NflN-f’ilifixffiFQLD FACTQQS 0F
EUCLEDEAN SPACES

That: for {he Dogma of Dim. D.

3.: ECHESAN STAE EEEV‘EESETY
Alfred John E50313
1967

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11 ntitled
p eeeeeee d b

has been accepted towards fulfillment
of the requirements for

PhoDe degree in Mafihgmatics

Major professor

 

Date—Mum

 

 

ABSTRACT
NON-MANIFOLD FACTORS OF EUCLIDEAN SPACES

by Alfred John Boals

This thesis is a study of a class C of decomposition
spaces which are shown to be factors of Euclidean space.

Suppose A and B are disjoint compact subsets of
En. Then we know it is possible to find disjoint compact
sets A* and B* such that A'Ciint A* and B.C int 3*. In
Chapter I we give sufficient conditions for A and B to
insure that A* and B* can be picked to be cells.

In Chapter II we define the class C of decomposition
spaces and prove that the product of any member of C and
a line is topologically En for some integer n.

In ChapterIII we prove that the product of any two

members of the class-C is topologically En+m for suitably

chosen integers n and m.

NON—MANIFOLD FACTORS OF EUCLIDEAN SPACES

BY

Alfred John Boals

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1967

ACKNOWLEDGMENTS

The author wishes to express his gratitude to Professor
K. W. Kwun for suggesting the problem and for his helpful

suggestions and guidance during the research.

ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . .
INTRODUCTION . . . . . . . . .

Notation and Terminology .
CHAPTER

I. SEPARATION THEOREMS .

II. A CLASS OF DECOMPOSITION SPACES

III. THE "DOGBONE SPACE" SQUARED IS E6

LITERATURE CITED . . .

iii

Page

ii

14
29
36

LIST OF FIGURES

Figure Page
2.1 . . . . . . . . . . . . . . . . . . . . . 21
2.2 . . . . . . . . . . . . . . . . . . . . . 22
-2.3 . . . . . . . . . . . . . . . . . . . . . 25

iv

INTRODUCTION

In 1957 R. H. Bing [3] gave an example of a decomposi-
tion of E3 into tame arcs and points such that the associ-
ated decomposition (the "dogbone space“) is not topologically
E3. In fact this space is not even a manifold (i.e. there
exist points which do not have Euclidean neighborhoods).

The "dogbone space" was constructed in answer to a question
of G. T. Whyburn [8, p. 70] which asked: Is it true that
if G is an upper semi—continuous decomposition of E3
into point like compact continua, then the decomposition
space is homeomorphic to E3?

In [5] Bing gave examples (i.e. "unused example" and
"segment space") of two other decompositions of E3 into
tame arcs and points. The "unused example“ is known to be
distinct from E3, however, whether or not the "segment
space" is E3 is still unknown. Bing [4] proved that the
product of the "dogbone space" and the line is 4-dimensional
Euclidean space (E‘). It is reported [5] that John Hempel
has proved that the product of the "segment space" and E1
is E4. In [2] S. Armentrout asked if the same were true of
Bing's "unused example". An affirmative answer is given to
this question in Chapter II.

In proving that the product of a dogbone space and a
line is E4, Bing showed that E‘ has non—manifold fac-
tors. J. J. Andrews ano M. L. Curtis [1] gave another fac-
torization of E4 into factors one of which is not Euclidean.

1

2
They proved that if a is an arc in En then En/o x E1

. 1 .
Is En+ . K. W. Kwun proved that If a CZEn and B CZEm

where o and B are arcs then En/o X Em/B is En+m.
Thus Kwun has shown that En where n 2.6 can be written
as the product of two non—manifolds.

In view of the relationship of these last two results,
it is natural to ask the following question. Is the product
of dogbone spaces, unused examples, or segment spaces top-

ologically E6? This question is answered in the affirma-

tive in Chapter III.
Notation and Terminology

Suppose X: is a topological space. A collection of
subsets G of X will be called a decomposition of X if
U[s|s e G} = X and s1 n 82 = v for any two distinct ele-
mentSof G. G will be called an upper semi-continuous de-
composition of X if for any element 9 e G and any open
set U CIX which contains g, there exists an Open set
V C'U such that g CZV and V is the union of elements
of G.

Suppose X is a topological space and G is a decompo—
sition of X. The decomposition space associated with G,
say Y, is defined as follows. The points of Y are the

elements of G and a base for the topology of Y is the
collection of sets of elements of G whose union in X is

open. More information on decomposition spaces can be found

in [8] and [6, Chap. 3].

3

A pseudo—isotopy of a topological space X into a
topological space Y is a continuous function H on~X x I
into Y such that Ht(x) = H(x,t) is a homeomorphism for
each 0 j.t < 1.

Any subset of a topological space which is homeomorphic
to In where I = [0,1] will be called an njgell. .Any
paracompact Hausdorff space in which every point has a
neighborhood whose closure is an n-cell will be called an

n-manifold.

CHAPTER I
SEPARATION THEOREMS

In proving that the product of the "dogbone space"
and a line is E4 Bing was forced to give a rather lengthy
construction of a sequence of 4-cells. Theorem 1.1 asserts
the existence of suitable n-cells in a more general con-
text. Theorem 1.1 is applied in proving the main result
of Chapter II. Theorem 1.2, in addition to being of inter-
est in itself, plays an important role in the proof of the

main result of Chapter III.

Definition 1.1: If 7X is a topological space and
D CZX then by int D is meant the set X - X - D, where

X — D is the set theoretic closure of X - D in X.

 

Theorem 1.1: Let C1, C2, ..., Cp be disjoint com—
pact subsets of a Hausdorff space X. Let D1, D2, ..., Dp
be (not necessarily disjoint) n-cells such that for each
i = 1,2, ..., p

Ci C int Di'
Then for any [a,b] CE1 and g > 0 there exist disjoint
(n + 1)-cells E1, E2, ..., Ep contained in X X (a-e, b+g)
such that for each i = 1,2, ..., p

(1) Ci X [a,b] C int Ei

5

where #1 is the projection of X x E1 onto X.

