
([p] U 8 (H)) and f. S a 3 defined by f(x1,x2,...,xn+2)
is a homeomorphism of St“.1 onto itself

= (-x1,X2, o o o ,Xn+2)
. 1 l

with f(S ) = S .

Let A = f({p} U g-1(H)) which is a Cantor set since
[p] U g'1(H) clearly is a Cantor set.

Now 0 E F, so {(0,...,0)] X {0] E H. Thus

-1 -1

p = f(-l,0,...,0) = f g ({0,...,0] X {0}) E f g (H). Also
f(p) # p. 80 p E A and Ac: 81 - [p]. Since any two Cantor

sets are homeomorphic there exists a homeomorphism a from C

onto A. So Lemma 2.7 applies and Sn+1 - C is homeomorphic to

Sn+1 ' A, WhiCh by f = f-I‘ is homeomorphic to Sn+1 -
Sn+1_A

([p] U 3-1(H)) which is homeomorphic by g to T which is

l-movable. Thus Sn+1 - C is l-movable.[3

Lemma 2.8 followed by Corollary 2.5 proves that a Cantor
set has an invert point (this could of course be shown directly).
The following theorem is a counterexample to the converse of

Corollary 2.5, in dimension 2 3; that is, Cc: S“, n 2 3, and C

29

has an invert point does not imply that Sn - C is l-movable.

Theorem 2.9: For each integer n 2 3, there exists a Cantor set

c in sn with sn - c not l-movable.

Proof: Let B be a wild Cantor set in Sn (such exist by [3]),
then where p E Sn - B and g: Sn - p 4 En is the stereographic

projection homeomorphism, B is wild in Sn - p and g(B) is wild

in B“. By Corollary 3 of [19], g(B) contains a Cantor set A

which is wild at each of its points in E“. There exists a trans-

lation homeomorphism f of En with f(A) CZE2 X (0,+m) X En-Bcz

E2 X E1 X E“.3 = En (as A is compact). Let L' be the

standard Cantor set in E1. Then L = L' X {(0,—l,0,0,...,0)]<:
1 -
E X Eu 1 = En is a tame Cantor set in E“.

Clearly f(A) U L is a Cantor set in En which is locally
wild at each point of f(A) and locally tame at each point of L.

Let C = g-1(f(A) U L). There is a constant, K, large

2
2

2 2 ,
[(x1,...,xn) E En‘ x1 +'x2 S K and x3 = x4 =...= xn = 0] gives

n
three connected components for E - T, the two bounded ones are

enough so that T = [(x1,...,xn) E En] xi +-x +...+ xi ' K] U

u' and v', containing reSpectively f(A) and L.

So Sn - g-1(T) has three connected components, with two

of them u = g-1(u') and v - g-1(v') containing respectively
g'1(f<A)> and s'lm.

Suppose Sn - C were lwmovable. let L = 1/4 min {dist
(g'1£(A>, g'1(L>), dist (g'lf<A), s'1<r>>, dist (s'1(L). g'1(T)>,

diam (g-1f(A)), dist (p,c), diam g-1(L)].

30

In the proof of Theorem 2.2, there exists a tj with
0
‘th
l/tjO < L and ht (Mt. ) C Sc,L W1 ht a homeomorphism of
jo JO 30
Sn - C. iFrom lemma 2.1, ht extends to a homeomorphism h
jo

n
of S onto itself. L was chosen so small that no component

of Sn - Mt contains points of both g-1f(A) and g-1(L)

j0
(as g‘1(T) separates g-1f(A) and g-1(L),while g-1(T)CMt ).
jo
But Sn - Sc L is connected and so contained in a single connected
3
component of Sn - h(Mt ). But L was chosen so Small that
jo
Sn - Sc L contains an open neighborhood of a point from each of
9

g‘luA) and 3‘10». say a e g'luA). b e g'la.) and a e 9,.

. n n
beeb,ea and 9b are Open In S sand eaUebcs -Sc,L

Thus h-1(ea U eb)<: some connected components of Sn - MC
1 1 1 jo
and so {h‘ (a), h' (b)} is a subset of either g' f(A) or
g-1(L). In the first case let a = h-1(b) E g-1f(A) and m = h,
and in the second case let a = a and m = h-l. Then m is a

homeomorphism of Sn and a E g-1f(A) with m(a) E g-1(L). AS

h is an extension of ht and p EMt (by the choice of L)
1 jo jo
- n
{9:CP(P)} C [Pal-1(1)) :h (9)} C S ' C.

n
Since 8 - C is a connected manifold, for any two

points (and so for p and m(p) in particular) of

Sn - C there is a homeomorphism Y' of Sn - C onto Sn - C

which is the identity outside of and on the boundary of an n-cell

in SD - C, and Y'(¢(p)) = p. Extend Y' to a homeomorphism

Y of Sn onto itself by Y] n = Y' and Y‘C = identity on C.

31

Thus Ym is a homeomorphism of Sn with Ym(p) = p and

so Ym‘ n is a homeomorphism of Sn - p onto itself. Hence
S -p 1
p = g(Y¢‘ n )g- is a homeomorphism of En onto itself which
8'?
sends g(a) E f(A) to an element B of L.

Then there exists an open neighborhood 9 of g(a)
with e c: 1:2 x (o,+m) x End and Me) c: E2 x (40,0) x En'3.
So M9 n(f(A) U 1.)) -- Me n f(A)) c L which is tame in s“
and so f(A) U I. is locally tame at g(a) E f(A). This is a
contradiction as f(A) U L is wild at each point of f(A). Thus
Sn - C is not l-movable.D

There is perhaps a chance of proving the following converse

of Corollary 2.5 when n = 2, because of Lemma 2.11 and Theorem 2.13.

Conjecture 2.1Q If C is a closed O-dimensional subset of 82

and C has an invert point, then 82 - C is l-movable.

2
Lemma 2.11: If C is a 0-dimensional compact Subset of S and
h is a homeomorphism of C onto itself, then there exists a

A

. 2 .
homeomorphlsm h of S onto itself so that h‘ =h.

uggggf: i) C is a subset of a 0-dimensional, compact, perfect
subset A of 82. First C is a second countable metric
Space (as 82 is such), and hence has at most a countable number
of isolated points, say [xi‘i E I} for some I<: [1,2,...].

For each i E 1, xi has a open neighborhood N with N c:Sx

1
1’1

and a homeomorphism g: N d E2 with g(xi) = (0,0). Let

A
xi

g-l(B) where B is the standard Cantor set in [0,1] X

2
[0} CZE . Then AX is a 0-dimensional, compact, perfect Subset
i

32

of 82 with x E A <: S 1 . Let A = C U U A . Then by

"i xi’i' iEI xi
Theorem 11 2, of [11], A is O-dimensional, A clearly has no
isolated points, and as diameter Ax < %' with A and C
. x

i
closed for all i E I it is quite easy to Show that A is

closed. Thus A is also perfect and compact in 8

ii) Let x E $2 - A. Then there is the stereographic
projection homeomorphism s: $2 - {x} 4 E2. Let B' be the
standard Cantor set in [0,1] X [0}, thus a perfect, compact,
O-dimensional set in E2, as is S(A). As components of 0-
dimensional sets are points, Theorem I of [1] applies and there
is a homeomorphism t of E2 onto itself with t(s(A)) = B'.

