EMBEDDING CANTOR $ETS 3N MANEFOLDS

Thesis for the Degree. of Ph. D.
MICHIGAN STATE UNIVERSSTY
Richad 53am Gsbome
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Mic‘bigan State
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THESlS

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thesis entitled
"Embedding Cantor Sets in Manifolds"
presented by

Richard Paul Osborne

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mathematics

Dr. John G. Hocking

Major professor

 

[hm January 22, 1965

 

 

 

 

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ABSTRACT

EMBEDDING CANTOR SETS IN MANIFOLDS

by Richard Paul Osborne

This thesis is a study of the positional properties of
Cantor sets in manifolds. Chapter I is essentially a general-
ization to En of Bing's work on tame Cantor sets in E3.
Characterizations of tame Cantor sets are given in terms of
neighborhoods whose boundaries do not intersect the Cantor
sets. It is also proved that the countable union of tame
Cantor sets is tame.

The principal result of Chapter II is that each Cantor
set in ED lies on the boundary of an n—cell in En.

In Chapter III a very wild Cantor set is constructed
in E“. This Cantor set is then embedded in 82 X S2 and it
is shown that it lies in no open M—cell in 82 x 82. This
shows that there is a simple closed curve in 82 x 82 which

bounds a disk but which lies in no open H—cell.

EMBEDDING CANTOR SETS IN MANIFOLDS

By

Richard Paul Osborne

A THESIS

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

Department of Mathematics

1965

ACKNOWLEDGMENTS

The author wishes to eXpress his gratitude to Professor
J. G. Hocking for suggesting the problem of Chapter III, for
his guidance and sympathetic ear, and for the use of his

library during the research.

ii

ACKNOWLEDGMENTS
Chapter
I. TAME CANTOR SETS IN an
Introduction
Examples of Tame Cantor Sets.
Characterization of Tame Cantor Sets
Union of Tame Cantor Sets.
Local Tameness
II. AN EXTENSION THEOREMIMfllHOMEOMORPIIISMS ON
CANTOR SETS . . . . .
III. A TAME CANTOR SET WHICH LIES IN NO OPEN
N-CELL . . . . . . .
Introduction
The Construction of. A . .u .
Homotopic non- trigialigy of E — . .
A Cantor set in S x E which lies in no
Open u— cell . é .
A Cantor Set in S2 x S which lies .in no
Open 4- cell .

Concluding Remarics and Conjectures.

BIBLIOGRAPHY

TABLE OF CONTENTS

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53

CHAPTER I
TAME CANTOR SETS IN E“

The surprising properties of the Cantor ternary set
and its homemorphic images have provided the topologist
with some of his most provocative examples. For instance,
the "necklace" of Antoine formed the basis of the first
counterexample to the Schoenfliess conjecture in dimension
three. With the current interest in the topology of n-
dimensional Euclidean space En, the positional properties
of Cantor sets have assumed new importance. Many recent
results concerning tame and wild imbeddings in Enedepend
upon such positional properties and it is the aim of this
thesis to extend the knowledge of these Cantor sets and to
apply this new knowledge to problems concerning En.

As far as possible throughout this thesis we will use
C to represent n—cells, A to represent Cantor sets or sets
used in the construction of Cantor sets, and superscripts
to denote dimension.

Definition 1.1: A set ACZEn will be called a Cantor
set if it is a homeomorphic image of the Cantor ternary

set on [0, l].

The following well known theorem [23] is the principal
tool used in constructing Cantor sets in En and will be
used freely throughout this thesis without Specific reference.
Theorem: Every O-dimensional, compact, perfect, metric
space is homeomorphic to the Cantor ternary set.

Definition 1.2: An arc°<CZEn is tame if there is

 

homeomorphism of En onto itself which.nmps A.onto the unit

interval on the positive Xl axis in En.

Definition 1.3: (Bing) A Cantor set ACEn will be

 

called tame if A lies on a tame arc in En.

Cantor sets in EU may have very strange properties.
The following examples are all tame Cantor sets in En. In
these examples we rely upon graphic illustrations of
the first few steps in the constructions instead of
attempting to write out the analytical expressions for the
sets involved. In every construction of interest of a
Cantor set in En the set is constructed as the intersection

of a sequence of compact neighborhoods.

Example I: A Canter set in the plane (E2) whose pro—

Jection onto the x-axis covers the interval [0, 1]“

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Step 1 Step 2
Step 3
. f‘—_j' . . .
.l__

 

 

 

 

.1,
.L.__l
"—1

 

 

With only slight modifications we can get a Cantor set
in the plane unit square which intersects every straight
line passing through the top and bottom of the square. This

example is essentially the one given by Bing in [7].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Step 1 Step 2
Step 3
l 1
LEE I
L O.
L i1
L J
L I
1 h
L 1
1: “I

 

Since the union of two Cantor sets is a Cantor set
we can easily obtain from the above example a Cantor set
in the plane which intersects every straight line cutting

two opposite faces of the unit square.

This example can be generalized to the unit cube

n

C in En. Let Ci, C5, and Cé be the three largest n-cubes

in the unit cube Cn whose projections in the Xn’ x plane

I
are the same as those in step 2 of the previous example.

i _ i v i
In each Ci we embed 3 n cubes Ci,l’ Ci,2 and 01,3 in a

fashion similar to the embedding of the Ci's in On except

that the roles of the X and X2 axes are interchanged.

1
After (n-l) such steps we get 3n-l n—cubes Ci’ i=l,2,---,
3

n-l, in On whose maximum dimension in the direction of

any axis is 2/3 and such that any straight line segment

passing thru the faces Xn=0 and Xn=l intersects the "top"

and "bottom" of one of the 3n-l embedded cubes. In each

I’ 2,—--, C3n-l we embed 3n—l cubes
2(n-l)

in the same fashion as before to get 3

of these 3n-l cubes C C
cubes whose
maximum dimension along any of the axes is (2/3)2. Con-
tinuing in this way we get a Cantor set A in On such that
every straight line intersecting the faces Xn=0 and Xn=1
intersects A. The union of n such sets (one for each

pair of faces) gives us a Cantor set in En which intersects
every straight line meeting two opposite faces of the unit
cube. As a matter of fact this example may be modified by
choosing the starting cubes 01’ 02’ and C3 properly to give
a Cantor set which intersects every straight line inter—

secting the interior of unit cube.

Although the following theorem appears to be trivial
and is well known I have not been able to find a proof
,given in the literature. The proof is worth presenting
because the technique involved here of constructing a homeo-
morphism as a limit of a sequence of homeomorphisms is a
powerful tool used throughout this thesis.

Theorem 1.1: If a Cantor set ACEn is tame then there

 

exists a homeomorphism h of En onto itself mapping A onto
the Cantor ternary set in the [0,1] on the X1 axis.

Proof: If A is tame than A lies on a tame arc «'and
since % is tame there exists a homeomorphism g of En onto
itself mapping «’onto the interval [0,1] on the Xl—axis. We
have, then, a set homeomorphic with the Cantor ternary set
imbedded in [0,1]. We may assume without loss of generality
that the extreme points of g(A) are 0 and 1. Since every
homeomorphism of [0.1] onto itself can be extended trivially
to En we need only show that there is a homeomorphism of
[0,1] onto itself taking g(A) onto the Cantor ternary set.
Choose a countable dense set {am} from [0,1] -g(A) with

the same order relation as the corresponding diadic

rationals in (0, 1), i.e., if m =bl + b2 - 2 + ... +bk2k-l
then am corresponds to the diadic rational b1.2‘1+ b2,2-2 +
—k
+ bk? , where bi = 0 or 1. Let Im be the

longest open interval in [0, I] -g (A) containing am.

We define hl to be an order-preserving homeomorphism of

[0, 1] onto itself mapping Il onto (1/3, 2/3). Suppose m +

1 = b. + b - 2 + ... + b 2k‘l. We choose h to be an order
2 k m+l

preserving homeomorphism of [0, 1] onto itself mapping Im+l

onto the interval

2b1 2b 2b 2hl 2b

Oh lo...ohl + + ... + +...+ 2
m m“ 2 k 3 k

3 3 3

h

 

 

 

and being the identity outside of

 

 

h 0 O O 2bl 2b2 2bk 2bl 2bk+l
h h + + .+
3 3 3 3 3

We define h (x) = lim h oh
m—+

m m m_lo...ohl(x). h is 1—1 by con—

struction and since h is the uniform limit of continuous
functions h is continuous.
Corollaryil.3: If Al and A2 are any two tame Cantor

sets in En then there exists a homeomorphism of En onto

itself mapping Al onto A2.

We now wish to prove the following theorem which
characterizes tame Cantor sets in En.

Theorem 1.3: A Cantor set ACEn is tame iff for each
e>O there eXists a finite number of disjoint, tame n—cells

/] =
{08,1} covering A such that diam Ce,i<€ and BJCE,i A E

To prove this theorem we shall need the following lemmas.

Lemma 1.A: C be an n-cell in the interior of an n-cell

 

On and let pe Int On and U be an Open neighborhood of
p in Int Cn. There exists a homeomorphism h of On onto
itself such that h BJCn = id , h(C)CU and p6 Int h(C).
2399:: Think of On as the set of points of En such
that Hxll : 1. If q and r are any two points of int Cn
there exists a homemorphism g of Cr1 onto itself which is
the identity on Ban and g(q) = r. Now let q be the origin
and let g(p) = q. We pick a point re Int C and let g'(r) =
q. We may now shrink g'(C) by a homemorphism g" of On
onto itself into g(U). Now g-lg"g' is the desired homeo-
morphism.