Proof: Let f:[-e, r+g]-——> [a-g, b+g] be the homeo-
morphism given by

a + x ; if x e {—5.0}

f(x) = )x + a; if x e [0,r]

 

b - a
< r

b + x - r ; if x e [r,r+g]

Let k:[-e,r+g] BEE93> [-e.2p-1+g] be the homeomorphism ,

given by

k(t) = 21p:i%8- 1 (t + e) - e

 

For each j = 1,2, ..., p let kj be a homeomorphism

of {-5,r+g] onto [~e,2p—1+g] with the properties:

1. kj(-g) = -g and kj(r + s) = (2p - 1 + e)
2. kj(0) = 2j - 2
3. kj(r) = 2j - 1.

Let A = U Di CZX and note that since A is a compact
i
.Hausdorff space it is normal. de A = A - int A and Cj

for j — 1, ..., p are closed sets. Thus there exist open
sets Uj for j = 1,2, ..., p satisfying

1. Ui n Uj = ¢ 1f 1 ¢ 3.

2. C. C Ui for all i = 1,2,..., p,

U. C int A.

By the Urysohn lemma there exists a continuous function g
mapping A onto I = [0,1] such that

1. g(: Ci)= 1 and

2. g(A- UUi)=O.
i

6
ConStruCt h:A X {-8. r+e] -—9 A X {-8, 2p-1+e] as follows:

(X: 9('X)kj(t) + (1-9(X))k(t)): on i?- x {-5. He]

J
h(x,t) =
(x, k(t»: on (A - uni) x {-e, r+s]-
I
For each j = 1,2, ..., p h = id x k on de E: X {-5, k+e],

3
hence h is well defined. h is continuous since 9, k and

kj are all continuous. Suppose
h(X1, t1) = h(X2, t2)

then' x1 = x2 = x.

Case 1:

If x e Uj then

9(X)kj(t1) + [1“9(X)]k(t1) = 9(X )kj(t2)+[1-9(X )]k(t2)
or

9(X)[kj(t1)-kj(t2)]+[1-9(X)][k(t1)-k(t2)] = 0
But g(x) and 1-g(x) 3.0 and both k and kj are order
preserving homeomorphisms, whence kj(t1) - kj(t2) and
k(t1) - k(t2) are both positive, zero, or negative together.
Therefore

k(t1) - k(t2) = 0

and t1 = t2.

If x e A - UUi then k(t1) = k(t2) and again
i
Thus h is one-to-one and continuous hence a homeo-
morphism. h can be extended to a homeomorphism of

X x {-3, r+g] -—9 X x {-3, 2p—1+e] by defining

7
h(x, t) = (x, k(t)) on X - A.
For each j = 1, ..., p let E3 be the (n+1)-cells

defined by

. . 1 . 1
Ej = Dj x [23 — 2 - Z" 23 - 1 +-Zj.

Now define

E. = (id x f)[h’1(E!)]

J 3
Clearly Ej n Ei = o if i # j and lej = Dj. Moreover
Ei c int A x (a - a, b + g).
If x e Cix [a,b] then
(id x f)-1(x) 6 Ci x [0, 2r-1]- and

h - (id x f)'1(x) 6 Ci x [2i - 2, 2i - 1].

But

1

. . . . . 1
Ci X [21 — 2, 21 - 1] C int (Di X [21—2dzu 21-1+Zj)

whence

_1(

(id x f)-h’1-h~(id x f) x) 6 (id x f)h‘1(ni x[2i-2--l 2i-1+%])

4i
and 'x e Ei' Thus the Ei i = 1,2, ..., p satisfy all
the claims of the theorem.

The above theorem will be applied in Chapter II in the

following form.

Corollary 1.2: Let A be.a compact n-manifold in En
and C1,C2, ..., CP be disjoint compact sets in int A
such that there exist (not necessarily disjoint) n-cells
D1,D2, ..., Dp with the pr0perty that Ci C int Di C int A.
Then for any [a,b] CE1 and g > 0 there exist disjoint
(n + 1)-cells E1,E2, ..., Ep contained in int A X

3

(a - e,b + 5) such that Ci x [a,b] C Int E1 and WnEi = Di

where ”n is the projection of En x E1 onto En.

8

Theorem 1.3: Suppose B is a compact subset of
int In and C is a compact subset of In disjoint from
B. Similarly suppose D is a compact subset of int Im
and' E is a compact subset of Im disjoint from D. Then
there exists an (n + m)-cell G with the following
prOperties:

(1) B x D c int G CIG c int In x int Im

(2) G n [(B xgs) u (c x D) u (c x E)] = o.

ggppfi: Let T C int In be an n-cell such that
T n (B U C) = ¢ and T is the product of its projections.
Such an n-cell exists since C n B 3 ¢ and they are each
closed. Similarly let R C int Im be an m-cell such that
R n (D U E) = ¢ and R is the product of its projections.
Let wiT = [ti’ ti] for each. i 8 1,2, ..., n and
ij = [rj, r5] for each j = 1,2, ..., m, where Va is
the projection onto the o-th coordinate.

Let 51 = min[distance from B to (de In U C),
distance from T to (B U C U de In)]. Let 62 = min[dis-
tance from D to (de Im U E), distance from R to
(D U E U de Im)]. Set 6 = min(61, 62). Let k, k1, k2,

., km be homeomorphisms defined as follows,
(1) k : [0,1] -—» [0,1] such that
k(0) = 0, k(1) = 1 and
k[d/2, 1-5/2] = [1/4, 3/4].

(2) For each i = 1,2, ..., m let
k. : [0,1]-—-> [0,1] such that

J.

ki(0) 3 0, X41) = 1 . and

ki[ri,ri] = [1/4, 3/4].

9
Let Un be an Open subset of (6/2, 1 - 6/2)n such
that B c Un and Un n c = ¢, Set wn = In-[d/a, 1 - 6/3)”.
By the Urysohn Lemma there exists a continuous function
g : In -—» [0,1] such that
g(B u de In) = 1 and
g(In - (Un u wn)) = 0.