Now let B = t(s(C)). If it were true that for every homeomorphism

. . 2
g of B CZE2 onto itself, there IS a homeomorphism g of E

onto itself with §|B = g, then we would be through. For given a

homeomorphism h of C onto itself, g = ts hS-lt-I‘B is a

homeomorphism of B onto itself, and so extends to homeomorphism

A

2 h A - - A
g of E onto itself with g‘B = g. Thus f s 1t 1 t

g s is

ll

2
a homeomorphism of S - [x] onto itself and so extends to a

2
homeomorphism h of S onto itself by h(x) = x and

* * -1 -1
h‘sz - f. NOW h‘C = f‘C = S t g ts‘c (and as tS(C) =

'{X}

'1 '1 _ -1 -1 -l -l _ -1 -1 -1 -1 _
B): s t gts‘c - s t (tshs t )ts|C - (s t ts)h(s t ts)‘c _
h\C = h.

iii) Thus it remains only to show that if g is a homeo-

morphism of B, a closed subset of the standard Cantor set C [0,1] x

x 2
{0] C1E2, onto B then there is a homeomorphism g of E onto itself

with §|B = g. 80 suppose such a B and g are given. Define

33

l 2
a homeomorphism r from B onto a closed subset of [0] X E C:E

by r(y,0) = (0,2) where g(y,0) = (2,0). Then by Klee's Theorem
3.3 [15], there exists a homeomorphism f of E2 onto itself
with f‘B = r. L: E2 « E2, defined by L(x,y) = (y,x), is a

homeomorphism of E2 onto itself and so let g = L - f.[3

Corollary 2.12: Let c C 3n - {p} c: s‘“+1 - [p] as in Lemma 2.7.
Then any homeomorphism of C onto itself extends to a homeo-

morphism of Sn+1 onto itself.

Proof: Let h: C a C be a homeomorphism of C onto itself. By
Lemma 2.7 there exists a homeomorphism a: C a A (Of C onto a
set A c 81 - {p]) and this homeomorphism extends to a homeo-

, ._ n+1 . -
morphism a of S onto itself. Then ah: C w A is a
homeomorphism satisfying the hypotheses of Lemma 2.7, so there is
a homeomorphism B of Sn+1 onto itself and B‘C - ah-

, - -l n+1 ,
ConSider the homeomorphism (a) B of S onto itself,

o'lah = h.l'J

(E)'la\c = (E)'1(e\c> = (Efloh -((E)'1‘A)oh
Theorem 2.13: $2 - C is l-movable when C is a "Cantor set".

‘nggf: Let us use the notation for Lemma 2.7 with n = 1. If p E C,
then there is q E $2 - C and there is a homeomorphism h of 82
onto itself with h(q) = p. Thus without loss of generality we may
assume p E C. So let g be the stereographic projection of

$2 - [p] onto E2. Then g(C) is a "Cantor set" in E2 and

g(S1 - [p]) is a line in E2. There exists a "Cantor set", say

B, on g(S1 - {p]). 80 by Theorem I of [1] there is a homeomorphism
f of E2 onto itself with f‘g(C) a homeomorphism from g(C)

onto B. Thus Q = g-lf g is a homeomorphism of $2 - [p] onto

34

itself and so has an extension, say Y, to $2 by Y‘ 2 = Q
8 -{PJ

and Y(p) = p. Then letting the C and n of Lemma 2.8 be

3-1(3) and 1 respectively, Lemma 2.8 tells us that S2 - g-1(B)

is 1-movable. But Y(C) = g-1(B) so Y‘ 2 is a homeomorphism
S -C

2 2 -
from S - C onto S - g 1(3). Hence 82 - C is l-movable.[]

CHAPTER III

SIMPLICIAL CDMPLEXES AND 1-MOVABILITY

In this chapter we will first prove some lemmas and a

theorem to enable us to prove the main re8ult of this chapter,
. . (n-2)

which is that ‘K‘ - ‘K j is l-movable for each n-dimen-

K(q) is the

Sional abstract simplicial complex K, where
q-Skeleton of K and n 2 3. We then observe that a l-movability
condition and engulfing theorems trivially give characterizations
of the n-sphere using a theorem of Doyle and Hocking in [7]. We
will then give many examples Showing that even when K is a
compact combinatorial n-manifold we cannot claim |K‘ - \L‘ is
1-movable, when L is a q-dimensional subcomplex of K, 0 S q S n
or even when L = K(q) for 0 S q 5 [Egg].

When a is simplex, a E ‘o‘, B C ‘0], a-B will denote
the set {to + (1-t)e|t 6 [0,1], 3 e B}, at will denote the
t-skeleton of 0 considered as the simplicial complex consisting
of all the faces of o, and <o> will denote the set ‘0‘ - ‘6‘
as in [22] with & = 80 = C(n-l).

Lemma 3.1: Let a be an n~simplex, C a compact set, c ¢ ¢,

o‘“'2)|, U \o‘“'2’\

in lad

C<: ‘0‘ - l a neighborhood of
with U'fl C = ¢. Let 6 be the barycenter of 0. Then there
is a pseudo-isotopy [Ht] of lo] onto ‘0‘ which is fixed on

”(WIN and has H1(C) c: a-qas‘ - E).

35

36

§r_og_§: We may consider [6‘ to be linearly embedded in En and
Bn to be the n.- ball with center 6 and radius large enough,
say R, so that ‘6‘: B“. There is a homeomorphism h of ‘0‘ onto
Bn induced by the radial projection of |°(n-1)‘ onto bdy 8“.
Now, h(U) n h(C) I ¢ (as 6.0 C 3 ¢) SO d = distance
(h(B),h(c))>0. Let Dn be the n-ball with center 6 and radius
R - d/2. Then (a - h(U)) - (a - (radial projection from a of
h(U) onto bdy Dn))is disjoint from h(C).
let {Gt} be the obvious pseudo-isotopy that shrinks

Dn radially down to 6 and sends each ray from 6 onto itself.

(l-t)(x-6) + 6 for ‘X-a‘ s R - d/2
ct“) = (it-a) R + (lx-6j-R)[R-(1-t) (R-d/2)] ) + ..
|x-a| ( ' ' d/2 °

for R 2 ‘x-6‘ 2 R - d/Z

where the points x and 6 are considered as vectors from the

origin and

2 2 2
1(y13y2’°°°ayn)‘ =(y1 +y2 +-~-+ yn)% .

Thus G1(h(C)) C 6. (bdy Bn_ h(U)) and so [Ht] where
Ht = h-lcth is the desired pseudo-isotopy on o as

h also preserves rays from 6.[3

Lemma 3.2: Let a be an n-simplex, 6 its barycenter, and F
a compact subset of ‘6‘ - ‘O(n-l)‘.
Then there is a pseudo-isotopy [Ht] of ‘0‘ onto ‘0‘

which is fixed on ‘O(n-l)‘ and has H1(F) = O.