Lemma 1.5: Let Cn be a tame n-cell in Er1 and let

 

pl, p2, ...,pk and ql’ q2’...,'qk he sets of distinct points
in Int Cn There exist tame, disjoint n-cells Cl, C2, ,
OR in Int or1 such that {p1}U{qi} c: Int 01 i= 1, 2, ...,
K.

Proof: For each i = 1,2, ..., K we pass disjoint

polyhedral arcs°<i from pi to qi and beyond so that pi and

q are not endpoints of4'1. Then we "swell up" eacha(i

i
into a tame n-cell. (This procedure of "swelling up" a
polyhedral are into an n-cell is a standard device in the

literature).

Lemma 1.6: Let C C ..., C be disjoint, tame

 

1’ 2’ k
n—cells in the interior of an n—cell On and let pl, p2, ...,
pk be any k points in (Int Cn -- §§QC1). Then there exist
disjoint tame n-cells c', ..., Ci in Int On such CiL){pi}

C: Int Ci for each i = l, 2, ..., K.

Proof: Choose k points ql, q2, ..., qk of On so that
qie Int Ci' By Lemma 1.2 there exist disjoint n-cells
C3, CE, ..., C; such that {p1}[/{qi}c:Int CE. Since C1 is
tame it is collared [11] on the outside by a collar A1.

Without loss of generality we may suppose (C iL/Ai)f\(CJL/A )

J

= 0 for i # j. Now Ci\) A1 is an n-cell with C being a

i
tame n-cell in Int (CiJ A1). By Lemma 1.1 we may shrink

" and

C1 by a homeomorphism h i

i so that hi(ci)CI Int C

hil Bd(CiL/Ai) = id. Now h;l(0£) =C' is the desired n-cell

 

i
for each i = 1,2, ..., k}
Lemma 1.7:” Let Cl’ C2, ..., Ck be tame, disjoint n-
n . n ‘k
cells in Int C , let pl, p2, ..., pk be pOlnts of C -121, C1

and let U,,U ., U be disjoint neighborhoods of pl,p2,

1 2’ k
. n U?
pk respectively in C - i=1 01'

h of On onto itself such that hlecn = 1 and pie h(Ci)C1Ui

There exists a homeomorphism

for i : l, 2, ..., k.
Proof: By Lemma 1.5 there exist disjoint tame n-cells

l l l l
01’ C2, ..., Ck such that {pi}l/ Ci CLInt 01' By Lemma 1.A

there exists a homeomorphism g of On onto itself such that

v n_Wi=
pie g(ci)c;Ui/7cic;ui and gl (C i=1 Ci) id.

10

Proof of the main theorem: For each k = 1, 2, 3

we choose a covering {C =1,...N of A with the following

k,i‘i k}

properties: 1) 0 [lo = v for i s j, 2) c is a tame
k,i k,j

k,i
n-cell with bicollared boundary for each k and 1,3) diam

c < 1/2k for each i = 1, 2, ..., N and u) c

k,i k k+1, i C—

Int Ck j for some j = 1, 2, ... Nk' The choice of such a
,

sequence of coverings of A may be done inductively. For

k = 1 choose a covering satisfying 1) thru A) by using the

hypothesis of the theorem. Suppose {Ck i:i=1,...Nk} has
N 3
been chosen. Since §:%IBdI (Ok i) /]A=O there exists N> 0
3
such that d ((Qk/ Bd Ok i)’ A)> N. Now we cover A by a set of
i=1 ’
disjoint, tame n-cells Of diameter less than min (l/2k+l, N).
I . =
This then gives us the desired set le+l,i'i l’°"’Nk+l}
/:ll k_// _
It is clear thatk ( i- l C k,i) - A.

Next we define a set of homeomorphisms {hk} of En onto

itself. Let c be a tame n-cell in E“ with\\§/C .CZInt c
l i-l 1,1 1

and let0(be a tame arc in C which intersects the interior

1
of each Cl,i' Define h1 = id. Assume now that hk has been
defined so that i)& contains points in the interior of
hkohk-10"' h1(ck,i) for each i = l, 2, ..., Nk’ ii)
hkl En (L/Ck-l,i) = id, iii) hk moves no point farther than
1/2k_l and iv) diam hkohk-lo"°0hl (Ck,i)< 1/2k. Now since
<£.intersects BdC for each i = l, 2, ..., N it follows

k,i k

11

that in each C

W10

there are points of°<in C . not in

k,l k,l

i=1 k+l,li' .Now we renumber the set {Ck+1,i:i = 1,2, ...,
’ C:

Nk+1} using another parameter j so that Ck+l,j,i Ck,j for

l = l, 2, ..., M3“ Let {pk+l, 3,1:1 = 1, ..., Mj} be pOlnts

J _
of (414 Int Ck’J_ i=1 ck+l’3’i and let{ Uk+l’ J’izi = 1,

2, ..., Mj} be a set of disjoint neighborhoods of the points
. _Mj

:i — l, 2, ..., M3} lying in Ck,3 - i=1 Ck+1,j,i.

{ pk+l,j,i
By Lemma 1.6 there exists a homeomorphism gJ of Ck j with the
3

desired properties i)-—-iv) for each j. We define hk+l =

0 0 U
g1 g2 ... gNk.

Finally we define h by h(x) = plE hkohk_lU...0hl(x).

N
If XAA then there exists N such that for k>N xt _ C .
i-l k,l
hence hk leaves x fixed. We see that h is a homemorphism

on En — A. Let x,y 5A and d(x,y)> l/2k then if Xe Ok i
’

thk’i it follows that h(x) # h(y). We must yet show that

h is continuous but this follows from the fact that h is the
uniform limit of a sequence of continuous functions. Clearly
then, h is a homeomorphism of En onto itself such that h(x)ed
for each X 6A. This follows from the fact that

0 o . < k
d(hk h ... hl(x),°<) 1/2

0
k-l

Note: In the hypothesis of the previous theorem we

Specified that the coverings of A be composed of tame n-cells.

II

This was not necessary, for, given any n-cell On, C can be

approximated from the inside by a tame n-cell. To see this

let B: denote the ball in En of radius r, let g be a homeo-

morphism of B? into En and let A be a compact subset of
Int (g(B?)). There is a6> 0 such that Int (g(B?_6))

contains A. g(B$_6)) is a bicollared (n-l) sphere in
En hence by the generalized Schoenflies theorem [10]
g(B2_5) is a tame n-cell containing A in its interior.

Theorem 1.8: The union of two disjoint, tame Cantor

 

sets in En is tame.

Proof: Let Al and A2

suppose that d(Al,A2) = 6 . Let e> 0 be given. Cover A

be disjoint tame Cantor sets and

1
by a set of disjoint open n-cells of diameter less than

min (6/2,e). Cover A similarly. We then have an

2
covering of AlL/A2 by disjoint n—cells. An application
of Theorem 1.7 completes the proof.

Theorem 1.9: Let A be a Cantor set in En. A is tame

 

iff A lies on a tame (n-l) - sphere in E“.

Proof: If A is tame then A lies on a tame arco<.
Let h be a homeomorphism of En onto itself mappingo<onto
the unit interval on the xl-axis. The boundary 8 Of the
unit cube in En contains h(x’) hence h_l(S) is a tame (n-l)
sphere containingo(.

Suppose now that A lies on a tame (n—l)-sphere 8. Let

(JCS be an (n-l)-cell such that ACZInt C. Since S is tame

C is tame so there is a homeomorphism h of En onto itself

13

such that h(C) is a subset of the hyperplane Xn = 0. Let
4 be an arc in h(C) containing A. Klee has given a homeo-
morphism of En onto itself mapping such an arc into the
xn-axis. Evidently theno(is tame, hence A is tame.

The next theorem establishes that the union of two
tame Cantor sets is tame. A generalization of the process
used in the proof of this theorem will be used later to show
that if a Cantor set is the countable union of tame Cantor
sets it is tame.

The following lemmas are needed in the proof of the
result mentioned above:

Lemma 1.10: Let h':I'+I' be a homeomorphism of I'

 

{leaving the endpoints of I' fixed, let I' be the unit

interval on the X -axis in En and let Cn be the n-cell in

l
n
E defined by
n _ . _ 2 2
C - {x. x — (x1, x2, ..., Xn) and x2 + x3
+ ... + xi 1 1 and 0 1 x1 1 I}. Then h' can be

extended to a homeomorphism h of En which is the identity

outside of On.

2 2

Proof: Let S ={xzx = (x1, x2, ., Xn)’ x2 + x3 + ...,
+ xi = l and x1 = 1/2}. For each point an and each reI'
let £x,r be the line segment joining x and r. Now define
h(y) = y if y it x r for same xeS and ref'

t ° t v v = t
y if yetx r where y €2x,h(r) and d(y ,I ) d(y,I ).

3

h is the desired homeomorphism.

1“

Lemma 1.11: Let Or1 be an n-cell in En, n # A.

 

on

can be approximated by a polyhedral n-cell, i.e., given
e> 0 there exists a polyhedral n—cell, P“, such that
d(x,Ban)<s for all xeBdPn and PnCCn.

Pgoofz We use the theorems due to Bing [8] and Connell
[18] which say that for n,# A stable homeomorphisms of En
can be approximated by piecewise linear homeomorphisms.
Letfih be an n-simplex in Int On. Now Int Cn - Int An is
a half open annulus [10] so there is a tame n—cell DnC-Cn
such that Dr1 — Int An is an annulus and d(x, Ban)<e /2 for
every xe Ban. Since Dn-Int An is an annulus there exists
a stable homeomorphism h of En mapping Bd An onto Ban.
Using the aforementioned theorem we approximate h to within
5/2 by apdecewiselinear homeomorphism f. Then f(An)

is the desired piecewise linear (polyhedral) cell.

Theorem 1.12: The union of two tame Cantor sets in

 

1’1

E , n # A, is tame.