Consider the following collection of maps

n

h. : I X Im n

> 1 x 1m , i = 1,2, ..., m.

 

For x e In and (y1, ..., ym) e Im

r’(X:(in---.yi_1.9(X)k(yi) +
[1-9(X)]ki(yi).yi+1.....ym) for

" m
(X,(yl,...,ym)) 6 Un X I o

 

hi(x.(yil---.ym))=.<((X:(y1.-o-.yi_1:g(X)yi +
[1-g(X)]ki(yi).yi+1.---.ym)) for

- m.
(x,(y1,...,ym)) e WnX I

(x.<y......yi_1.ki<yi>.yi+1.....ym>>

Lfor (x,(y1,...,ym)) e [In—(Wn U Un)]XIm.

 

Each hi is well defined since Uh n Wn = ¢,

m _ . .
wjhi/de Un X I - ”j for all j # 1

m _ . .
wihi/de Un x I - k w. , where again Hi Is the

i 1
projection onto the i—th coordinate axis. And

In ._ . .
wihi/de Wn X I - ”j for all 3 ¢ 1
m
wihi/de Wn X I kiw..

Clearly each hi is continuous and onto In X Im. Suppose

10

for x, x' e I and (y1, ..., ym),(zl,..., zm) 6 Im

hi[x,(y1, ..., ym)] = hi[x',(z], ..., zm)]

then x = x' and yj = zj for j # i.

Consider the three cases:
(1) x 6 Un ,
(2) x e Wn , or

(3) ern-(U uw).

-)

Case 1: 9(X)k(yi)+[1-9(X)]ki(yi)=g(X)k(Z-)+[1-9(X)]ki(zl

1
and 9(X)[k(yi)-k(zi)]+[1-g(X)][ki(yi)-ki(zi)] = 0-

Note that g(x) and 1 - g(x) 2.0 also

[k(yi) - k(zi)][ki(yi) - ki(zi)] 2.0 since k and .ki

preserve order. Thus y1 = 21.

Similar arguments show that for cases 2 and 3 yi = 21.

Thus for each i, hi is an injection consequently a homeo-

morphism.

n

Define H : I x Im‘——> In

X II“ to be the homeomorphism

h1' h2° ... - hm.
Set J = {5/2, 1- 5/2]n x [1/4, 3/4]m CiIn x Im.

If (x,y) e B ij then x 6 Un and H(x,y) e J. Thus

H(B ij) CZJ. Let (x,y) e [c x (D u E)] then x e I - Un

and there exists a j such that wj(y) e I-[rj,r3]. ‘If

x e wn then H(x,y) ¢ J. If x e In - (w x Un) then

n

wjhj(x,y) e I - [1/4,3/4] and H(x,y) ¢ J. Thus

H[C x (D u 2)] n J

¢.
Note that H/B x Im = id x k*, where k* is the m-fold

product of k; (i.e. k* = (k x k x ... x k) with m factors).

11
Thus it follows that w; H(B x D) and w; H(B x E) are
disjoint compact subsets of [1/4, 3/4]m C Im, where w;
is the projection of In X Im onto Im. Also w; H(B X D)
C (1/4, 3/4)m. Let y = min(distance from w; H(B x D) to
de[1/4, 3/4]m, 5/2). Let Um be an open set in
(1/4 + 7/2, 3/4 — y/2)m‘ such that a; H(B x n) cum and
and Um n w; H(B x E) = ¢. Let wh = {1/4, 3/4]m - A
[1/4 + y/3, 3/4 - y/3]m. There exists a continuous function
f , {1/4, 3/4]m ——a {0,1}

such'that

(1) f/w;[H(B x D) u de{1/4, 3/41m] = 1

(2) £/{1/4,3/4]m - (Um u Wm) = 0

Let Y, Y1, Y2, ..., Yn be homeomorphisms defined as
£01lows

(1) w : {5/2, 1-5/2]-——> {5/2, 1-5/2] such that

2(5/2) = 5/2 , 2(1-5/2) = 1—5/2

and
i{5/2 + y/2, 1—6/2-7/2] = {1/4, 3/4]
(2) For each i = 1,2, ..., n let
Yi : {5/2, 1-5/2] -+> {5/2, 1-5/2] such that
gfl5/2) = 5/2 , 21(1-5/2) = 1—5/2
and

witti. ti] = [1/4. 3/41.

Consider the following collection of maps

die; In x Im -e In x Im , i = 1,2, ..., n.

For (x1, x2, ..., xn) e In and y 6 Im

12

(id for ((x1,...,xn),y) e (InXIm)-int J

((X1.---.Xi_1.f(Y)Y(Xi) +

[1-f(y)]‘l'(xi),xi+1
((x1,x2,...,xn),y) e {5/2,1-5/2]n x Um

,...,xn),y) for

f(y)x. +

9.[(x1,...,xn),y]= 4 ((x1,...,x 1

i-I'
[1-f(Y)]Yi(xi)lxi+1leo-txn)IY) for

((x1,x2,...,xn),y) e {5/2,1-5/2]n x Wm

((xl’""xi—1’wi(xi)’xi+1’""Xn)’Y)

on {5/2,1-5/2]nx({1/3,3/4]‘“-(Um u Wn))
k.

 

 

Each 91 is well defined Since Wh n Um - ¢.

n = . .
vjei/{5/2,1-5/2] x de um wj for 1 # J
njei {5/2,1-5/2] x de Um = 2.7
and

njei/{5/2,1—5/2]nx ({5/2,1-5/2]m-(um u wm)) is wj if

j # i and Vivi if j = i. By an argument exactly like

the one given above for hi’ each 91 is a homeomorphism.
Define a = 91-92- ... '9n° Set J' = {1/4,3/4]n x
{1/4 + 7/2, 3/4 - y/2]m ch. If (x, ) e B x D and

y
X = (X1: °--: Xn): Y = (Y1vY2: .... Ym)-
Then e{H(x,y)] = 6(x,[k(y1), .. k(ym)])
[(Y(xi).---.Y(xn)).(k(yi)'k(y2).-.-k(ym))].