37

Proof: Construct Bn and h as in Lemma 3.1 and let d = distance
(bdy Bn,h(F)).Then d > 0. let 1)“ be the n-ball of radius
R - d/2 and center 6 with {Gt} as in Lemma 3.1. Then as

h(F) c n“, Gl(h(F)) = a, so letting

1

HC = h Gth’ [Ht] is the desired pseudo-

isotopy on ‘0‘ .[3

Lemma 3.3: let a be an n-simplex with barycenter 6 and
Tl’T2’°"’Tn+l the (n-l)-faces of a. Let iGt be a pseudo-
isotopy of ‘Til onto lTi‘ fixed on ‘Ti(n-2)| for each

i = 1,2,...,n+l. Then there is a pseudo-isotopy [Ht] of lo]

onto M with Ht‘l'iil =iGt for tE[0,l] and i=l,2,...,n+l.
Proof: Let [Gt] be the pseudo-isotopy of \c(n-l)‘ defined by
Gt‘lTil = iGt for i = 1,2,...,n+l and t E [0,1]. Using polar
coordinates in lo] (as in [22], p. 117), [Ct] extends to the

pseudo-isotopy [Ht] by
Ht([dsl]) = [Gt(o)s k]-

[Ht] clearly has the desired properties. [3

Theorem 3.4: Let K be a simplicial complex (not necessarily

(“‘2)‘

 

locally finite) of dimension n, A be a subset of ‘K| - ‘K
which has A n ‘6‘ compact for each 6 E K. Then there are two

pseudo-isotopies, [Ft] and [Gt], of ‘K‘ onto \K‘ which are
(“'2)‘

fixed on ‘K and have

38

c1F1(A)cT= U 6-? U U {’r)
sex-K(“'1) ¢€K(“'1)-K(n'2)

T is (n-l)-face of o and A n.<T> # ¢
and A n <o> E ¢

l2:gg£; For each (n-l) simplex T of K, obtain, from the con-
struction in.Iemme.3.2(with ¢ # F = A n ‘7‘ = A n \T‘ - ‘T(n-2)"
T as the o' of lemma 3.2) U1. = h.1(Bn - D“) a closed neighbor-
hood of |T(“‘2)\ in \¢| with ¢ = UT n F = UT n A n |T\ = UT n A
and let {THt] = {Ht} (the pseudo-isotopy of Lemma 3.2 for the above
F) when A n [T] E ¢; when A n ‘T‘ = ¢, let THt = identity on ‘T\

and UT = ‘7‘. TH1(‘T\ - UT) : {6]. Now, for each n-Simplex o of K,

n+1
let TlsTZ....,Tn+1 be its (n-1)-faces. Then U = 9:1 UTi is
“*1 (n-2) (n-2)
a closed neighborhood of U ‘Ti | = ‘o \ in 130‘: By Lemma 3.1
i=1

(With 0 = o, U = U, and C = A n ‘0‘) there is a pseudo-isotopy
{OFt] (the pseudo-isotopy [Ht] of Lemma 3.1) of ‘0‘ onto

‘6‘ which is fixed on ‘g(n-1)‘ and has
CF10. n M) c 5 ~ (\o(“‘1)| - if)

when A H <d> E ¢. ‘When A n <o> = ¢, let {oFt} be the pseudo-

isotopy which is fixed on ‘0‘. OFt = identity on ‘6‘ for all

t E [0,1]. Hence there is a pseudo-isotopy {Ft} of ‘K‘ onto

‘K‘ defined by Ft‘|1“ = identity on ‘7‘ £01? each T e ROI-1)

K(n-1). Now by Lemma 3.3

and Ft‘lo‘ = opt for each a E K -

(letting a = o, 6 = 6, Ti = Ti for i = 1,2,...,n+1, iGt = THt)
we obtain the pseudo-isotopy [Oct] = the [Ht] of Lemma 3.3.
Hence, there is a pseudo-isotopy [Gt] of ‘K‘ onto ‘K‘

defined by

39

K(in-1)

II
C)

Gt“o‘ o t for each 0 E K -

G l = H for each T E K(n-1) - K(n-2)
t l'ri

Tt
= , , (n-2)
Gt“T‘ identity on ‘T‘ for T E K
G and F are both clearl fixed on K(n-2) and
{t} it} y I

G1F1(A) c T is easily verified.[3

Lemma 3.5: let A be a compact Subset of a normal space X,
and [Ht] a pseudo-isotopy of X.

If there is a homeomorphism, g, of X onto itself with
g(H1(A)) n H1(A) = ¢, then there is a homeomorphism, f, of X

onto itself with f(A) n A = ¢.

‘25232: As X is normal, g(H1(A)) and H1(A) closed, we have
disjoint open sets U and V with U 2 g(H1(A)) and V 2 H1(A).
9 = 8-1(U) 0 V is open and H1(A) C e C V, g(e) C U, and

U<: X - 9. Let H: X X [0,1] a X be the pseudo-isotopy {Ht}.
Then H-1(9) is open and H-1(e):3 A x {1] which is compact

in X X [0,1]; so there is a S E [0,1] with H-1(9)23 A x (S,l].
Let p = (1+S)/2. Then Hp is a homeomorphism of X onto

itself and Hp(A)<: 9. Let f = Hglg Hp. Then f(A) = Hglg Hp(A)
c Hglgm) c H1310» c H;1(X-e) c Hglcx-upm) c Hglmpa-A» = x-A.

So f(A) nA = ¢.EJ

Lemma 3.6: Given an n-Simplex 6, n 2 3, with its vertices ordered

as indicated a = {a1,a2,...,an+1], let Ti, i = 1,2,...,n+1 be

the (n-l)-faces of o with their vertices ordered as induced and
indicated by the ordering of the vertices of o,

T. = [alsazsooosa Then there is a well defined

1 i-l’ai+l’°°°’an+l}'

40

set [a,51,82,...,5n*1] Cl‘o‘ dependent only on the ordering of
the vertices of o with a E <o>, Bi E <Ti> for i = l,2,...,n+l,
n+1 n+1
and (U a-Bi) 0(U C-Ti) =¢.
i=1 i=1

Proof: We will use barycentric coordinates throughout. u E 6-?

n+1
for some i, when u = ( 2 [fl+£] aj) + 15—F- i for some

i

 

=1 (n+1) n n+1
i

t E [0,1], and so each coordinate of u is one of at most two

+1

. l-t t 1-t “ 21
_ + - — o =
p0351ble values, (n+1 n) and n+1 let oz .121 (n+1) (n+2) aj E <o>
1-1 2
and for each 1 = l,2,...,n+l let Bi = (.121 n—j_(n+l) aj)
n+1 ZII'E

+ j;§+1n(n+l) aj E <Ti>' Then v E 6-31 for some i, when for

some 3 E [0,1]

 

 

i-l
_ st 2(l-S)j, 2(1-s)i
V ' (j21(n(n+l) + (n+1)(n+2)) a3) + (n+1) (n+2) ai

, “*1 (amuse, 2(1-8)j )
j=§+1 n(n+l) (n+1) (n+2) j

Observe the coeffic1ents of a1,a2,...,ai_1,ai+1,.

the barycentric coordinates of v, are all different values.

.. a in
’ n+1

But this is n different values and n 2 3, SO
n+1 n+1
( U 8 °‘?) (l( U 0" B.) = ¢ .[3
. 1 . i
i=1 i=1
Theorem 3.7: Let K be an n-dimensional simplicial complex (not
necessarily locally finite), n 2 3.

Then ‘K‘ - ‘K(n-2)‘ is l-movable.