Proof: Let Al and A2 be tame Cantor sets in En. A1

lies on a tame aTCcKl and A2 lies on a tame arcc(2. We

may think of<x2 as being on the x -axis in En and we assume

1
that the endpoints Of0(2 are not in All/A2. Let s> 0 be

given and let A =¢(2r\(AlL/A2). A is a subset of a tame

3 3

Cantor set ono(2. Blowo(2 up into an n-cell Cn given by

15

x:d(xl,al):e/3 . Let x1, x2, ..., xk+l be a finite set
of points Of°(2 - A3 such that 0<x1+1 — Xi<€/2 for i = l,
2, ..., k and x1 and xk are the endpoints OfCX2. We define

- O = n
Ci ~ {y.y (yl, y2, ..., yn), Xiiyiixi+l, yeC }for

i = l, 2, ..., k. Evidently the Cis are n—cells of diameter
less than c which coverq{2. Shrink each Ci to form
a new n—cell CXClCi so that Ci — Int 0; is an annulus and

A3/7C3 = A3/7Ci. By Lemma 1.11 there exists a polyhedral

(with reSpect tocfll) n-cell Ci such that Ogc;1nt Ci and

CiCLInt Ci. If necessary we rotate C slightly to get an

i

n-cell C? such that C"CZInt C*<:Int C! and Bd C¥r7
l i i 1 l

a finite set Pi' For each peP

n—cell containing p contained Int Ci - CE and so that for

any qui Cq(\Cp = 0. Let hi p be a homeomorphism of
3

Cp/74 1’ onto itself which leaves the endpoints of Cp

x1 is

i choose Cp to be a small

1

fixed and maps a point of (Op/7X l) - A onto p. By Lemma

1
1.10 hip can be extended to a homeomorphism hip of En which

is the identity outside of Cp.

= = -l *
i pEPi hip and set 01 hi (Ci).

The set {Cizi = l, ..., K} is a disjoint collection of n-

Finally we define h

cells which cover x2!\(AlL’A2) such that Bdqu(AlL/A2) = E.

l

The remaining points of AIL/A2 -— géfi Ci can be written as

the union of two disjoint tame Cantor sets hence by Theorem
1.8 is a tame Cantor set. Since Alt/A 2 — $51 C1 is tame it
may be covered by a disjoint system of n-cell of diameter

09

less than e:whose intersection with i=1 Ci is void.

16

A study of local prOperties of embeddings
of Cantor sets in En is of some interest in its own right
and will enable us to prove the global theorem on the union
of tame Cantor sets which is a generalization of Theorem 1.12.

Definition 1.1:: A Cantor set ACEr1 is said to be

 

locally tame at st if there exists a neighborhood NX of

x such that A/7Nx is a tame Cantor set.

 

Theorem 1.13: A Cantor set ACEn is tame iff it is
locally tame at each of its points.
Proof: Suppose A is locally tame at each of its

points. It follows from the definition of local tameness

that A may be covered by a finite set N1, N2, ..., Nk of
open subsets of En such that Ni/lA is a tame Cantor set
k
= l = l =3 __ U I
and Ed Ni/IA Q. Define Ni N1 and N1 N1 j=l N3.
Since Ni CNi and Mimi is tame, it follows that NiflA is
tame (or empty). The collection {Nizi = l, 2, ..., k}

is an open cover of A such that Ni/7%'= Q for i # j. Thus
A = §§4(Ni/]A) is a decomposition of A into a finite number
Of disjoint tame Cantor sets which by Theorem 1.8 is tame.
The proof of the converse is trivial.

Definition 1.5: An arco(CEn will be called locally

 

tame at xecc if there exists an open n-cell neighborhood

NX of x and a homeomorphism h:NX—>En such that h(NX) =

EU and h(NX/M) is the x axis in E“.

l

17

One might have expected that local tameness of a
Cantor set would be defined in terms of local tameness of
arcs. That such a definition is equivalent to that given
is shown by the following theorem.

Theorem 1.14: A Cantor set ACEn is locally tame at

 

xeA iff A lies on an arc which is locally tame at x.

Proof: Suppose A lies on an arco<which is locally tame
at x. Let NX be a neighborhood of x and thX+En be a
homeomorphism of NX onto En such that h(Nx/7df) lies on the
xl-axis. Now let N; be a compact neighborhood of h(x) such
that Bd N§/]h(A/7NX) = 0. Then h(A)/7N; is a tame Cantor
set. Let e> 0 be given and choose 6> 0 such that d(x,y)

l(x), h-l(y)) <6 for each x and y in

< 6 implies d(h_
Nk. Cover h(A/iNX)/7Nk by a disjoint family {Ck} of n-cells
of diameter less than a such that (Usd Ck)” h(A/INX) = (a
The family {h-l(Ck)} is a covering of A/Yh—1(N;) by a
disjoint collection of n-cells of diameter less than e
such that (U Bd h‘l(ck))n A = D. By Theorem 1.3
A/7h_l(N£) is a tame Cantor set.
Conversely suppose A is tame at x. Let NX be a neighbor—
hood of x such that NX/)A is a tame Cantor set. We may
suppose that NX is an n-cell such that Ed an A =0. Let
X be a tame arc containing NX/)A. Letc(’ be an arc in

En

... I . .
— NX contalnlng A — Nx° x ando< are diSj01nt arcs so
they may be connected to get an arcCK"° 5% " contains A and

is locally tame at x.

18

In recent papers by Cantrell and Edwards [16] and by
Cantrell [1“] it has been shown that if an arc in En, n 1 A,
is Wild it must fail to be locally tame at an entire Cantor
set of points. Papers [12], [13] and [15] have been written
by Cantrell in which a principal objective is to establish

n-l n

the analogous result for S in E , n l A,

i.e., if an (n-1)-sphere Sn"l in E“, n_3 A, is wild then

Sn"l fails to be locally flat on a Cantor set of points

(see [Lllfor the definition of local flatness for spheres)
Although this statement has not yet been proved it has been
shoWn to be related to a generalized annulus conjecture [15].
One might wonder what sort of wildness properties a Cantor
set in En could have. Could a Cantor set be wild at just

one point? The following set of theorems is aimed at
answering such questions. Although these theorems are the
same in statement to those established by Bing [7] for E3

the proofs used by Bing could not be generalized to the case

1'1

of E , n>3.

Theorem 1.15: If A is a Cantor set in En which is

 

locally tame at each of its points with the possible exception
of a single point xosA, then A is tame.

Pgoof: In [7] Bing established this theorem for
n = 3; consequently we assume that n1“. Let {Ni} be a decreasing

sequence of open neighborhoods of XO such that Ed Nif)A = Q

l9

and diam Ni<l/i for each i = l, 2, 3, .... Now since Ai

= (N1 — Ni+l)n A is locally tame at each of its points A1

is a tame Cantor set for each i. For each i let Ci 1, C
2
., C1 k be a disjoint collection of n—cells such that
’ 1

1,2’

_ _ cEi/
Bd C fiA - fl, Ci,jC:Ni N1+1 and Aic‘j=l Ci,j' It

1,3
is easily verified that the n-cells may be so chosen. For
each 01,3 letafii,J be a tame arc in Cij containing Cijfl A.
Define Bk = {xzx eEn and d(x, x0): l/k}, let > be an ordering
on pairs of integers defined by (i, j) >(m,t) if i>m or
i = m and j>m, and choose arcs B as follows: let 8
,w. k | ij 11

n ,g

be a tame arc in E —- i=1 j=1 joining an endpoint ofx 11

to an endpoint Oftii2. Suppose now that arcs B have been

i,J
_ Kal
chosen for ijch i<m, let em? be a tame arc in Bm—l (i,j)>(m,£+l)
Cij) - (i,j)<(m, ) (C(i,j Bij) jOlng the free endpoint of

for i = l, 2, ..., k —l

qu,z with an endpoint ofn m

~m,2+1 /
and let Bm,km be a time are in Bm— (i,j) g(m+l,1)Ci,j-

b . . /
(1:3) <(m,km) (Ciij Bij) Joining the free endp01nt quim,km

to a free endelnt Of3tm+l,l'

(See Figure 3.)

9\9

 

2O

Leta(=i=kj1 \Jfiijl (xijUBij') U {x0}. Theno<is a metric

continuum with exactly two non-cut points, hence is an

arc. By the method of constructiono(is locally flat except
possibly on a countable set. By the result of Cantrell [1A]
such an arc in En, ngu must be tame.

Corollary 1.16: The set of points at which a Cantor

 

set ACEn is wild can contain no isolated point.

Proof: Suppose a Cantor set A is wild at the point
x0 and suppose there is a neighborhood Nx0 of XO such that
A/lNX6 is locally tame at each point except x0. The Cantor
set A/NX0 contradicts the previous theorem.

Corollary 1.17: The set of points at which a wild

 

Cantor set ACLEn fails to be locally tame is a Cantor set.

Proof: Let W denote the set of points of A at which
A fails to be locally tame. From the definition of local
tameness it is clear that W is a closed subset of A and by
Corollary 1.16,W contains no isolated points. It follows
that W is closed and dense—in-itself hence W is a Cantor
set.

Theorem 1.18: If a Cantor set ACEn is locally tame

 

at each of its points with the possible exception of the

points Of'a tame Cantor set B in A, then A is tame.