Thus 6 °H(BXD)CJ'. If (x,y)e {(ch) u(ch)]

then H(x,y) K J hence 6 ' H(x,y) é J'. Suppose

(XIY) 6 B x E then r; H(x,y) e [1/4, 3/4]m - Um and there

13
o . ' *-
exists a 3 such that wj(x) fl [tj, tj]. If me(x,y) e Wh
then 9 ° H(x,y) ¢ J" since 7&9'H(x,y) = v;H(x,y) e Wh

-

* . = * . * _ _
and wm n va 5. If an(x,y) e ,nm(J) wh Um then

ajs;n(x,y) ¢ {1/4, 3/4] and 6°H(x,y) ¢ J4.

Therefore 9-H{(B x E) u (c x D) U (c x E)] CIn x Im - J'.
Define G = H-1-9-1(J'). G is the (n + m)-cell
contained in int In X int Im satisfying properties 1 and

2 of the theorem.

Note that J' defined in the above proof is a product
of cells. Thus a proof similar to that of Theorem 1.1 would

prove the following corollary.

Corollary 1.4: Suppose Bi’ i = 1,2, ..., p, are
disjoint compact subsets of int In and C is a compact

subset of In disjoint from U Bi = B. Similarly suppose

1
Dj' j = 1,2, ..., q, are disjoint compact subsets of int Im
and E is a compact subset of Im disjoint from U Di = D.
i
Then there exist (n + m)—cells Gij’ i = 1,2, ..., p ;
j = 1,2, ..., q, such that
(1) Gij n Grs = ¢ 1f 1 # r or 3 # s,

. . n m

(3) Gij n {(c x D) U (B-X E) u (c x E)].

CHAPTER II
A CLASS OF DECOMPOSITION SPACES

In this chapter we define a class of upper semi—
continuous decompositions of En and prove that the
associated decomposition Spaces are factors of En+1.
This class contains the decompositions for each of the

spaces (a) "dogbone space", (b) "unused example" and

(c) the "segment space" [see 5].

Definition 2.1: Suppose a is an arc in En (i.e.
o - h[0,1] for some homeomorphism h : I -—> En) such
that P = vl/a is an injection, where n1 is the pro—
jection of En onto the lst coordinate. In this case a

will be said to have property 05.

Let Q be an arc with property 08 and assume that

#1 h(l) = b and 71 h(O) = a with a < b. Define the

continuous function f : E1 -—9 En by
P-1(a) for t.: a
—1
f(t)= P (t) for aitib
P_1(b) for b fi.t .
. . 1 n-1 1 n-1
Define the homeomorphism k : E X E -—> E X E

by ‘k(t,x) = (t, x - f(t)). For any 5 > 0 let
Ci - {zlzeEn: H Z -all :8}
c2 = [z|z 6 En, I] z - b|| : 8]

C3 = {zlz 6 En, a i.wlz j,b and [[2 - vlz[[.: e}
14

15
then Q8 = C1 U C2 U C3 is an n-cell containing w1(o).
The n-cell k-1(Qe) shall be called an g-radial neigh-

borhood of a .

Remark 2-1: Note that if c is an arc with property
05 then for any a >p0 the e-radial neighborhood of o

intersects the planes w;1(t) = Rt = [(t.y)[(t,y)e t X En-l

J

in the void set, a point, or an (n-1)-cell.

Remark 2.2: Suppose a is an are which has property
05. Since the homeomorphism used to define radial neigh-
borhood is uniformly continuous, it follows that for any

a > 0 there exists a 6 > O and a collection of planes
-1

R1 = w (ti) with t1 = a < t2 < ... < tp = b such that
the Ri cut the o-radial neighborhood of a into (p + 1)
n-cells ci ; i = 0,1,..., p and diam Ci': 5.

Let A1, A2, ... be a sequence of compact n-manifolds

(not necessarily connected) in En satisfying

P1. Ai+1 C intAi for all i = 1,2,3,

P2. Each component of A05 = n Ai is an arc with
i

property 08.

Lemma 2.1: Suppose e > 0 and A1 and A00 are as
defined above, then there exists a finite collection of
n-cells Ui satisfying

1. For each Ui there ex18ts an arc oi CIAG) 0 int Ui

such that the distance from x to the boundary of

Ui is less than S for-all x e ai.

16
2. There exists an integer m such that if A is a

component of Am then A C int Uj for some j.

Proof: For each arc o 6 Ac) let No be the g—radial
neighborhood of a. For each ’No there exists a neighbor-
hood Vfi CINa with the property that if an arc 5 C:Aoo
intersects Va non-trivially then B CINQ. The existence
of such Vd's follows from the fact that the decomposition
of En into the arcs of A00 and the points of En IjAoo
is an upper semi-continuous decomposition. The collection
of sets [Vd|o C A00] is an Open cover of the compact set
A00. Thus there is a finite subcollection V1, V2, ..., V

P

which cover A00. Let N1, N2, ..., Np be the correspond-

ing Na's. Note that by the choice of the Va's ‘we have
each are a CIAOD contained in the interior of at least one
Ni' For each arc o C:Aoo there exists an integer m(o)
such that

C!

1. a CIA0 C Am(a) , where Am(a) is the compo-

m(a)
e of containin ;
n nt Am(a) g a

2. C int Ni , for some i = 1,2, ..., p.

o

Am(o)
. . a .

The collection [int Am(a)|a CZAGD] is an open cover of
A00. Therefore there is a finite subcover. From this col-
lect'on of o
1 Am(o)
m(o). m = m(a) is the desired integer. Each Ni is the

'3 there is one with largest subscript

g-radial neighborhood of some a C Aoa‘ Therefore the col-

lection Ui = Ni satisfies the claims of the Lemma.