Proof: Let A<: ‘K‘ - \K , A compact. We need only Show

 

a homeomorphism, f, Of ‘K‘ - ‘K(n-2)| onto itself with

f(A) n A = ¢. Since A is compact, A H <d> # ¢ for at most a

41

finite number of shmplices a E K (by Corollary 19 of [22], p. 113),

also A and K satisfy the hypotheses of Theorem 3.4. Let

{01’02’°°°’Op} = {o E K-K(n-1)‘<o'> n A if (15} and

(n-1)_K(n-2)|A n.<T> E ¢ or T is a face of a.

[AISA2’°°°’>\q} = {T E K for some i E {1,2,...,p} 1}

with o, E q if i E j and xi # I for i E j. Now let
1

J

q
T= U 8.°i)U(kU{i})o
(I=l,2,...,p 1 k =1 R

i f
3k s a face 0 Ci

J

Then by Theorem 3.4, followed by lemma 3.5 applied twice, the
existence of a homeomorphism, h, of [K] - ‘K(n-2)| onto itself

with h(T) 0 T = ¢ would imply the existence of the required f,
and hence the theorem.
Let S = {v E K‘v is a vertex of either oi for some
i - 1,2,...,p or some xk for some
k = 1,2,...,q] .
S has only a finite number of elements,so they can be ordered

by counting them, v1,v2,v3,...,vL. This ordering of S induces

an ordering of the vertices of each Oi’ i = 1,2,...,p and of
each xk’ k = 1,2,...,q.

Now, for each k = 1,2,...,q, either is not the face

"k

of any oi’ i = 1,2,...,q, in which case choose 6 to be any

k

element of <xk> - [1k], or xk is an (n-l)-face of Oi for an
i E {1,2,...,p}. In the latter case let Oi be the o of

Lemma 3.6, then xk = T for some j E {l,2,...,n+l]. Define

J

6k = the Bj of Lemma 3.6. Note that 6 is well defined for

k

42

each R since the ordering of the vertices of xk induced by
S is the same as that induced by the ordering of the vertices

of any a which has

t xk as a face, t E [1,2,...,p].

As Xk and 6k are both elements of <xk>, it may be
easily shown using Lemma 2 of [22], p. 116, that there is a

homeomorphism gk of ‘xkl onto itself which is fixed on

<n-2) ,
‘xk | and gk(Xk) 6k. Define a homeomorphism g of

(“‘1)‘

‘K onto itself by

g‘lka = gk for each k = 1,2,...,q

and g‘

(n-l) _
\K ‘

q
(n-l)‘ - = identity on [K Eil‘xk‘ .

all
A

k=1 k
For each n-simplex, o 6 K, let Og be the homeomorphism

of ‘0‘ onto ‘0‘ which has defined

8‘ = 8‘ s
o (“-1) (“-1)
\o l \c l
by Cg([a,t]) = [g(a),t] (using polar coordinates in ‘0] as
in [22], p. 117). Then define the homeomorphism, g, of ‘K|

onto itself by g‘|K(n-l)| = g and g‘\o\ = og for each

s e K - K(n-1).
Now for each a E K - K(n-1), let do = 8 if
c E [61,oz,...,op] and a0 = the a of Lemma 3.6 for o = Ci

1
under the ordering of the vertices of oi induced by that of S.

Now define the homeomorphism ah of ‘0‘ onto itself given by

(again using polar coordinates)

Oh([a,t]) = t a0 +(1-t) a .

(“'1)‘

Note h is the identity on ‘6 . Define the homeomorphism,
o

43

h, of ‘K‘ onto itself by

6| = h for each a e K - K("'1)
\ol 0

and 5‘ |r(“'1>‘ .

\K

= identit on
<n-1>| y

Now let h = hg. h is clearly a homeomorphism of \K] onto

itself and hl'o‘ is a homeomorphism of ‘0‘ onto itself and

h‘<o> is a homeomorphism of <o'> onto itself for each 0 E K.
Thus to show h(T) n T

¢, we need only show

(*) h(T n <o>) n (T n <o>) ¢ for each s E K.

This is clearly true for each 0 E K with T n <o> = ¢.

$232.1: 0 E K(n-2). T 0 \K(n-2)‘ = ¢ so T H <O> = m- (*) is
true 0

§§§g_2. a E K(n-1) - K(n-2) and T H <d> E ¢. Then a = 3k
for some k E {1,2,...,q]. But T n«<xk> = Xk and

h<1k> = fia<xk) = fi<g(xk)> = fi<gk<xk>> = fi<5k) ‘ 5k

(n-l)‘.

as 6k E [K But 6k E Xk, so

h(T n <a>) n (T n <o>) = {5k} n {xk} = ¢. (*) is true.

_ K(n-l)

Case 3. o E K and T fl-<o> E ¢. Then a = oi for

some i E [1,2,...,p]. So T H <oi> C T n ‘01]

= U 6, - 1k . Now for A

1 a face of o.
3k is face of Ci 1

k

h(oi-ik) - fi§(ai-ik) = fiél‘oi‘(ai-ik>

fi<oog<a,-ik>> = fi<ai-g(xk)> = fiei gk(lk))
1

= h(°i'5k) = h|\oi\(°i°5k) = c:(°i°5k) = 00,6k

44

= a°6k. (where a is the a of lemma 3.6 for our ordering
of vertices)
= a-Bj (where Bj is the Bj of Lemma 3.6 for our ordering

of vertices and 01 = o and 3k = Tj).

n+1
So hCT ni<oi>) c {J “'51 (the a and Bj being those of
J=1
Lemma 3.6 with our ordering of
vertices and Ci 8 o)
A n+1 A
and T n <O->'§ U 6o°kk ‘ U 6wt, . (the a and T 's
1 - 1 J J
xk is a face of 61 j=l
and k E {1,2,...,q] being those of Lemma 3.6
with our ordering of
vertices and Oi = o).
n+l n+1
But lemma 3.6 gives us ( U a-B ) n (lJ 6-T,) = ¢, hence
i=1 1 i=1 3

h(T n <oi>) n (T n (01>) = ,5. (a) is true. [I]

Let us consider Open n-manifolds that have been obtained by
the removal of a closed set from a closed n-manifold. The follow-
ing theorem is trivially given by the sphere characterization

theorem of Doyle and Hocking [7].

Theorem 3.8: Let Mn be a closed n-manifold, A be a closed sub-
n n n n
set of M , A C V C M with V homeomorphic to E , and M - A

be l-movable. Then Mn is an n-sphere.

‘nggfz Let M“, A, and V be as in the hypothesis. Then A is
compact and so contained in the interior of an n-ball, B C:V, with
bicollared boundary. Let B be the initial n-ball with bicollared
boundary in the proof of Theorem 1 of [7] for the manifold it“.
Then Mn = U U R with U homeomorphic to En and U U R a

"standard decomposition" as in [7] and B C U. Now

45

My - A = (U - A) U R and R is compact in M“, hence compact in

Mn - A. ‘Mn - A is 1-movable so there is a homeomorphism h of

Mn - A onto itself with h(R) n R = q). Thus h-1(U - A) is a
neighborhood of R and U - A is a Euclidean domain (can be
imbedded in En),so h-1(U - A) is also a Euclidean domain. But
then Theorem 4 of [7] says that Mn is homeomorphic to S“. [3

Thus any engulfing theorem that says a closed subset A
of a closed n-manifold Mn is engulfed by an open n-cell tells us
that if Mn - A is 1-movable then. Mn is an n-sphere, also if
Mn is not an n-Sphere then Mn - A is not l-movable. SO the

engulfing theorems of Newman in [18] and Zeeman in [25] give

Corollaries 3.9 and 3.10 respectively.