21

Pgoofz Let s> 0 be given. Using Theorem 1.7 we
shall show that A is tame by covering it by a set of disjoint
n-cells of diameter less than 6 whose boundaries do not
intersect A. Cover B by disjoint n-cells 01’ C2, ..., C
of diameter less than 9/2 such that B/7(§:1Bd Ci) = O.

k

Let 0 < 2 6<min d(Ci, C ). We restrict our attention
ifj 3

now to a particular C Let 0<61 < d(B, BdCi), letni =

i'
min (s/2,6,6 ) and let N1 = {x:d(BdCi,x)fni . The

Cantor set A/7Ni is tame at each of its points hence it is
tame by Theorem 1.13. Cover A/INi by disjoint n-cells of

diameter less than ni whose boundaries do not intersect

1,1: 01,2: --~, Ci,m be the set of all such

n-cells containing a point of BdCi/]A. Next we choose

A/lNi. Let C

homeomorphisms hi,l’ hi,2’ ..., hi,m with the properties:
n

1 h E - C = id and 2 h A C A BdC =

) 1.3 ' 1.3 ) i,J( ”113) 1 ‘3
= o o .- _

of diameter less than s such that Bd h;l(Ci)/?A = Q. If

we have defined hi for each i and we define h = hi h20...
l

0h ,then h' (Cl), h‘1(c

>,
k 2
by disjoint n-cells of diameter less than e such that

., h-1(Ck) is a covering of B

A/1(;§é Bdh_l(Ci)) = v. Since A-(£§€h_l(ci)) is tame it

may be covered by disjoint n—cells of diameter less than B

k
which do not intersecti=1 h-l(

Corollary 1.19: Each wild Cantor set in En contains a

Ci)'

 

Cantor set which is wild at each of its points.

22

Pooof: Let A be a wild Cantor set in En and suppose
A fails to be locally tame on a Cantor set W. Then W must
be wild at each of its points, for if W were locally tame
at xew then there would be a neighborhood Nx of x such that
NX/7W is tame. But then A/leis locally tame except for
the points of a tame Cantor set, contradicting Theorem 1.17.

Corollary 1.20: The set of points at which a Cantor

 

set is wild is empty or is a Cantor set which is not locally
tame anywhere.

Using the previous results we may prove the following
theorem on the union of tame Cantor sets in En. This theorem
was given by Bing [7] for E3 and the proof now generalizes
easily. It is repeated here for the sake of completeness.

Theorem 1.21: If the Cantor set ACLEn is the countable

 

union of tame Cantor sets A A then A is tame.

1’ 2, ...
Proof: If A were wild then by Corollary 1.18,A would

contain a Cantor set A' which is wild at each of its points.

The Baire-Moore theorem tells us that no compact Hausdorff

space is the union of a countable number of closed subsets,

no one of which contains an Open subset of the space (for

a proof see[23]). So A' must contain a Cantor set A"

which is open in A' and which lies in one A1. But the A'

is not locally tame at any of its points. This contradicts

the fact that A1 is locally tame at each of its points.

CHAPTER II

AN EXTENSION THEOREM FOR HOMEOMORPHISMS
ON CANTOR SETS

In 1921 L. Antoine [3] gave an example Of a Cantor
set in E3 whose complement was not simply connected.
This then was the first known example of a wild embedding
of a Cantor set in En. Shortly thereafter (192“) J. W.
Alexander [1] showed that the Cantor set of Antoine, Often
called Antoine's necklace, was contained in a 2-sphere
in E3 disproving the Schoenflies theorem for E3. Con-
currently Alexander [2] gave an example of a 2-sphere in
E3 which was wild at a tame Cantor set of points. In
19U9 APtin and Fox [5] constructed 2-Spheres in E3 which
were wild at a single point. Shortly thereafter (1951)
Blankinship [9], a student of Fox, published a paper in
which be generalized the construction of Antoine's neck-
lace to En for any n13, i.e. be constructed Cantor sets
in En whose complements were not simply connected. In
this same paper he showed that these generalized necklaces
must lie on the boundary Of a K-cell, OfKin; thus giving
a method for constructing wild K-cells and spheres in En.

In this chapter we shall Show that every Cantor set

n

in E , n12, lies on the boundary of a K-cell in En. This

23

theorem is a direct extension for EU of the well known
theorem [25]: Any o-dimensional, compact subset of a
Peano space lies on an arc.

We shall need the following lemmas.

Lemma 2.1: Let U be a component of the set V in a

 

locally connected space X. Then BdUCZBdV and if V is
open then U is open.

Pooof: Let XeBdU,then for each neighborhood NX of
x in X Nxcontalns points of U and U' (the comphament in
X Of U). If x were not a boundary point of V then there
would he an open connected neighborhood NX of x which was
contained in V. Then UuNX is a connected subset of V
prOperly containing U, contrary to the assumption that U
was a component of V. If V is open then BanV = 9 so BdUnU =
W and U is open.

Lemma 2.2: Let U be a bounded, connected, open

 

subset of En and let A be a compact subset of U. Then
there exists a polyhedron P;U such that ACLInt P and Int P
is connected.

Pooof: LetE = min d(x,BdU). Triangulate
XEA

Ell

by a triangulation T of mesh less than 6/2 and let P'
be the polyhedron composed of all simplexes of T contained

in the star of a simplex containing a point of A. Let

P1’ P2,---, Pk be the closures of the components of the
interior of P'. Since U is connected there is an arc CK
joining each of the polyhedra P P °°°, P

1’ 2’ k”

Let 5' = min d(x, BdU), let T1 be a refinement of T of

mesh lessxgghn 6l/2 and define P" to be the set of all

' simplexes of T5 which are contained in the star of a simplex
which contains a point ofo(. Finally define P = Pit/B".

P is then the desired polyhedron.

Lemma 2L3: Let s>0 be given and let A be a compact,

 

O-dimensional subset of En. Then there exists a finite

collection of disjoint, open, connected subsets {U :1 = 1,2’-°°

i

’,K} of En which cover A and such that 1) diam U <e , 2)

i

U: is a polyhedron and 3) En - Ui'is connected.

Proof: First we select an open neighborhood Nx of

each point xhof A of diameter less thanisuch that Ed Nx/lA =

Q. Select from this cover of A a finite subcover N1, N2,
G) C”)

Np' Now let N = i=1 N£'£=l (Bng). By Lemma 2.1 each
component VJ of N is an Open set whose boundary is in

P
BdN = £§{ IBleg thus BdVJ/TA = U. The set{VJ} is an open
cover of A hence there is a finite subcover {Viz i=l,2,°",m}
of A by sets of {Vj}° We now have a finite covering
{Vi: j=l,2."°,m} of A by disjoint, open, connected subsets

of’E? each of diameter less than e such that BdV n A = a.

1
Now for each i apply Lemma 2.2 to get a polyhedron in V1

whose interior W is connected and contains Vi/IA. Finally

1

for each i let U be the complement of the unbounded com-

i
1. If UiCLUJ for i # j drop this-set from the
list of polyhedra covering A. We now have the desired

ponent Of En-W

covering of A.

26

Lemma 2.4: Let P and Q be disjoint Compact polyhedra in
En both of which are the union of n-simplexes and leta(be
a polyhedral arc with endpoints p and q such that
4 4 P = {p} and q/7Q.= {q} and qiis in the interior of
an (n—1)-simplex;a-of Q. Thencx can be "blown up" into
a polyhedral n-cell C such that 0/)P and C/lQ are (n-l)-
cells in BdP and BdQ respectively and for a given e>O
d(x,o()<e for any xeC.
ProoP; Letc{1 be a line segment in En and let P1

and P be (n-1)-dimensiona1 hyperplanes in Er1 intersecting

2

(x l at its endpoints a and a, respectively. Leta”l be an

1

(n-l)-simplex in P containing p and not intersecting P2.

1

Then the set C consisting of all points lying on line

1
segments parallel tOO(l with one endpoint inazl and the
other endpoint on P2 is a polyhedral n-cell. To see

that C is, indeed, an n—Cell is a straightforward but

1
messy computation in analytic geometry.

NOw, starting at p, number the linear segments ofa(in
order: 41,42, "',o(k. Let p = a1, a2, "', ak-l = q
be the set of endpoints ofa(1,o(2-, "’,o(k where a1 and
a1+1 are endpoints Of0(i. Letor-l be an (n—l)—simplex in
BdP such that dfamCTi<s, pea; andtxl does not lie in the
(n-l)-dimensional hyperplane determined bycri. Let P2 be
a (n-l) dimensionalghyperplane‘”intersecting“l andc{2 at a2.

Applying the remarks of the above paragraph blowa(l up into

27

an n-cell Cl such that BdlelPl and Bd02/)P2 are (n-l)-
simplexes. Now apply the entire process again toa(2 and

BdCl/iPZ. We continue expanding in this fashion until we

come toa(k., Chooserk+1cr so that qEU—ki-l and so the

necell generated by lines parallel toa(k with endpoints in
k+l

0-k+1 intersects Ck-l in an (n—l) = simplex. C = $41 C1

is the desired n-cell.