17
Lemma 2.2: Suppose Ai’ i = 1,2, ..., are defined as
above and A is a component of Ar for some r. Given
a > 0 then there exist integers y(1), y(2), ..., y(m+1)
and sets Kij CIA x E1 i = 1,2, ..., s: j = 1,2, ..., m,
which satisfy the following conditions.
1. For each i , Kio is an (n+1)-cell and Ki' is

J
the disjoint union of (n+1)-cells K.

ijk'
k = 1,2, ...,u(i,j).
2. Kio n Keo 2 ¢ if i # e.
3. U K.. C .A . n A ', 2m+1-'
i 13 ( y(J) ) X [J 3]
int A . 0 A ', 2m+1—' U K..
( 7(3) ”(3 J>Di ”+1
for each j
4. For each i Kio can be written as the union of
(n+1)-cells Die , e = 0, 1, ..., m, such that

Die n Div = de Die n de Div 18 an n-cell if
[e - v] = 1 and is void if | e - v[ > 1.

5. Diameter of w*(D

. *
. < for all i e where
n 1e) 5 . . fin

'is the projection En X E1 -¢ En.

6. Die n Div 0 Kijk is either VOid or an n-cell.

§£22£3 Let the e of Lemma 2.1 be the min(e, distance
from A030 A to de A) hence there exists a finite set of
n-cells Kio , i = 1,2, ..., s and an integer y(1) sat-
isfying;

a. Kio C int A for all i.

b. If A' is a component of Ay(1) n A then

A' Ciint‘Ki'O for some i.

18

Note that the 'Kio may not be disjoint. By Remark 2.2 '

each n-cell Kio can be chosen so that there is a finite
. = . , .
set of planes Rij’ 3 1,2, ..., mi which cut Kio into
(m. + 1) n-cells DE. such that
i 13
I I = I I
Dij n D iv deDij n say Div

is an (n - 1)-cell if [j - v] = 1 and is void if [j - v[ > 1.
Without loss of generality assume mi = m for all i.

Similarly apply Lemma 2.1 to each component of

Ay(1) n A to obtain an integer y(2) and sets .Kil , where
Kil is the union of n-cells Kiik , k = 1,2,... u(i,1),
satisfying;

i. If A* is a component of Ay(2) n A then

A* C int K!

I ' I I ' I
11k CKilk C int A CA C int Kio

for some k and some component A' of Ay(1)n A.

ii. K!

11k n Rij is either v01d or an (n — 1)-cell.

Condition ii actually follows from the proof of Lemma 2.1.

Continue this procedure to obtain the integers 7(3), y(4).

..., y(m + 1) and sets 'Kij and n-cells 'Kijk satis-
fying conditions analogous to i and ii above.

For each i and j define Wijz to be the union of
the components of Ay(j+1) n A which are contained in Kijz
but not in Kijp for any p < z. Note that Wijz are
compact and Wijk n Wijz = o if k # 3. Let [W ] and

i01
{Kio} be respectively [Ci] and [Di] of Theorem 1.1 and

let a - s = 0 and b + g = 2m + 1. Then define Kio = Ei

of Theorem 1.1. By the proof of Theorem 1.1 we see that

Kio can be written as the union of (n + 1)-ce11s Dig such

19

that v; Dig = Dig' Further the Dig satisfy condition 4.
In general let {Wijk] and [Kijk] be reSpectively
{Cik} and {Dik} of Theorem 1.1 and let a =«%, a = j
and b = 2m + 1 - j. If Kijk - Eik of Theorem 1.1 and
Kij = E Eik then conditions 1 through 5 are clearly

satisfied and condition 6 follows from ii above.

Remark 2.3: Note that if i # r and A' is a com-

ponent of A ( n

v j+1)

A' x E1 = ¢ since K3. n K'. . . Also K3.
13 r3 v(3+1) lap
Kijq = o if they are not contained in the same n—cell

of Kij-i'

a . I
n A contained in Krj then Ki,j+1

CIA - A n

The proof of the next lemma is based on the follow-

ing known result.

Theorem: Suppose that A is an n-cell which is the
union of two n-cells A1 and A3 with the properties that
A1 n A2 and de A1 n de A2 are (n-1)-cells and A1 0 A2 C
de A1 n de A2. If B CIA, B is compact and B n de A CIA2
then there exists a homeomorphism h of A onto A whiCh

is fixed on the de A and such that h(B) CZAZ.

Lemma 2.3: For g > 0 and A a component of Ar

(where Ai i - 1,2,... are defined as above) let
y(f), Diz’ Kij’ and Kijk be defined as in Lemma 2.2.
Then there exists a homeomorphism h : En X E1--> En X E1

such that the following hold.

1. h = id on the complement of U Ki1
i

20
2. h = id on the complement of

U([Ki1 n (D.

1 10 U D11)] U [Ki2 O (D. n D. )] U

11 12

U [K. n (D

im im-i U Dim)])

3. If .A' CIA

. d .. ' ',2 +1-'
y(3+1) n A an K13k TIA X [J m 3]
then
h((Di0 U ... U Dij) n A x ([3,3+1] U [2m-j,2m-3+1]))
C Di? U DiY+1

where Y = min(j, max[e|Kiek n Die # ¢, Kiek :IA']).

Before reading the proof of Lemma 2.3 it may be help-
ful to look at Figures 1 and 2. The homeomorphism h will
be obtained as the composition of homeomorphisms hm-1.hm-2.

- h1. Figure 1 illustrates how the hj will be con-
structed. The shaded region of Figure 2 is that part of

A>([O.2m+1] which is not moved by h.

Proof: Let h1 En x E1 -—> En x E1 be a homeomor—

phism defined as follows

.- n 1-
h1 — id on E X E g (Ki1 n (Dio U Dil))

For each i and A' a component of ~A7(2) with

A' X [1,2m] CCKi1 then

kl

n D. = ¢ or K. n D. = ¢ then
10

a. If K 11k 11

iik
h1 = id on Kilk
b. If Kilk n Ki0 n Di1 # ¢ then

h1 = id on de Ki1 n (Di0 U Dii)

h1(A' x [1,2m] n (Di0 U D. )) CDi1

11

h1 as defined exists since A' X [1,2m] is compact,

 

‘ A(Y1) n K

 

ilui

21

1 hl

Figure 1.

stars

11UI
izp.2

i4u4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E .W/fl ////// W/flW/fl/ //////

. ///////// 7 .