Corollary 3.9: let X be a locally tame closed set of dimension
s n-3 in M, a p-connected closed topological n-manifold without
boundary (thus n 2 3). Then M - X is l-movable implies that M

is an n-sphere.

Igrggf: Let X and M be as in the hypothesis. Then since there
is an open set V, homeomorphic to En and so (p-l)-connected, in
M and M - V is compact the main theorem of [18] applies to insure
the existence of a homeomorphism, h, of M onto itself with X<:.h(V).
But h(V) is homeomorphic to En and so Theorem 3.8 implies that

M is an n-Sphere. D

Corollary 3.10: Let Mn be a connected closed combinatorial
n-manifold, n 2 3. Let q s n-3 and ni(Mp) = 0 for i = 0,1,...,q.
If Mn - ‘Ll is l-movable where -L is some subcomplex of dimension

q of some triangulation of M“, then Mn is homeomorphic to S“.

46

2:22;: Assume the hypotheses. The result may then be obtained as

a corollary to Corollary 3.9. An alternate proof is to Observe

that Theorem 1 of [25] is that ‘L‘ lies in the interior of an

open n-ball in M“. A8 ‘L‘ is closed in My and an Open n-ball

is homeomorphic to En, Theorem 3.8 implies that Mn is an n-sphere.EJ
We now have some examples that restrict the possible improve-

ment of Theorem 3.7 in certain ways.

Corollary 3.11: Let q and n be integers such that 0 S q S n-3
and q 5 [Egg]. Then there exists a finite n-dimensional simplicial
complex K (that is even a combinatorial n-manifold without boundary)
having ‘K‘ - |K(q)‘ not l-movable.

q+l X n-q—l

Proof: There is a finite triangulation, K, of S S that

is a combinatorial n-manifold. As n-q-l 2 q + 1,

wild) = view) x view"1

(Q)‘

) = 0 for i = 0,1,2,...,q. Suppose

[X] - lK were 1-movable. Then by Corollary 3.10, \K‘ is

homeomorphic to the n-Sphere. But [K] is homeomorphic to

q+l n-q-l

S X S , which is not homeomorphic to the n-Sphere. This

(Q)‘

is a contradiction, so ‘K‘ - \K is not l-movable.[3

For an n-dimensional simplicial complex K, we have the

(Q)‘

question of whether or not ‘K‘ - ‘K must be l-movable. The
answer is affirmative for q = n (since ¢ is l-movable),

(n-l) . . . .
q = n - 1 (since ‘K‘ - ‘K \ is the diSjOint union of open
n-simplexes, each homeomorphic to E“, and so clearly l-movable),
and for q = n-2 (by Theorem 3.7). The answer is negative when

n 2 3, and 0 s q 5 [Egg] (by Corollary 3.11). When

n-2 .
[—§—] < q < n-Z, the answer is unknown at present.

47

Remark 3.12: Observe that when the codimension, n-q, is 0,1,

or 2 we have [K[ - [K9[ is l-movable. However when n-q 2 3,
if we let n = 2q +-2, then Corollary 3.11 gives an example where
[K[ - [K(Q)[ is not 1-movable. Hence for each p > 2 we cannot
have the theorem that [K[ - [K(n-p)[ is l-movable, so 'Theorem

3.7 has the best fixed codimension possible.

Example 3.13: There is a n-dimensional,infinite, locally finite
simplicial complex, K, that is a combinatorial n-manifold and a

triangulation of En = E1 X En"1 (so that those simplices whose

carrier is a subset of E1 X ([O,+m) X En-Z) form a triangula-
tion of [O,+m) X En-l) with the following property. Every
homeomorphism h that is a translation along the first factor
by some integer d (i.e. h(x,y) - (x +-d,y)) is a simplicial
map (that is, h(v) is a vertex of K for each vertex of K
and whenever v1,v2,...,vk are the vertices of a simplex of K,
h(vl),h(v2),...,h(vk) are vertices of a simplex of K). Thus
such an h has h([K(q)[) = [K(q)[ for each integer q,

0 S q S n, and so h[ is a homeomorphism of

lK\-IK(")\
[K[ - [K(q)[ onto itself.

Let q be an integer, 0 S q S n-1, and A be a compact
subset of [K[ - [K(q)[, then A<: [-a,a] X En-1 for some integer

a > 0. So the homeomorphism h that is a translation by the

integer 3a along the first factor has h a homeo-
K[-[K(q)
( \ \
morphism of [K[ - [K q)[ onto itself and A n h(A) = ¢. Hence
[K[ - [K(q)[ is l-movable, and so examples of l-movability occur

for all integers q and n with 0 S q S n-l.

48

Suppose we consider (as a relaxation of the previous ques-
tion) the possible l-movability of [K[ - [L[ where K is an
n-dimensional finite simplicial complex that is even a compact
combinatorial manifold, L is a subcomplex of K(q), L E K(q).
There are three basic possible relationships possible for K and
L as follows:

a) K has no boundary

b) K has non-void boundary, 5K, and L n 5K = ¢

c) K has non-void boundary, 3K, and L 0 5K # ¢ .

For each of a), b), and c) we have an example with [K[ - [L[

l-movable and another one with [K[ - [L[ not l-movable (except

for b), where no l-movable example could exist), for each pair

of integers n and q, n 2 2, 0 S q S n.

1) For a) l-movable, let X = 6, for an (n+l)-Simplex o, L be a
q-simplex of K, with its faces. Then [L[ is cellular in
[K[, so [K[ - [L[ is homeomorphic to [K[ less a point, SO

n

to S less a point, so to En and so 1-movab1e.

2) For a) not l-movable, let K be a triangulation of S1 X Sn-1
and L be the faces of a fixed q simplex of K. Then [L[
is a subset of an open subset Of [K[ that is homeomorphic
to E“, and S1 x S".1 is not homeomorphic to 8“. Thus
Theorem 3.8 would be contradicted if [K[ - [L[ were l-movable,
and so [K[ - [L[ is not l-movable.

3) For b) not 1-movable, let K be a complex that triangulates
S1 X [Dn-1[ where DH”1 is a (n-l)-dimensional simplex and

K has an n-simplex o with 0 0 3K = ¢. Let L be the

faces of a fixed q-dimensional face of 0. Then [K[ - [L[

4)

5)

49

n-2

is a n-manifold with boundary homeomorphic to S1 X S
but then the boundary is compact so Theorem 5.1 implies that
[K[ - [L[ is not l-movable.

For c) l-movable, let K be the first barycentric subdivision
of the faces of a fixed n-simplex. When q S n-l let L be
the faces of a q-simplex of boundary K, and when q = n let
L be the faces of a n-simplex of K that has exactly n of
its vertices in boundary K. In either case [K[ - [L[ is
homeomorphic to [K[ less a single point, and so homeomorphic
to [0,+m) x En'l. Thus [K[ - [L[ is l-movable.