Theorem 2.5: Let A be a Cantor set in En. Then A

 

lies on the boundary of an n-cell CnCZEn. Furthermore Or1 can
be any chosen that A is tamely imbedded in Ban. (Note
that On itself may well be wild in En and in fact Cn must
be wild if A is wild.)
£3993; Let Co be a polyhedral n—cell in ED whose

P P be dis-

11’ 1,2’ ’ 1,kl
joint polyhedra of diameter less than 1/2 such that Int

distance from A is 1. Let P

U
1,1 is connected and.ACi=1 Int Pl,i (Lemma 2.3). Let a(

be a polygonal arc in BdCo. let.Xl,1, X1,2’ ’ xl,kl

P

be k1 distinct points ofo( and let yl’l, yl’g: : yl,k

be kl points such that]l i lies in the interior of an
S

l

(n—1)-simp1ex on the boundary of P Choose disjoint

l,i'

polyhedral arcs0( , o( , "',0( so that the end-
1,1 1,2 1,kl

points oféfl’i are xl i and 371,1 and so thatdl’in Co =

D
k
- L/ _
{x1,1} ando{l’i/7‘ j=1 Pl,i) — {Y1,i} . Applying Lemma 2.A

blow eacha(l 1 up into a polyhedral n-cell Cl 1 such that
. ’ 3 a

Cl,irK \)

J¢i 01.3) = a’ 01,1 Co is a polyhedral (n-1)—cell

28

,r] _ _ =
and 01,1 ‘P1,i is a polyhedral (n 1) cell. Let TO Co. Let

h1 be a homeomorphism of CO onto C1 = Co(§;€ Cl,i) such

that hl (X ) = Y

l,i l,i'
Suppose now that the sets Pm,l’ Pm,2’ , Pm,km,
Cn,1’ Cm,2, , C m,km and Cm together with Tm and hm

have been defined. For the moment we restrict our attention

to a polyhedron Pm i' Applying Lemma 2.3 we get disjoint
’

i i . i
.0 2'
m+1,l’ P P in Int Pm

pOlyhedra P m+1,2’ ’ m+l, 1
meter less than 1/2

,1 of dia—

m+1 whose interiors cover P A A. Let

m,i
= h -°h °"'°h and choose distinct oints x iv

rm m m-l 1 p m+l,1’

X i ... i

m+1,2’ Xmi-l,9.,i Of fm (00/le Pm,i' For each j =

' ... 1
1,2, ’23- let Ym+l,i

(n-l)—simp1ex on the boundary of P

be a point in the interior of an

i
m+l,i‘

in order to avoid ever increasing numbers of subscripts

1 i
m+1,j 5’ 3

i = 1,2, ---, km lexicographically in (i,j) (and correspond-

i , i
ingly the Xm+1,j s and Ym+l,j

ordering using two subscripts P

At this point,

or superscripts,we shall reorder the P = 1,2, ..{,21,

'3). So we now have an

m+1,1’ Pm+1,2’ "' Pn+1, km+l°

m+l,l, °(m+1,2’ —--, mm”, kmh be disjoint polyhedral

and“m+l,i are the endpoints of

Now letc(

arcs such that l) xm+1,i

°< m+l,i’ 2)°(m+l,ic Pm’ji for some ji, 3)0( an Pmiji .-_

{xm+1,i}’ u)°‘m+l,i an<Pm+l,i) = {ym+l,i} and 5) °( m+l,in

(2&3 Pm+l,£) = O (See Figure A). Next we wish to define the

m+l,i

set Tm Let SE x ={yz yeCO and d(x,y)_fe} and let 2 =

+1' , m+l,i

29

.: onsmfim

 

 

 

 

3O

Zm+l,i

$1 (Xm+1,i)° Choose €m+1>0 small enough so that S€m+l,
C:Int Tm, €m+1 <1/2 m+1 and S€m+l, Zm+l,i /)S m+l,j =

U for i # j. k

Define Tm+l = :gil S€m+l, Zm+l,i' Applying Lemma 2.4 blow
cache/“1+1,i up into a polyhedral n=cell C’m+l,i such that
l) Cm+l,i /)Cm+l,j = U for i # j, 2) Cm+l,i/jc m’ji is

an (n-l)-cell in fm (Tm+l), 3) Cm+l,fq Pm+l,i is an
(n-l)-cell. Let Cm+1 = CmL) ( ggil Cm+l,i) and choose a
homeomorphism hm+lz Cm—HCm+l of Cmonto Cm+l such that
h(xm+l’i) = ym+l,i and hm+l |Cm — f(Tm+l) = id. Finally

we define f(x) =

m+oo

lim fm (x).

Since f is the uniform limit

of a sequence of continuous functions f is continuous.

Because the domain of f is CO, a compact set, we need only

show that f is 1-1 to establish that f is a homeomorphism.

co

= /3
Clearly Tm+lC:Tm and T m=l Tm

is a Cantor set in BdCO.

For any point stO=T there exists an N such that for m>N

x t Tm thus for all m>N fm(x) = hm(f

(x)) = f

m-l

m—l

(x) so

f = fN in a neighborhood of x and f is a homeomorphism in

a neighborhood of x.

f is continuous f(BdCo) is compact.

d(a, fm (BdCO))< 1/2m hence d(a, f (BdC )) =

nA=

and Ac:f(BdCO). Because fm(BdCO)

We see that f is 1—1 on C0 - T.

Now for any point aeA

0 so aef(BdC)

Q for each m

and hm, m=l,2,--- is eventually the identity on each a t

T it follows that A<:f(t).

a sequence {Zm} of points from the set {Z

such that d(Zm,Z)

Since for each as

m,i

Since

T there exists
i=l,2,--—, Xm}

<1/2m and d (fm(Zm),A) <1/2m. Let e>O be

31

given, by uniform continuity of f there is a 6>0 such that
for d(x,y)<6 d(f(x),f(y))(€/2. Choose m large enough so
that 1/2m<6and l/2m‘2< 8/2. We have
d(r<a>,A>_<_d<f(s>, f<zm>) + d<f<-zm)., rm <2 mm
+d (Fm(-Zm), A)

1 8/2 + 1/2m-l

+ 1/2m.
It follows that f(a)c;A hence f(T)C.A: f(T) :A. Now
let y ¢ Z be another point of T and let {Ymg m=l,2, ———}

be a sequence of points from @%m i=l,2,---km, m=l,2,—-—}

:i.
such that d (ym’y)< l/2m. There exists N such that for
m>N d(s m’ ym) >J >0. Now since $12 fm (Zm) = f(B) andmlim
ffl(yfi) = f(y) and from the fact that fm(ym) and fmGZm) are
eventually in distinct, disjoint polyhedral neighborhoods it
follows that f(y) x f (5).

Finally we want to show that Ac:f(x). This follows
from the fact that d(fm(w), a)< 1/2m for each acA.

Note that f(CO) is an n-cell which is polyhedral
except at the points of A.

At first glance the above theorem may not so appear

but it is an extension theorem which may be stated thus:

Corollary 2.6: Let f1 be a homeomorphism mapping

 

the Cantor ternary set on the xl-axis in EU into En. f1

can be extended to a homeomorphism f of the unit cube On

in En into En.

32

If f1 can be extended to On it can surely be extended
to any face of On, thus:

Corollary 2.7: Each Cantor set in En is tamely imbedded

 

in the boundary of a K—cell in En for O<K:n.

Note that if f1 could be extended farther to a neighbor-
hood of the unit interval on the x1- axis then A would be tame.
It is not difficult to see that Theorem 2.5 may be
generalized to piecewise linear manifolds. The analogs of the

Lemmas 2.2, 2.3, and 2.4 are easily established and the proof

of Theorem 2.5 follows as before. Thus we have

Theorem 2.8: Let Mn be a piecewise linear n—manifold.

 

Then each Cantor set in Mn lies on the boundary of an n-cell
in M“.

It might have been tempting to assert that each Cantor
set in an n—manifold lies in some Open n—cell. However the
arguments of [21] show that this is not the case.

Without the piecewise linear structure and Lemmas 2.2,
2.3, and 2.4 are meaningless so a proof of Theorem 2.5 for
manifolds without piecewise linear structure would involve a
proof quite different from the one used for piecewise linear
manifolds. Perhaps the local linear structure could be used.

Corollary 2.9: Let A be the Cantor ternary set on the

 

real line and let f:A——+ Mn be a homeomorphism of A into a
piecewise linear manifold Mn. Then f is homotopically trivial.
Proof: f may be extended to a map F of the unit square

D2 to M“. Since F(BdD2) is homotopically trivial f is.

33

A generalization of Corollary 2.9 to continuous maps of
Cantor sets into Peano continua is possible. One need only
show that each map of the Cantor set into a Peano continuum

can be extended to the cone over the Cantor set.

CHAPTER III

A TAME CANTOR SET WHICH LIES IN
NO OPEN N-CELL

Characterizing spaces by certain prOperties of the
set of homeomorphisms of the space onto itself is not new
in tOpOlogy. Such properties as homogeneity and near—
homogeneity have long been used to characterize simple
closed curves in the plane. In 1960 Hocking and Doyle
characterized the n-sphere [19] by a prOperty called
invertibility. (A space S is invertible if for every Open
set UCZS there is a homeomorphism h of 8 onto itself such
that h(S - U)CLU.) They showed that an invertible n—
manifold is an n-sphere and that a weakly invertible Open
n-manifold is En. (A space S is weakly invertible if for
each Open set U<ZS and each compact set CCls there is a
homeomorphism h of S onto itself such that h(C)<ZU). As a
natural generalization of weak invertibility Hocking and
Doyle undertook the study of what was called weak dimensional
invertibility [21]. (An n-manifold Mn is weakly k-invertible
if every compact subset of Mn of dimension k lies in an
Open n-cell in Mn). By the use of a theorem of Stallings
[26] it is easily shown that if k:>[n/2] then a.k-invertible,

compact, combinatorial n—manifold is an n-sphere. A very

34

35

surprising theorem proved in [21] states that a 0—invertible
3-manifold is S3. This theorem may be restated as follows:
Let M3 be a compact 3-manifold such that each compact,
0—dimensiona1 set in M3 lies in an open 3-cell. Then M3.

is a 3-sphere. In [21] it was observed that in all of the
decided cases an (n - 3)—invertible, compact, combinatorial
n—manifold is an n-sphere, the only undecided case being

n = 4.

It is natural then to attempt to find an example of

a compact, combinatorial 4-manifold M4 with the prOperty

that every 0-dimensiona1, compact subset of MI4 lies in an

Open 4-ce11 in M“. In [21] Hocking and Doyle indicated

4

that M would have to be simply connected.

With these facts in mind it is natural tolconjecture
that each compact, 0-dimensional subset of S2 x 82, the
topological product of 2-spheres, lies in an open 4—cell.
It is the purpOse of this chapter to show that this is not
the case, i.e. thatS2 x S2 contains a Cantor set which
lies in no Open 4-ce11.