/////) //

//////I

i 7/////

2. / {/////{////A

//////////////////
.////// / ////2 (4/) 7457/1 /

 

Figure 2.

23

n (D. U D.

10 11) is the union of two (n + 1)-cells which

intersect in an n-cell in their common boundary and

A X [1.2m] fl de [Kijk n (Di0 U Di1)] is contained in Di1

 

and Kijk n Kijz = ¢ If k # I.
Now proceed inductively to define hj for j =
2,3, ..., m-l. As a notational aid define Lij =
(Di0 U Di1 U ... U Dij) n Kij. Define
hj : En X E1 > En X E1 as follows
_ - n 1 _ . . .
hj — id on E x E (hj_1 hj-z ... h1(g Lij))
For each i and A' a component of Ay(j+1) n A With
A x [3,2m+1-j] CiKijk then
a. If H n Dij-i = ¢ or H n Dij = ¢ then hj = id
on H where H = (hj_1' hj-z. ... ° h1(Kijk)).
b. If H n Dij_1 n Dij ¢ 5 then
hj(hj_1° hj_2- ...-h1(A x[3,2m+1-3] n Dij_1U Dij)))
CD...
1)

hj exists since A' X [j.2m l-j] is compact,

hj-l. °h1(A X [3,2m+l-j] n (Dij-l U Dij)) is the union
of two (n + 1)-cells which intersect in an n-cell in their
common boundary and

h. °...'h1(A' x [j,2m+1-j]) n de(hj_ n

J_1 °...'h1[kij

1 k

(D UD..)1)

ij-l ij
is contained in Dij'

Define h : En X E1 -> En x E1 as h = h 'h

m-1 “1-2.00. hi.

Clearly conditions 1 and 2 are satisfied by h. To see

that condition 3 is satisfied let

24

x e (Ay(j+1) n A) x ([j,j+1] U [2m-j,2m+1-j]).
There exists some component A' C Ay(j+i) n A such that
x e A' x ([j,j+1] U [2m-j,2m+1-j]) and a unique kijk con-

taining x. Let Y = min(j,max[e|Kiek n Die # ¢’Kiek :>A )).

Case 1. If T < j then

h(x)

ll
13‘

m_1...hw...h1(x) = h

h(x) = hm_1 3+1 hj Ih1(x) = hj+1 hJ h1(x) c
1] U Dij+1

Lemma 2.4: Suppose a > O and A is a component of

Ar (where Ai i = 1,2, .... are defined as above). Then

there exists an integer N and a uniformly continuous
homeomorphism h : En x E1‘-> En X E1 which is the
identity on En+1 — (A x E1) and such that for each
w e E'

(1)1r (h(Axw))C[w—2m—1,w+2m+1].

n+1

(2) diam (w; (A' X W)) < 48.

where A' is a component Of An 0 A, w is the projec-

n+1
tion Of En x E1 onto E1, and w; is the projection onto
En.

Figure 3 shows how to apply Lemma 2.3 to prove Lemma

2.4. In Figure 3 only one sequence Kiik’ Kizk’ ""Kim+1 k,

containing a component of AN n A, is shown. The (n + 1)-
cells in the figure are shown as if they intersect each of
the (n + 1)-cells Dio’ D11, ..., Dim' This may not be the

case, however, an analogous figure is Obvious.
O

H H H H H H H H H
H to w as OI OI «I on to
\

H
C)

C) hi ED (0 I5 0| 6) ~J (D <9

25

 

 

x\\\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ \\\\\

 

\\\‘

 

\\W

\\\\

 

 

 

 

 

 

Rum

 

 

 

I I 1 l/\
I
D10
2

 

26

Proof: Apply Lemma 2.3 to A X E1 and integers m

and y(m+1) and sets Diz’ Kij and Kijk' Set N =
y(m+1) for g = 0, i1, i2, .... let
= 2 + 2 ' = + + 1
x9 g( m ) xg xg m
y9 = g(2m + 2) + 2m + 1 yé = y9 + m + 1.
Note that Diz’ Kij’ Kijk CZA x [xo, yo], by suitable

translations Of E1 we get sets analogous to Diz”Ki£’

and Kijk in A x [x , yg] for each 9. Apply Lemma 2.3

9

. yg] for each g. Define D: = D and

to A x [x 12 im-Z

g
apply Lemma 2.3 to A X [xk. yk] using IBIZ in place of

D. Thus there exists a homeomorphism h:En X E1-> En X E1

13'

which is uniformly continuous. By the choice Of xg, yg.

xé and yé and Remark 2.3 there exist integers i and

k such that

U D U Dim_k+2 U D

I
”E h(A X W) C”Emil: ik+1 im-k+3)
for each component A' CIA 0 AN and w e E1. Note that
i and k depend on A' and w. Diam ”h(Diz) < e for
all i and 2. Thus condition 2 is satisfied. For

w e E1 there exist x and yé+6, where

. .X'
g yg 9+6
6 = 0, -1, such that w e [xg, yg] n [xg+6’ yg+6]' "Thus

”n+1(Al X w) C [xg' Y9] U [xé+5' yé+51

ard condition 1 is satisfied.

Theorem 2.5: For each component A CAr (where Ai'
i = 1,2,... are as defined above) and each a > 0 there
exists an integer N and an uniformly continuous homeomorph—

ism h : En x E1 ——> En X E1 such that

27
1'. h = id on En+1— A x E1
2- IFn+1[h(X)] - Wn+1(x)l < e

3. For each w e E1 diameter of each component of

AN X w is less than S-

ggpgf: Let 5' = g- then by Lemma 2.4 there exists
a uniformly continuous homeomorphism hl and an integer
N satisfying

a. h1 = id on En+1- Arx E1

b. lwn+1h1(x) - wn+1(x)l < 4m + 2 for some positive

integer m.

c. diam w;(A' X w) < 48'
for all w e E1 and components A' CIAN n A.