For c) not l-movable, let K be the second barycentric Sub-

n-l[

division of a complex that triangulates S1 X [D where

D“.1 is a (n-1)-dimensional Simplex. For q S n-l let L

be the faces of a cellular q-simplex of boundary K, and for

q = n let L be the faces of a n-simplex with exactly n of
its vertices in boundary K and L n boundary K cellular in
[5K[. In either case, Remark 5.2 tells us that if [K[ - [L[
is 1-movable, then so is B = boundary ([K[ - [L[) =

[aK[ - [L[, which is homeomorphic to [3K[ - [p] for a point
p E [5K[. But p has an open neighborhood homeomorphic to

En-l. When n > 2, Theorem 3.8 says that [5K[ is homeomorphic

to Sn-l. But [aK[ is homeomorphic to S1 X Sn-Z, thus not
homeomorphic to Sn-l. Hence [K[ - [L[ is not l-movable.
When n = 2, [5K[ - [p] is homeomorphic to (SIX [0,1]) - {x}
(for x E S1 X [0,1]), which has exactly one compact component.

This contradicts Theorem 5.1 and so [5K[ - [p], hence B, hence

[K[ - [L[ is not l-movable.

CHAPTER IV

PRODUCTS AND k-MOVABILITY

In this chapter we prove a theorem about products and
then give some examples showing that the non-compactness hypothesis
of the theorem cannot be dropped and that even under fairly strong

restrictions the converse of the theorem does not hold.

Theorem 4.1: If X is a non-compact k-movable topological Space

 

and all compact Subsets of X are closed, and Y is any tOpological

Space, then X X Y is k-movable (k any positive integer).

Proof: Let X and Y be as in the hypotheses. Given a compact
subset, K, of X X Y, L = p1(K) is compact in X (where pi is
the projection of X X Y onto the ith factor). So there exist

compact subsets of L, say L1,L2,...,Lk, and homeomorphisms of
k
1,h2,...,hk, so that L = U Li and
i=1

1,2,...,k.

X onto itself, say h

L U hi(Li) = w for i
Now for each i = 1,2,...,k we have Li is compact,
thus closed, and so p;1(Li) is closed and K1 = K O p;1(Li)
is closed in K and so compact. Define the homeomorphism fi
of X X Y onto itself by fi(x,y) = (hi(:)’Y)'
It is now easily seen that K = U Ki’ the K

1
i=1
-1
and fi(Ki) : fi(p1 (Li)) : fi(Li x Y) c: hi(Li) x Y. But

are compact,

K c: L x Y, so K n fi(Ki) : (L x Y) n (hi(Li) X Y) =

(L Ohi(Li)) XY=¢XY=¢.D

50

51

Remark 4.2: The non-compactness of X when k = l in Theorem 4.1
cannot be dropped (even if we require X and Y to be manifolds)
as Sm is l-movable and Sm X Sm is not l-movable. Note that a
closed n-manifold is l-movable if and only if it is invertible
(this is easily seen by the definitions), and invertible if and

only if it is a n-sphere (see [5] Theorem 1, and in [6] the state-

2
ment on p. 959), and Sm X Sm is not homeomorphic to S m.

Remark 4.3: The converse of Theorem 4.1 with k = 1 does not
hold as there exists a non-compact non-l-movable metric space G
with G X G homeomorphic to E6, and E6 is l-movable.

Let G be the decomposition Space G of E3 of R.H. Bing
in section 2 of [2]. The union of the family of non-degenerate
inverses of points G is a subset of the set A (which is a
topological solid Sphere with two handles) of p. 485 of [2]. A
is clearly compact. But G is a quotient space of E3, say by
the quotient map g, so g(A) is a compact subset of X. Assume
G is l-movable, then there exists a homeomorphism h of G onto
itself such that A n h(A) = ¢. Let 2 be one of the points
of G which has a non-degenerate inverse. Then h(z) has a
degenerate inverse and so h(z) has a neighborhood homeomorphic
to E3, and hence 2 has a neighborhood homeomorphic to E3,
so G is locally E3 at 2. But this contradicts Theorem 13
of [2].

Now, A. Boals has shown in [4] that G X G is homeo-

morphic to E .

52

Remark 4.4: The converse of Theorem 4.1 does not hold for any k

 

even if we require X and Y to be manifolds, either one of which
may be Specified to be compact and either one of which may be
specified to be l-dimensional.
This may be seen by using the following facts.
3) If an n-manifold Mn has compact boundary then it is not
k-movable for any positive integer k (see Theorem 5.1),

b) (0,1) X [0,1) is homeomorphic to [0,1] X [0,1).

Now (0,1) x ([0,1) x s“) is homeomorphic to ((0,1) x [0,1)) x s“,
which is homeomorphic (using b)) to ([0,1] x [0,1)) x s“, which
is homeomorphic to both [0,1] X ([0,l) X S“) and

[0,1) x ([0,1] x s“).

For each positive integer k, (0,1) is k-movable, so
Theorem 4.1 tells us that (0,1) X ([0,1) x S“) is k-movable. But
[0,1],[0,l),[0,l) X S“, and [0,1] X Sn. are all manifolds with
compact boundary, and so they are not k-movable by a).

Thus counterexamplesto»the converse of Theorem 4.1 are

x = [0,1], Y = [0,1) x sn and x = [0,1), Y = [0,1] x s“ .

CHAPTER V

MANIFOIDS AND k-MOVABILITY

In this chapter we prove that if M is a l-movable n-manifold
with boundary then the boundary of M is not compact and number of
compact components of the boundary of M is either zero or infinite.
Further statements concerning k-movability and manifolds with boundary
are followed by the theorem that a connected Hausdorff Space that
is the union of k Open subsets homeomorphic to En must be k-
movable. This theorem has the corollaries that every open connected
triangulable n-manifold is n-movable and that for n 2 5, every
contractible open n-manifold is 2-movable. A closed k-movable n-mani-

fold is the union of (k+l) open n-cells.
Theorem 5.1: If M is a kamovable manifold with boundary then the

 

boundary of M is not compact and the number of compact components

of the boundary is either zero or infinite.

2322;; Let M be a k-movable manifold with boundary and suppose
that the boundary of M, B, is compact. Then ¢ E B = .U Bi with
the B1 compact, B1 E ¢, and there exist homeomorphism;—1
h1,h2,...,hk of M onto itself with B H hi(Bi) = ¢ for
i = 1,2,...,k. But then B H h1(B1) = ¢ and this cannot happen
Since a homeomorphism of a manifold onto itself has the boundary
as the image of the boundary.

Now let M be a k-movable manifold with boundary and

suppose that the boundary of M, B, has a finite non-zero number

53

54

of compact components. Let D be the union of the compact com-

k
ponents of B. Then ¢ # D = U D1 with the D1 compact and
1'1
D1 f ¢, and there exist homeomorphisms h1,h2,...,hk of M onto
itself with n n hi(Di) -= ,5. So D n h1(D1) = ,5. But h1|B, the

restriction to the boundary,of a homeomorphism of a manifold onto
itself, is a homeomorphism of B onto itself,and as such,sends
a compact component of B onto a compact component of B. Hence
¢ ,4 hlml) . h1|B(D1) c D and D n 111(1)) 5‘ ¢. This is a con-
tradict ion. B

Each homeomorphism of a manifold onto itself when restricted
to the interior is a homeomorphism onto the interior and when
restricted to the boundary is a homeomorphism onto the boundary.