Definition 3.1: In the space Sn x Em any set of the

 

form {x} xEm where xeSn will be called a parameter m-plane.

If P is a parameter m-plane in Sn x Em and {ht} is an isotopy

2

of Sn x E onto itself then hl(P) will be called a curved

. - . n .. m
parameter m—plane. In S x S a parameter m—sphere and a

curved parameter m—Sphere are similarly defined.

36

Theorem 3.1: Let ACSn x Sm be compact. If A

 

intersects every curved parameter m-sphere then A lies in
no open (n + m) ~cell in Sn x Sm.

ProoP: Suppose A lies in an Open (m + n) «cell C,
then given e>0 there is an isotOpy {ht} of Sn x’Sm onto
itself such that hl(A) has diameter less than c and
htISn x Sm — C = id. Let Sn and Sm be metrized in the

metric which they inherit as unit spheres in Er”.1 and

Em + 1 respectively. Metrize Sp x Sq by the standard
product metric i.e. d((x,y), (X',y')) = ([dn(x,x')]2 +
[dm(y,y')]2)l/2 where dn and dm are the metrics for Sn and
Sm respectively. Now d({x} x Sm, {x'} x Sm) = dm(x,x').
If we choose x' so that dn(x,s') >c then {x} x Sm and

{x'} x Sm cannot intersect the same set of diameter less

than c, i.e. they cannot both intersect hl(A). Suppose

-l
l

is a curved parameter m-sphere which does not intersect A.

{x} x Sm does not intersect hl(A) then b ({x} x Sm)

' l
A similar construction will establish the following:

Theorem 3.2: Let ACISn x Em be compact. If A

 

intersects every curved parameter m-plane then A lies in
no Open (p + q)—cell in Sn x Em.

The above theOrem makes it clear that if an example
could be given of a Cantor set which intersects every
curved parameter plane in Sn x E2 then such a Cantor set
could not lie in an Open (n + ?)-cell. Such a Cantor set-

"approximating" S2 in 82 x E2 will be constructed.

37

In giving the construction of a Cantor set in E”, which
will be used in "approximating" a 2-sphere and in verifying
the desired properties of it we shall use the following
lemmas, the first is due to Blankinship [9], the second is a
generalization of Artin's work in [4].

Lemma 3.3: Let dr’ds’Ds be arbitrary real numbers
with 0<dS:Ds. Let S be a compact set in En contained in the

set defined by x = dr and O<dxixsips' Let S be the set

1'"

.generated by rotating 8 about the 0142)wfihuu3defined by xr 8

r’ XS = 0, or more explicitly

IS = {XI x e E)1 and there exists y s S and there exists

0 such that xi = y1 if i # r or s and XP = dr + yS sin e,
x8 = y8 cOS'Ol
Then

a) for each (y,o) s S x E' (mod 2n), the correSpondence

(y,O)-—+x where xi = yi, i # r or s, x

I"

= dr +yssine, x8 =

y8 cosO is a homeomorphism onto S. We can therefore use the
pair (y,O) as a set of coordinates for S.

b) if U is the set in S consisting of all points with
representations (u,@) for which u s U C-S,d 5 e i 8,3 «a: 2n
where U # a then max {diam U, dS p(8 — a)} idiam U 5 11am
U + Ds p(B-a)
where

p(G) = 2sinl/2 0 if 0 i O i n

2if1r_<_Of_21r

38
c) if x c S then dr - Ds 1 xr 1 dr + DS.

n + 1 defined

Lemma 3,4: Let En be the hyperplane in E
by xn + 1 = 0. Let E? be the half space in En defined by
xn are, let S be a set in E? and let G be a simple closed
curve in the complement of 8. Let S'be the set in En + l

which we get by rotating S about the hyperplane defined by

xn . xn + l I 0, i.e.
S = (x: .3 ye S and JO30_<_G<2-nland xi = y1 for i 7‘
n or n + 1, xn = yn cosG, xn+1 = yn sine}

n +1 _ ”

Then C is null homotopic in E 8 iff C is null homotopic

in E2

Proof: If C is null homotOpic in E? - S then certainly
n + l ‘
+ _

- S.

C is null homotopic in E
n + 1

8. Conversely we define the

continuous map f: E -—4E2 by f(x) = y if x is the image

0f y under a rotation of.E$ about the hyperplane xn = xn + l:

0. It is clear from the definition that x]: S iff f(x)e S.

n + 1 — S, i.e. that C

n+1-S. LetgzD2---—-?En+l-~

2

Suppose C is null homotopic in E
bounds a singular disk in E

be a continuous map such that ngdD
2

is a homeomorphism of

BdD
2

onto C. Then because f is the identity on C,fg:

D 2

——+E2 - S is a continuous map such that fginD is a

homeomorphism of BdD2 onto C. Thus C bounds a singular

disk in E2 - S and hence C is null homotopic in E? - S.
It should be remarked that the properties of linked

sets established in [24] and [9] will be heavily relied

39

upon in verifying the prOperties of the Cantor sets constructed.
In a sense the Cantor set constructed will be a generalization
of Antoine's construction and indeed Antoine's necklace is

the Cantor set used in E3 to approximate the one sphere.

The Cantor set to be constructed here is not, however, the

same as that constructed by Blankinship in [9]. The Cantor
sets Of Blankinship could be used to "approximate" a surface

hOmeomorphic to S‘ x S’x - - - xS' (n - 2 factors) in En.

The Construction

1
Let S' be the unit circle in the xl,x2-p1ane in E“.

1
Rotate S" about the 2-dimensional hyperplane defined by x2 =
x“ = O to get a 2-sphere in the x1, x2, xu-hyperplane.

Direct calculation using Lemma 3.3 shows that we get the

2 2 2

2-sphere whose equations are x1 + x2 + x4 = l and x3 = O.

This will be the 2-sphere which is approximated by the Cantor
set to be constructed.

1 1
Expand S into a solid torus T3 homeomorphic to S X D2 Where
2 . .
D is; the two dimensional disk, T3 lying in the x1, x2, x3-
hyperplane. We shall use only the half of T3 with x2 1 0 and

we shall refer to it as T3.
3

In T3 embed four cyclically linked solid tori Tl’ — - -,

3
1 2 is less than

2/3 the diameter of T3. Rotate T3 about the plane defined

by x2 = xu = 0 to get T“ which is homeomorphic to 32 x D2.

4

T3 (figure 5) so that the diameter Of T and T

In T the rotated images of T3 = Ti/7T3 T3 = TgflTand T3

2’ 2 3
4 4

T and T: respectively where T?

, 4
will be TI’ 2 and T2 are

.
if

Xx

40

X3

 

 

 

'Figure 5.

41

homeomorphic to T“ and T: is homeomorphic to S' x S' x D2.
In T3“ construct a generalized Antoine's necklace A,
as done by Blankinship [9]. Let g be the linear map of E“
which shrinks TLl to the size of TlLl and T2L4 and leaves the
center of the image at the origin. Let fl and f2 be
Euclidean motions such that flg maps Tu onto T1“ and f2g
maps Tu onto T2”. Define gk = gogo...og (k factors) and
let fk,i = gk fi(g—l)k. Now set
A'1= Al
A2 = Al U fo,lg(’£‘l) U fo,2g(Al)
A3 = A2 U fl,l fo,l gain) U f1,2fo,lg2(‘£‘l)Ufl,lfo,2g2(£‘li
~ Ufl,2fo,2g2”,
A4 = A'3 U i,§J,x=l,2 f2,i f1,j f0,k 953(Al)
Define the sequence {Aaz d = 1,2,3, ... } by
A0 = TLI
A1 = A'lquL‘uTZ“
A2 = Aéufongwlu) U fo,2g(TlLl)Ljfo,1g(T2Ll)U fo,2g(T2u\’
A = A' L) ( iJ _ f f . g2(T 4) )
3 3 i,j,k—1,2 l,i o,j k

00

Finally define A =aJT Aa. A is the desired Cantor set. Note
that A is the union of a countable union of linked generalized
necklaces together with limit points. These limit points

constitute a tame Cantor set in the x xu-hyperplane.

l,

42

Compactness of A follows from compactness of AG for

each d and from the fact that AQC1A

d-l'
A is certainly 0-dimensiona1 at each point of iglAl
and for each point x of A — iglAl there is a neighborhood
of x of the form h h ———h (Tu) which has diameter

11 12 1x

less than (2/3)k times the diameter of T“ and whose boundary
does not intersect A. It follows that A is 0—dimensional at
each of its points; hence A is 0-dimensiona1.

That A is perfect is not difficult to establish although
it is Of no interest to us in the arguments to follow.

Let C be the circle in the x

x —p1ane which is the

1’ 3
3

boundary of the intersection of this plane with T .

Lemma 3.5: C is not null homotOpic in E“ - A.

 

Proof: By the theorems Of [17] we see that C is not

_ 3
null homotopic in E3 - EngiB'

that C is not null homotopic in E“ - inTiu.

Lemma 3.4 then assures us

n _ n n n
Let Bl — Tl L)T2 L/T3

[I1
I

_ n n
Tl L/fo,lg(Bl )L/fo,2 g(Bl

CD
II

f f g (B

1’1 ' n l
3 T1 L/(igl,2fo,ig(T3 ) (i,j=1,2 l,i O,j

n = 3 or 4

Set Bn = aQI Ban. Note that BLl is like A except that the

generalized Antoine's necklaces of Blankinship have not been

substituted for the 4—tubes.