Let h2 be the homeomorphism ha : EN X E1 -> En X E1
given by
4m + 2
"—T’T'

'h2(x.t> = (x. t)

The homeomorphism h = hglhl hz is the desired homeo-
morphism.

The homeomorphism h is isotOpic to the identity since
the homeomorphisms of Lemmas 2.3 and 2.4 were.

SuppOse Ai i = 1,2, ..., be a sequence Of manifolds
as defined above (see page15). Let Gn be the upper
semi-continuous decomposition of En into the arcs of A03

and the points of En - Aan' Denote the decomposition space

of Gn by X‘n'

28

Theorem 2.6: Xn x E1 = En+1

 

This theorem follows from Theorem 2.1 and the follow—
ing theorem which is due to R. H. Bing [4].

Theorem: Let Ai i = 1,2,..., and. Xn be defined
as above. Further suppose that for each i and e > 0
there is an integer N and an isotopy u Of En+1 onto
En+1 such that no is the identity U1 is uniformly con—
tinuous and

1. ut = id on En+1 - (Aj X E1)

2. an+1 Ut(x) - vn+1(x)| < e where Fn+1 is the

n+1 onto the (n+1)-st coordin-

projection Of E
ate.

3. For each w e E1 the diameter Of each component
“Of u1(AN x w) is less than 5.

Then .x.n x E1 = 12:"+1 .

Remark 2.4: Note that there exists a countable col-
lection Of compact sets Ri such that

1. AxE1=URi.
i

2. h(Ri) CZRi for all i = 1,2,
3. h/bdy R1 = id for all i = 1,2, ...
4. diam.[vn+1(Ri)] < c/S.

5. diam h[Ri n (An X E1)] < 5/2.

Where h. is the homeomorphism Of Theorem 2.5.

CHAPTER III
THE "DOGBONE SPACE" SQUARED IS E6

In [7] Kwun showed that there exists two non-manifolds

whose product is En for n 2.6. In this chapter we give

6

another factorization Of E into non—manifold factors.

Let Bi be a collection Of m-manifolds in Em whiCh

are analogous to the Ai defined in Chapter II. That is Bi

(i = 1.2.‘°°) is a collection Of compact manifolds in Em
satisfying P1 and P2 (see page 15) and Ba) 3 0 Bi'
1
Throughout this chapter let Ai (i = 1,2, ---) be

as defined in Chapter II.

 

Lemma 3.1: Given A and .B components Of ‘Ar and
BS respectively and e > 0 then there exists an integer
N > max(r,s) and a homeomorphism h: En X Em —-> En X Em
such that;

1. h = id on En+m

- (A x B)
2. DiamiKA' X B') < e for each component
A' CANOA and -B' CBNOB.
Proof: By Lemma 2.1 there exists integers J and K,
a set of n—cells E1, E2, ---, Ep, and a set of m-cells
F1, F2, ---,Fq such that

1. Ei C int A for each i

ll
...:
N

'0

ll
H
[0

ID

2. Fj C int B for each j

29

30
3. For each component A"CAJ n A there is at
least one i such that A' C int Ei. .
4. For each component B' CZBK n.B there is at
least one j such that B' C int Fj'
Let N = max(J,K) and note that for each component
A' X B' C(AN X BN) 0 (A x B) there exist integers i and
j such that A' X B' C int Ei X int Fj'
By Theorem 1.2 there exists a collection of (n+m)-cells

G1, G2, ..., G such that;

3
1. For each component A' X B' Of (AN X BN) 0 (A X B)
there exists a unique k such that .A' X B' C
int GR and A' X B' n Gj = ¢ for all j:# k.

2. Gk C int Ei x int Fj CIA X B for some i and 3.
Note that even though i # j it may be the case that

Gi n G. # ¢. Since (U G-) n Gi is compact for each

3 #1 '3
i = 1,2, ...,fi, there exists an (n+m)—cell Qi =
{z e Em+n/Hz - zilLfi 5, for some 2i 6 int Gi and O fi.e/2]
C Gi such that Qi fl Gj = ¢ for i # j. -For each compo-

nent A' X B' C:[(AN X BN) n (A x B)] there exists an

n+m +m

integer i and a homeomorphism hi : E -—o En such

that
I I
1. A X B CGi

2. h. = id on En+m - G.
i i

3. hi(A' x B') C Qi.

Define h = hl ' hz ° ...° h£° (Even though the Gi's

are not disjoint, hi is the identity on Gj n (AN X BN)

31
for j # i. Thus h satisfies conditions 1 and 2 of

the theorem.

Remark 3.1: Since the homeomorphism h of Lemma 3.1
is the identity outside a compact set h is uniformly con-

tinuous and isotopic to the identity.

 

Theorem 332: Let Ai, i = 1,2, ...; Bj’ j = 1,2,
be defined as above then there exists a pseudo-isotopy
H : En+m X I -—> En+m such that ;

a. H(x,0) = x

b. If Ht(x) = H(x,t) then for all t < 1 Ht is

+m onto itself which is the

a homeomorphism of En

identity outside a compact set.

n +m

c. H1 maps E onto itself and maps each compon—
ent of A00 X BOD onto a distinct pOint.
m+n
d. If er -(ACDXB®) then
-1
H1 (H1-(X)) = X.

Proof: Let 60 = diam(A1 x B1) and 8i = 1/21 for
i = 1,2, ... . A sequence of integers 1 3 N(1), N(2),

and isotopies.

i n+m i-l n+m

i

for i = 1,2, ... which satisfy

32

 

 

3. diam H1(A' X B', 1:1) < 8i for each component
A “3 CAN(i+1) XBN(i+1)
4. Hi(x t) = Hi-1(x i;l) for x e En+m - (A . x B . )
' ’ i N(i) N(i)
and i = 2,3, ...
5. l|H1(x,t) - H1(x,t')][< Ei-l for all x e En+m
, i-i i
and t,t € [ i li+1]P

are defined inductively as follows. Let Ar and BS of
Lemma 3.1 be A1 and B1 respectively and let 5 of
Lemma 3.1 be 51. Then there exists a uniformly continuous

isotopy

+
h1 : En m X I -—> En+m

and an integer N(Z) such that
h1(X,O) = 0,

diam h1(A' x B',1) < 51 for each component

A' X B' CAN(2) XBN(2)I

h1(x,t) = x on En+m - (A1 X B1).