Thus we have the following remark.

Remark 5.2: If M is a k-movable n-manifold with boundary then

 

both the interior of M and the boundary of M are k-movable

spaces.

Remark 5.3: If we let M be the n-ball, then the interior of M

 

is homeomorphic to En and so k-movable while the boundary of M
is Sn and so k-movable. But M has a compact boundary and so
(by Theorem 5.1) M is not k-movable. Hence the converse of

Remark 5.2 is not true.

Remark 5.4: By Theorem 5.1, no compact manifold can be the boundary

 

of a k-movable manifold. However if N is any non-compact k-

movable n-manifold, then N is homeomorphic to the boundary of

the (n + 1)- manifold N X [0,1). N X [0,1) is k-movable by Theorem 4.1.

55

Theorem 5.5: A connected Hausdorff space X that is the union

of k open subsets homeomorphic to En is k-movable.

k
Proof: Let B be a proper non-void compact subset of X = U X
i=1
is open and homeomorphic to En by a

i,

where for each 1, X1

a En. For each b g B, b E Xj for some
b

} ( ) for some real t > 0.

b

b
1 -1
B c U gJ (80 t ), g.1 (S0 t ) is open in x for all b E B, and
bEB b ’ b b ’ b N
-l

B is compact implies that B G U g (S ) and b 6 B for
j 0,t L
L=1 bL bl,

L = 1,2,...,N, and N some positive integer. For each

homeomorphism gi: X1

jb E {1,2,...,k} and so b E g SO,tb

i = 1,2,...,k let X 3 maximum {tb ‘L E {1,2,...,N}} and

L k
-l . .
g C O B 0
Bi B H gi (SO,X) Xi Then B QiiBi Since B 18 a proper

subset of X, there exists an x E X - B, and there exists for each
i - 1,2,...,k, a homeomorphism fi of X onto itself with
fi(g;1(0)) = x. But X - B is Open and fig;1 are continuous
functions for i = 1,2,...,k so there is a real number t > O
with f gT1(S ) CZX - B for i = 1,2,...,k. Let p be a
i 1 0,t
homeomorphism of En onto itself that is the identity outside
some compact set and ”(S

) c:S . Then for each i = 1,2,...,k

0,), 0,t

ii = g; pgi is a homeomorphism of X1 onto itself that is the

identity outside some compact subset of Xi. So

Ji(x) for x 6 X1
pi(x) = is a homeomorphism of X onto
x for x E X - X1

itself. Let hi 8 fiui a homeomorphism of X onto itself. Then

- -1 -1 -1
“1031) '3 fiuimi) " fi“i(Bi) '3 gigi ”31(31)C figi ”31(81 (804))

-l -1
Cifigi “(80,x) Cifigi (SO,t)(: X - B. -Hence B n hi(Bi) ¢ for

56

i = 1,2,...,k.EJ

Corollary 5.6: Every Open connected triangulable n-manifold is

n -movab 18 0

Proof: The theorem of [13] says that every open connected triangul-
able n-manifold is a union of n open n-cells. The corollary

follows immediately by applying Theorem 5.5.[3

Corollary 5.7: If Mn is a k-connected n-manifold without boundary
and q is the minimum Of k and n - 3 then Mn is ([qn —] + 1)-

movable.

Proof: Theorem 1 Of [20] (also Theorem 2 Of [16]) gives the result
that Mn can be covered by [J —] +-1 open n-cells. The corollary

follows immediately by applying Theorem 5.5.[]

Corollary 5.8: For n 2 5, a contractible Open n-manifold is 2-
movable.

Proof: Let MP be a contractible Open n-manifold with n 2 5. SO
n

M is n-connected and Corollary 5.7 (with q = n - 3) says that

M“ is ([n“ 7] + l)-movable. As n 2 5, [;§§] = 1.[J

Theorem 5.9: Let Mn be a closed n-manifold. If Mn is k-movable,

 

then Mn is the union of (k+l) Open n-cells.

‘PEggf: Let Mp be a k-movable closed n-manifold. Let U<: Mn and
U be homeomorphic to an Open n-cell. Then R = Mn - U is a proper
compact subset of MF. SO there exist compact subsets R1,R2,...,Rk
of R and homeomorphisms, h1,h2,...,hk, Of Mp onto itself with

R = S’Ri and R n h 1(Ri) = ¢ for i = 1,2,...,k. SO for each

i=11
-1
i 8 1 ,2,...,k, h. (R. 1)(: U, and so R. Clhi (U). But then

57

u, h11(U), h;1(U),..., h;1(U) are (k+1) Open n-cells and clearly
k

n

M =UU u h'1(U).D
i=1j

CHAPTER VI

CERTAIN OPEN 3-MANIFOLDS AND l-MOVABILITY

The l-movability of certain classes of open 3-manifolds is
considered in this chapter. The complexity of the proof Of Theorem
6.1 is to insure that the homeomorphism Obtained has compact support.
This gives the result that the monotone union of 3-dimensional non-
trivial products of domains is l-movable if E1 is one Of the
factors an infinite number Of times. Examples both Of a l-movable
union and Of a non-l-movable union exist when E1 appears at most
a finite number of times. We then prove that every W-Space (con-
tractible Open 3-manifold that can be triangulated by a countable,
locally finite simplicial complex which is a combinatorial manifold
without boundary and each compact subset of which can be imbedded
in S3) is l-movable. This has a corollary that all contractible

domains in E3 are l-movable.

Theorem 6.1: Let A be a proper non-void compact subset of E1 X B,
where B is an Open manifold. Then there is a homeomorphism h
Of E1 X B onto itself such that A D h(A) = ¢ and h is the

1
identity map outside a compact subset Of E X B.

Proof: Let A and B be as in the hypothesis, and for i = 1,2
let pi be the projection onto the ith factor from E1 X B. Now
p1(A) is compact in E1, so for some real number a > O,

p (A)<: -a,a . (A) is compact in B and so has an open
1 p2

58

59

neighborhood N with compact closure, N, in B. Let X = distance
(p2(A), B - N) > O, and L = maximum distance (b, B - N). let
bEB

dis a e - - -
t “c eih’B N) , a continuous function from B to the real num-

b 2
M) x
bers. Then 0 s has) s-l; and 1 s p,(b) for b e p2(A) while
g(b) = 0 for b E B - N. Define a homeomorphism gb Of E1

onto itself by

 

 

for t S -Za t
for -2a s t s -a (t:za)-(3au,(b) + a) - 2a
gh(t) = for -a s t s a t +-3ap(b)
3a t-a 3a;
f a s t S-4L'+ 28 + a - 3a b + 3a + a
or x 1.1, + a A M > Mb)
A

for 3—:”-+2a_<t

Define the function h from E1 X B onto itself by
h(t,b) = (gh(t),b). h is clearly continuous and has as inverse
the continuous function f given by f(t,b) = (ggl(t),b).
If (t,b) E A then b E p2(A) and so p(b) 2 1, while
t E p1(A) and so -a < t < 3. Hence p1(h(t,b)) = gh(t) =
t + 3au(b) > -a + 3au(b) 2 -a + 3a = 2a. Then h(t,b) E A.
If (t,b) E E1 X B - [-2a, 2%L-+ 28] X d§), then either
t E (-m,-Za) U (giL'+-Za,-+m) or b E B - N. In the first case
gh(t) = t for all b E B and in the second case g(b) = 0 so
gh(t) = t for all t E E1. Hence in either case h(t,b) = (gh(t),b) =