43

The sets Ba“, d = 1, 2, 3, -—- and B“ can be con—
structed in yet another way: by rotating Ba3 and B3 about
the xl,x3—plane. Now by Theorem 1 of [17] we see that C is
not null homotOpic in E3 - 8&3 for each . It follows that
C is not null homotOpic in E3 - B3. For if it were then it

3

— B3. But such a disk

3

would bound a singular disk D in E
would lie a positive distance from B hence it would not
intersect E3 - Ba3 for a sufficiently large a. This contra»
dicts the fact thatCiis nOtrnUJ.homotOpic in E3 — B03.

Applying Lemma 3.4 we see that C is not null homotOpic in

E“ — B“.

The homotopy relations computed in [9] assure us that

replacing T3” in B“ by the generalized Antoine's necklace

does not change UnahomotopicnOH-tPiViality 0f C in Eu * B“.
(This can be proven by an argument similar to that used in

Lemma 3.10.)

Define:
_ 4
H1 ‘ B1
4
H2 - (T3 OA)U(iyl,2fo,is)
H3 - (T3 U (121,2fo’igfl3 ))«IA)D (iga'jfilflfi’jfo’lg (Bl >>

By repeated use Of the above remark we can conclude that C

is not null homotopic in E“ — Ha, a = l, 2, 3, ---.

44

Suppose now that C is null homotopic in E“ — A, i.e. that

C bounds a singular disk D in E“ - A. Let the distance from

D to A be greater_than.diam-Tu ' (2/3)k. Then D cannot

intersect Hk’ a contradiction. This proves the lemma.

Definition 3.2: Let h : T“ ——+ S2 x D2 be a surjective

 

homeomorphism n = 3,4, let Ac: T“ be the Cantor set constructed

above and let 05 Int D2. Then h(A) will be said to approximate

Sn x {o} in Sn x D2. h(A) also approximates Sr1 x {o} in

Sn x Int D2 = Sn x E2.
As an immediate consequence of the construction of

approximating Cantor sets we get the following theorem.

Theorem 3.6: S2 x E2 contains a Cantor set which lies.

 

in no open 4-cell.

Proof: Let A approximate 82 x {O} in S2 x D2. Then

the one-sphere C which is the boundary of {p} x D2 for ps Sn
is not null homotopic in (S2 x D2) - A, hence neither C not
any of its homotopic images bounds a disk in the complement
of A. Thus each parameter disk in 82 x D2 intersects A.
Applying Theorem 3.1 we get the desired result.

Corollary 3.7: 82 x E2 is not C—invertible.

 

Bing [5] has given an example of a simple closed curve

2

in S‘ x E, which bounds a 2—cell but lies in no 3-ce11. We

may now prove the following:

Theorem 3.8: There is a simple closed curve in 82 x E2

 

which bounds a disk but lies in no open 4—cell.

45

3399;: Let A be a Cantor set in 52 x E2 approximating
a parameter 82. Using Theorem 2.8 construct a disk D2 whose
boundary C contains A. Since A lies in no 4 cell C lies in
no 4—cell.

The following theorem provides a negative answer to
the question which was the genesis of this paper, namely,

is S2 x S2 0-invertib1e?

2

Theorem 3.9: In the 4-manifold S x182 there exists

 

a Cantor set which lies in no Open 4-ce11.

2 2

Proof: Let S and 82 be 2-spheres, let K be an

1

annular region about the equator of SE, let D1 and D2 be

the closures of the complementary domains of K and let

S1 and S2 be the boundaries of D1

Let Al be a Cantor set in 812 x D1 which approximates
2

S1 x {pl}, pl 5 Int DI’ and let A2 be a Cantor set in

2 . 2
S x D2 which apprOleates Sl x {p2} , p2 6 Int D2.
A

and D2 respectively.

1
Finally let A.c.Sl2 x S22 be given by A = l (J A2. We shall
2 2

Show that A lies in no Open 4—cell in S1 x 82

step in establishing this we need the following lemmas.

Lemma 3.10: Let f:D2———+ M be a continuous map of a

As a first

 

disk D2 into a space M and let C be a simple closed curve in

D2 bounding the disk B in D2. If f(C) is null homotopic in
subspace N of M then there is a map g:D2 ——»—M such that
g(x) = f(x) for x e D2 — Int B and g(B) <; N.

PoooP: Assume that f|C:C ~+ N is null homotopic. By

a well known result of Borsuk (see for example [23]) fIC can

46

be extended to a map f' on pC, the join of C with a point,
so that f'(pC)C:N. Since such a join is homeomorphic to B

we define gzD2———+ M to be

g(x) = f(x) for x (D2 - Int B)
f'(x) for x B
Since the two definitions agree on C, g is continuous.
Lemma 3.11: The simple closed curve C = {q} x 81’

2 x 892) — A.

 

qul2

is not null homotopic in (S1
Proof: If C were null homotopic in (812 x S22) - A

then C wauld bound a singular disk D' in 812 x S22 — A.

Give 812 x 822 a polyhedral structure so that 812 x D1,
2

S1 x D2 and 812 x K are polyhedral. Let d(A,Da) >2 and

using the analog of Lemma 2.4 for manifolds let N1 and N2

be polyhedral neighborhoods of Al and A2 respectively such

1 and d(x,A2) <2 for each XEN2.

and [(812 x D2) — Int (N2)](J(Sl2x K)

that d(x,Al) <2 for each st

2 S
Then Sl x Dl — Int Nl

are polyhedra in which C fails to be null homotOpic. An
application of the simplicial approximation theorem produces

2 l , m
l x S2 — Int (NlfiJ N2) whose

boundary is C, i.e. we get a simplicial mapping 5 of a poly—

hedral disk D into 812 x 822 — Int (NlCJ N2) Such that

s(Bd D) = C. Let C1’ C2, : . . , CK be polyhedral simple

closed curves in D which bound disks B
l

a singular polyhedral disk in S

1’ B2, . . . , Bk in D

(s(D)/1 (812 x D2)) and s—1

respectively such that Cic: BdS-
k k
I 2 ' '
lJ » =
(SKD)/7 Int (Sl x D2)) (:i=lBl° If for each i 1, 2,
2 1

k, s (Ci) is homotopically trivial in S1 x K then x

applications of Lemma 3.9 produces a singular disk in

2 2
(81 x DlL) S1 1

known properties of A. Assume that for some i, say i = j,

x K)- N whose boundary is C, contradicting

s(CJ) is not null homotopic in S 2 x K. Then s(C ) is not

1 J
2
null homotopic in 81 x D2 - Int N2. Let Cj,l’ 03,2’ . . .,

CJ 1 be polyhedral simple closed curves in BJ which bound
3

disks B in B respectively such that

B . . . B
1.1: 3,2: ’ 4.1 3

0J ’12-. Bdfsf1(:'(BJ)n (Sax 01))] and s‘1(s(BJ

2
(S1 x D2))c:i:& BJ,1. If for each i = 1,2 . . ., 2, s(C

) Int

3,1)

is null homotopic in S 2 x K then we have a contradiction.

1

If some C3 1 is not null homotopic in 812 x D we continue as
3

before using C and B This sets up an infinite regres-

J,1 l,i'
sion, which is impossible due to the polyhedral structure of

D. Hence we have a contradiction. This completes the proof
of the lemma.

12 x S22 were contained in an open

4—ce11. It is an easy exercise to show that A would lie

in a collared 4—cell c” in S 2 x S22.

1
curve C' in 812 x K - Cl4 which is not null homotopic in 812 x K

then we would have a contradiction. For we would have a curve
2 2
2 2
4 4

C' in S x S2
in S x S
C C' would be homotopically trivial in the complement of C .

Now suppose ACLS

If we could find a

- A which is homotopic to C or a multiple of C

1
1 2 - A: but since C' would lie in the complement of

Hence C' would be homotopically trivial in the complement of
A. We now proceed to show that (812 X K) - CLl does, indeed,

contain a closed curve which is not homotopically trivial in

2
81 x K.

48

Let 82 be a polyhedral 2-sphere, let f be a homeo-

2 2

'
morphism of 82 onto a curved parameter sphere S in S x S2

1
— Cl4 and let s : 82——+(Sl2 x S22)- C“ be a simplicial map

which is homotOpic to f.

 

Lemma 3.12: The l-skeleton of s(S2)n(Sl2 x K1) con-

tains a closed curve which is not null homotopic in 812 x 81“

Proof: Suppose to the contrary that each closed curve

in the l-skeleton of s(S2)r)(S2 x K)) is null homotopic in

812 x K. Let Ul be a component of s-1

D1)) and let Cl be a simple closed curve in BdUl such that Ul
lies entirely in one component Bll of S2 - 01' Let Bl2 be

the other component of S2 - 01' Since s(Cl) is homotopically

trivial in 312
2 2 2
S into Sl x S2

s|(Ble cl)= s|(Bl2\Icl), s

(s(SZ) 0 (Int <le x

x K there exist simplicial maps S11 and S12 of
2

such that $110311) (:81 x K, sll|(Bl2UCl) =

2
12(312)‘Sl

Let 112(812 x 822) be the 2-dimensional homotopy group

~2 2 . 2 2
of S1 x 82 . Let t.S ———+ S1

2
phism. Then n2(Sl

x K and 512(BllUCl)=s'(BllUCl
x {p} , pe822, be a homeomor—
x S22) is the abelian group generated_by
[s] and [t], where E 3 denotes homotopy class. Since [511] *
[$12] = [s] (* is homotopy juxtaposition) it follows that
either [$11] or [812] contains a non—zero multiple of [s],
i.e. either [s11] or [812] can be written as pft] * q[s]
where q? 0, p and q integers.