Define H1(x,t) = h1(x,2t) , o .<_ t 1%.

Suppose HF and are defined. Since H:

Nk+1

is uniformly continuous for w = kgi' there exists a

5 > 0 such that if the diameter of v c En+m is less

than 6 then the diameter of H:(V) is less than €k+1

Lemma 3.1 implies the existence Of an integer Nk+2 and

an isotopy such that

h (x,0) - x on En+m’.

k+1

+
h‘k+1(x’t) x on En m ' [AN(k+1) X BN(k+1)]'

33
diam (A* X B*, 1) < O for each component

and hk+1 is uniformly continuous.

Define
Hk+1(x,t) = a]: hk+1[x,(k+1)(k+2)(t — Elf—1»);
for EET-i.t j_%$%.. Clearly 1 and 2 are satisfied.
Now
Watts = H1: new)

thus by choice of 6 condition 3 is satisfied.

+m

h x,t) = x for x e En

k+1( [AN(k+1) x AN(k+1)] hence
condition 4 is satisfied. h.k+1(A" x B",t) CA" x B"

for each component A X B CZAN(k+1) X BN(k+i)'

Diam[H:(A" X B")] < 8k by condition 3 thus condition 5

is satisfied.

 

Define
_ i n+m i+1 i
H(x,t) — H (x,t) on E x [ i , i+1] for
i = 1,2,
Define
H1(x) = lim H(x,t).
t—> 1
H1(x) is continuous map Of En+m onto En+m by condition

5. Clearly 1 implies that a. is satisfied by H. Con-
dition 4 along with the definition of H1 implies b. is
satisfied by H. Suppose a > 0 and o X 6 is a component

of A00 x BOD then there exists an integer p such that

1 _ iIL. '
25-_ SP < e. For all t > p+1 . diam H(A* X 3*: t)< 8p

34

where A* X B* is the com onent Of A X B con—

9 NM NM
taining o X 6. Thus H(o X 6, 1) is a point.’ Let x e
En-m — Aoo X BOD then there exists an integer N(q) such

n+m ’ . .

that x e E - B 4 m lies that
[AN(q) X N(q)] thus 1 p

H(x,t) = H(x, ail) for all t >.3&l.. H/En+m X [Opgéi] is
an isotopy thus H;1[H1(x)] = x and d. is satisfied by H.
Let o1 X 61 and Q2 X 62 be distinct components of
A00 X Ba) then there exists an integer N(j) such that
al X 51 CIA' X B' and o2 X 62 CIA" X B", where A' X B'
and A" x B" are distinct components of AN(j) x BN(j)°
Thus H1(Q1 X 51) # H1(Gz X fig) and Co is satisfied.

Therefore H is the desired pseudo-isotOpy.

Corollary 3.2: Suppose F is an upper semi-continuous

 

+m

decomposition of En consisting of the 2-cells o X 5,

n+m_ (A XB ) .

where a CIA and 5<:B , and the points of E
a) oo oo 00

If Z is the decomposition space associated with F then
Z is topologically En+m. Moreover, there exists a uni-

formly continuous homeomorphism carrying Z onto En+m.

Let Gn be the decompOSition of En into the arcs
Of A and points Of En - A . Similarly let G be

a: co m
the decomposition of Em into the arcs of B00 and points
of Em - BOO. Suppose that Xi (i = 1,2) is the decomposi-

tion space associated with Gi (i = 1,2).

Theorem 3.3: Xn XXm is tOpOlogically En+m.

35
ggggg: By Corollary 3.2 there exists a pseudo-isotopy
H of En+m onto itself which shrinks each of the 2—cells
o X B for a CIACO and 5 CIBOO. Let f = H1. The proof

will be completed by constructing a pseudo—isotopy K Of

f(En+m) onto itself which shrinks each of the arcs f(a x y),
f(z X S) where o is an arc Of AG). 5 is an arc of
B00’ z 6 En and y e Em.

Let

U1 = U f(int Ai x {Em - 31])
l
and

U2 = U f({En - Ai] x int Bi).
i

Note that each arc f(o X y) CIU1 and f(z x S) C‘Uz.
Also U1 n U2 3 ¢.
The pseudo-isotopy K can be constructed by amending
the construction Of the pseudo—isotopy in [7] as follows.
(1) Replace the compact neighborhoods Ti and Ti
with Ai and Bi respectively.
(2) In the proof Of the Lemma replace Theorem 1 of
[1] with Theorem 2.6 Of this thesis. And further

replace the Ri by Ri of Remark 2.4.

 

[1]

[2]

[3]

[4]

[5]

[5]

[7]

[8]

[9]

LITERATURE CITED

J. J. Andrews and M. L. Curtis, n-space modulo an arc,
Ann. of Math. 75 (1962) 1-7.

S. Armentrout, Monotone Decompositions Of E3, Top—
ology Seminar Wisconsin, 1965, Princeton Press

(1966)1-25.

R. H. Bing, A decomposition Of E3 into points and
tame arcs such that the decomposition Space is
topologically different from E3, Ann. Of Math.
65 (1957) 484-500.

, The cartesian product of a certain non-
manifold and a line is B‘, Ann. Of Math. 70
(1959) 399-412.

 

, Decompositions Of E3, Topology Of 3-

 

manifolds and related Topics, Prentice-Hall
(196275-21.

J. G. Hocking and G. 8 Young, Topology, Addison—
Wesley, 1961.

K. W. Kwun, Product of euclidean Spaces modulo an
arc, Ann. of Math. 79 (1964) 104-107.

G. T. Whyburn, On the structure of continua, Bull.
Amer. Math. Soc. 42 (1936).

 

, Decomposition Spaces, O 010 Of 3-mani—
folds and Related TOpics, Prentice-Hall (1962)

2-4.

36

‘IIIIIIIIIIIIIIII