(t,b), and h is the identity Outside the compact set

[~2a, 3—2’9 + 2a] x N.U

CD
Corollary 6.2: Let X = U Xi where X is a manifold, and for
i=1
each 1, X1 is open in X and homeomorphic to E1 X B

 

i With Bi

60

an open manifold, X.i C Xi+1. Then X is l-movable.

giggi: Let X, the Xi's, and the 81's be as in the hypothesis
with g1: X1 4 E1 X Bi 3 homeomorphism for each i. Let A be
a proper non-void subset of X. We have A<: Xj for some j.
Applying Theorem 6.1 to gj(A)<: E1 X Bj’ there is a homeomorphism
h of E1 X Bj onto itself with gj(A) fl h(gj(A)) = ¢ and h
is the identity Outside a compact subset, L, of E1 X Bj' SO
gglh gj is a homeomorphism Of Xj onto itself that is the
identity outside the compact Subset g31(L) and has

-1 -l

j h gj)(A) = ¢. But Xj and X - gj (L) are Open in X

and so g-lh g, extends tO a homeomorphism f of X onto
J J

A n (g

-1
itself, defined by f(x) = X_ for X E X ‘ gj (L)
gjlh gj(x) for x E Xi

A n f(A) = ¢. Thus x is 1-movah1e.[]

Let D be a domain in E3 which is the monotone union
Of Open non-trivial products of domains Ai X Bi for i = 1,2,...
For each i let dimension A1 3 dimension Bi’ then as
l S dimension A1 5 dimension Bi 3 2 and dimension A1 + dimension
Bi = 3, it must be true that dimension A1 = 1 and dimension Bi = 2.

The only connected l-dimensional manifolds are E1 and 81, so

either A1 = E1 for an infinite number of i or Ai = S1 for
all but a finite number Of i. In the first case, Theorem 6.2
tells us that D is l-movable. In the second case, there is an

example where D is l-movable and another where D is not 1-

movable (Examples 6.3 and 6.4 respectively).

Example 6.3: Let D = (-m,m) X B where B is any Open manifold

 

Of dimension 2. D is l-movable by Theorem 4.1 and

61

 

Q
n = u (-i,i) x B.
i-l
” 1 1 2
Example 6.4: Let D = u s x Bi = s x (E - {(1,0),(-1,0)})
i=1

 

l 2 2 . .
c: S X E , where Bi = E - (8(1’0)‘,5)1US(_1’0)Kai) TO show

that D is not l-movable, consider the compact set A of D
given by A = 81 X T where T = ([-2,0,2} X [-2,2]) U
([-2,2] X {-2,2}).

If D is l-movable, there exists a homeomorphism h Of
D onto itself with A n h(A) = ¢. There is a strong deformation
retraction, r, of D onto A, so the inclusion map, i, of A
into D induces an isomorphism, i*, from n1(A,a) onto n1(D,a)
for a E A. h|A is a homeomorphism from A onto h(A) and so
induces an isomorphism (hIA)* from n1(A,a) onto n1(h(A),h(a)).
n1(A,a) is isomorphic to n1(Sl,S) X n1(T,t) = Z X G where
(s,t) = a, G is a free group on two generators and Z is the
group of the integers under addition.

Now, A is connected and so h(A) is a subset of U,
one of the components of D - A. Observe that each component of
D - A is S1 X V where V is a component of E2 - ({(-l,0),
(1,0)} U T). Let V be that component Of E2 c ({(-1,0), (1,0)} U T)
for which 0 = 51 x v. Let h(a) = (u,v) e s1 X(E2 - {(-1,o),(1,0)}).
But then V has a strong deformation retraction onto F where
v E F and F is homeomorphic to 81. SO there exists a strong defor-
mationremraction, x, of S1 X V onto 81 X F. Then the induced homo-
morphism, X*: from. 111(S1 X V,(u,v)) onto 111(31 X F,(u,v)) is an
isomorphism. But n1(U,(u,v)) = 111(81 X F,(u,v)) = n1(Sl,u) X

n1(F,v) = Z X Z.

62
Let j be the inclusion map from, h(A) into D.
h
D ----)-D

i j commutes.

A -—h—+h(A)
A

\

SO

h
nlo,a)——*—>n1(n.h<a>>

T“, j* commutes .

n1<A.a) W n1<h<A).h(a>>
*

Since the other three homomorphisms are isomorphisms, so is j*,
the homomorphism Of fundamental groups induced by j.

Where the homomorphisms o and B are those induced by

set inclusion, the following diagram commutes.

nICDsh(a))
Ti, n1<u,h(a>>

n1<h(A>.h<a)) °’

But j* is an isomorphism, so a is injective. So a is an
injection of Z X G into 2 X 2. But 2 X G is not abelian and
Z X Z is abelian. This is a contradiction. Hence D is not

l-movable.

Let us now consider the 1-movability of a certain class

Of 3-manifolds, the W-Spaces.

Definition 6.5: A.W-sBace is a contractible Open 3-manifold that

can be triangulated by a countable,locally finite simplicial complex

63

which is a combinatorial manifold without boundary, and each compact

subset of the Open 3-manifold can be imbedded in S3.

 

Theorem 6.6: Every W-Space is l-movable.

Proof: Let U be a.W-space. Theorem 1 Of [17] is the result that

Q
U = U H where each H is a cube with handles, H c interior H .
i=1 i i i i+l
Let A be a compact subset Of U, then there is a positive integer

j so that A.C interior H . There exists a pesudo-isotopy,

J
{Gt} t 6 [0,1], Of Hj so that 61(A) C T c Hj and a N so that for
t 6 [0,1] Gt is the identity map outside, N, the compact closure of
a neighborhood of AUT (where T is the union of a finite number
of line segments and is homeomorphic to 3 $1 ; L is the
number Of handles on the cube with handle: fhat is Hj). SO this
pseudo-isotopy extends to one of U by defining Ct(x) = x for
all x 6 U - Hj and t E [0,1]. But since T is the union of a

finite number of line segments in H , there is a homeomorphism

J
h of U onto itself so that T n h(T) = ¢. Now, 61(A) n h(Gl(A))
: T D h(T) = ¢ so G1(A) n h(Gl(A)) = ¢. Hence Theorem 3.5 applies

and there exists a homeomorphism, f, of U onto itself with

f(A) n A = ¢. Therefore U is l-movable. U
Corollary 6.7: All contractible domains in E3 are l-movable.

“3529;: let D be a contractible domain in E3. It is well known
that any domain in a Euclidean space can be triangulated by a
countable,locally finite simplicial complex which is a combinatorial
manifold without boundary. D, and so each compact subset Of D,

can be imbedded in 83. Thus D is a W-Space, and Theorem 6.6

applies . U

B IB LIOGRAPHY

10.

ll.

12.

13.

14.

15.

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