IfEsll] contains a non—zero multiple of [s] we continue
as above with 511 and another component U2 of s11"l (sll(S2)

2 2
(\(Int 31 x 131)). Notice that sll(Bll) c s x K. If [sll]

l

“9

contains no non—zero multiple of [s] we proceed as follows:

-1
12

let 02 be a simple closed curve in Bd U2 such that U2 lies

entirely in one component B21 of S2 - C2, and let B22 be

the other component of 82 - C2. We should note that C2c:

_ _ 2 _

B11 and that 812(812)‘: Sl x K. Since 512(02) — s(C2)C;
2 2

S1 x K, sl2(C2) is homotopically trivial in S1 x K.

Hence there exist simplicial maps s21 and s22 of S2 into
2 2

2
S1 x S2 such that s2l(B2l)(:. Sl x K, s21

* =
Sl2|(B22 L)C2). Thus we have [821] [$22] [$12]. Hence

2 2
let U2 be a component of s (512(8 ){1 Int (Sl x D2)),

(1322 u 02) =

either [521] or [$22] is a non-zero multiple of [$12]. Con-
tinuing these arguments in a finite number of such steps we
2 2 2

must arrive at a simplicial map sk i:S ——+ Sl x 82 which
3
is a non-zero multiple of [s] and sk i(82) is a subset of
3

either 812 x (D1(J K) or 812 x (S2 L} K). In either case we

have a contradiction because either of these spaces can be

continuously deformed onto 812 x {p} , p a point of D1 or D,,

hence any map of 82 into one of these subspaces is homotopic
to a multiple of [t] only.

This completes the proof of Theorem 3.9.

We should observe that the Cantor set A defined in the
proof of the previous theorem may be considered to lie in one

of the half spaces 812 x D1. To verify this we need only note
2 2

that there is a parameter 2—sphere in S1 x 82 which does not

2 2 2 2
l — p) = Sl x E .

It is easily seen that there is a homeomorphism mapping A into

2
S1 x D1.

intersect A, so A is a subset of S x (S2

50

Theorem 3.1M: Let A be a Compact set in Int(Sk x Dm),

 

let S be a parameter k—sphere in Sk.x Dm and let h be a homeo—

k x Dm into an n—manifold M“, n

morphism of~S k+m. If h(S)

lies in an open n-cell in Mn then h(A) lies in an open n-cell
in Mn.

Proof: Let Cn be an open n—cell containing S. Then

k x Dm. There exists 2

0 such that if d(x, S) <2 then x e h‘1(cn). Let p:sk x

h“1 (on) is a neighborhood of s in s

Dm ———+ Dm be the natural projection and let p(S) = x a Dm

Let B be an m-cell in Int Dm such that p(A) C.B. There is a
homeomorphism gl of Dm onto itself such that gl(B) is in an 2

-neighborhood of x and g Bd Dm = id. Let (x,y) be the co-
1

ordinates of a point of Sk x Dm. Define the homeomorphism

k x Dm ———+ Sk x Dm by g((x,y)) = (x,gl(y)). We can easily

g:S
-l n k m

see that g(A)4: h (C ) and ngd(S x D ) = id. Finally we

define the surJective homeomorphism f:Mn ——+—Mn by

f(t) = h g h—l(t) for t e h (SK x Dm)

t otherwise
f maps h(A) into Cn: so h-l(Cn) is an n—cell containing A.

Corollary 3.15: Let A be a Cantor set in SR x Dm,

 

let Mn be an n-manifold with or without boundary, n = k+m,

k x Dm ———+ Mn be a homeomorphism. If.A c; Int

and let h:S

(Sk x Dm) is a Cantor set and if h(Sk x {p} ), p e Dm, lies

in an open n-cell in Mn then A lies in an Open n—cell.
Although it appears likely the converse of the above

corollary has not yet been established for 82 x 82. A proof

51

of this theorem would seem to depend on establishing the

analogs of Lemmas 3.11 and 3.12 for non-trivial embeddings

of 32‘x D2.

1
Theorem 3.9 could easily be generalized to S‘{ x S2 if

k x D2 with properties

analogous to those of the Cantor set constructed in S2 x D2.

one could construct a Cantor set in S

The proofs of the remaining lemmas do not depend on the
dimension of SR and could easily be generalized. I strongly

suspect that the Cantor set of Blankinship could be used for

this purpose in the following way: let 811 x 821 x

x Skl be the Cartesian product of k l—spheres where each 1—

sphere is parameterized by the real numbers modulo 2n , let

f' :S l x S l x . . . x SK1

1 2
f' (01, G2, . . ., Gk) = x e Sk where x is the point on S

with polar coordinates 01, 02, . . ., 6k and let fzsll x

S2l x . . . x Skl x D2 ——+ Sk x D2 be the natural extension

of f'. It is not difficult to see that if A is the Cantor

set of Blankinship in 811 x 821 x . . . x Skl x D2 then f(A)

is a Cantor set in Sk x D2. The generalization of Theorem

———+Sk be the map defined by
k

3.9 then depends on proving that f(A) has the desired proper—
ties in sk x D2.

The foregoing results lead quite naturally to another
conjecture. If one can approximate a k-sphere in a campact,
piecewise linear n-manifold Mn, n a k + 2, then why not ap-

proximate the k-skeleton of Mn by linking many Cantor sets

together? 'If it could be shown that the k-skeleton lies in an

52

open n-cell iff the Cantor set does then)by a theorem of
Stallings [26JJMn is the n-sphere, n > 2. Thus we are led
to the following:

Conjecture: The only compact, piecewise linear n-

 

manifold, n > 2, which is O-invertible is the n—sphere.

A proof of this conjecture would lead to a generaliz-
ation of the characterization by Bing [6] of the 3-sphere,
namely: If Mn is a compact piecewise linear n—manifold such
that each simple closed curve in Mn lies in an open n-cell
then Mn is an n-sphere. Bing's result was originally proposed
as a weakened form of the Poincare conjecture for 3—manifolds.
If one looks at the above conjecture from this point of view

it is indeed surprising!

BIBLIOGRAPHY

J. W. Alexander, An example of a simply connected surface
bounding a region which is not simply connected, Proc. of
Nat. Acad. of Sciences, U.S.A., 10 (1924), pp. 8—10.

J. W. Alexander, Remarks on a point set constructed by
Antoine, Proc. of Nat. Acad. of Sciences, U.S.A., 10
(1924), pp. 10-12.

L. Antoine, Sur l'homeomorphisme de deux figures et de
leurs voisinages, J. Math. Pures Appl., 4 (1921), pp.
221-325.

E. Artin, Zur Isotopic zweidimensionaler Flachen im R4’
Abhandlungen aus dem Math. Seminar, Hamburg IV (1926),
pp- 174-177.

E. Artin and R. H. Fox, Some wild cells and spheres in
three dimensional space, Ann. of Math., (2) 49 (1948),
pp- 979-990-

R. H. Bing, Necessary and sufficient conditions that a
3—manifold be 33, Ann. of Math., 68 (1958), pp. 17-37.

R. H. Bing, Tame Cantor sets in E3, Pacific J. of Math.,
11 (2) (1961), pp. 435—446.

5 can be approxi-

R. H. Bing, Stable Homeomorphisms of E
mated by piecewise linear ones, Amer. Math. Soc. Notices,
10 no. 7 (1963), p. 666.

W. A. Blankinship, Generalization of a construction by

Antoine, Ann. of Math, 53 (1951), pp. 276—297.

53

10.

ll.

l2.
l3.
14.

15.

l6.

17.

18.

19.

20.

54

M. Brown, A proof of the Generalized Schoenflies Theorem,
Bull. Amer. Math. Soc., 66 (1960), pp. 74—76.
M. Brown, Locally flat embeddings of topological mani—

folds, Topology gf 3-Manifolds and Related Topics, M. K.

 

 

Fort Jr. editor, Prentice-Hall.

J. C. Cantrell, Almost locally flat embeddings of Sn—1
in 3“, Bull. Amer. Math. Soc., 69 (1963), pp. 716—718.
J. c. Cantrell, Non-flat embeddings of s“‘1 in s“,
Mich. Math. J., 10 (1963), pp. 359-362.

J. C. Cantrell, n-frames in Euclidean k—space, mimeo-
graphed notes.

J. C. Cantrell, Some relations between the annulus con—
jecture and the union of flat cells theorems, mimeo—
graphed notes.

J. C. Cantrell and C. H. Edwards, Jr., Almost locally
polyhedral curves in Euclidean n—space, Trans. of Amer.
Math. Soc., 107 (1963), pp. 13—19.

R. P. Coelho, On the groups of certain linkages,
Portugaliae Math., 6 (1947), pp. 57-65.

E. H. Connell, Stable homeomorphisms can be approximated
by piecewise linear ones, Bull. Amer. Math. Soc., 69
(1963), pp. 87—90.

P. H. Doyle and J. G. Hocking, A characterization of
Euclidean space, Mich. Math. J., 7 (1960), pp. 199—200.
P. H. Doyle and J. G. Hocking, Invertible spaces, Amer.

Math. Monthly, 68 (1961), pp. 959-965.

21.

22.

23.

24.

25.

26.

55

P. H. Doyle and J. G. Hocking, Dimensional invertibility,
Pacific: J. of Math., 12 (1962), pp. 1235-1240.

H. Gluck, Unknotting SE in S“, Bull. Amer. Math. Soc.,

69 (1963), pp- 91-94.

J. G. Hocking and G. S. Young, Topology, Reading, Mass-

achusetts, 1961.

(A. Kirkcr, Wild O-dimensional sets and the fundamental

group, Fund. Math., 45 (1958), pp. 228—236.

R. L. Moore, Foundations 9: Point Set Theory, Amer.

 

 

Math. Soc. Colloquium Publications, Vol. XIII.
E. C. Zeeman, The Poincare conjecture for n i 5,

Topology pf 3-manifolds and Related Topics, Prentice—

 

Hall, pp. 199—204.