THE ESSENTIAL P. L. CHARACTER
OF CERTAIN COMPACT SETS IN
TOPOLOGICAL MANIFOLDS

Thesis for the Degree of Ph. D.
MICHIGAN, STATE UNIVERSITY
DAVID EMANUEL GALEWSKI
1959

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"The essential BL.character of certain
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presented by

David E. Galewski

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ABSTRACT

THE ESSENTIAL P.L. CHARACTER OF CERTAIN COMPACT SETS
IN TOPOLOGICAL MANIFOLDS

by David Emanuel Galewski

Chapters I and II of this thesis are a study of the
neighborhoods of o-dimensional sets and arcs in topological
manifolds. We prove that a compact o-dimensional set in
the interior of an n-manifold has a connected neighborhood
which embeds in euclidean n-space and that an arc in the
interior of a topological n-manifold has a P.L. n-manifold
neighborhood. These theorems enable one to extend results
on the properties of embeddings in P.L. manifolds to
I topological n-manifolds.

A topological embedding f : Sn-lutsn is said to have
a P.L. homotopy approximation if given any neighborhood N
of f(Sn-1) there exists a P.L. embedding g : 811-148n
such that g(Sn-l) C'N and f is homotopic to g in N.
In chapter III of this thesis we prove that every embedding

f : Sn-1-+Sn with n # 4 has a P.L. homotopy approximation.

THE ESSENTIAL P.L. CHARACTER OF CERTAIN COMPACT SETS
IN TOPOLOGICAL MANIFOLDS

BY

David Emanuel Galewski

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1969

TO

The Lioness

ii

ACKNOWLEDGMENTS

I would like to thank Professor Patrick Doyle for
suggesting my research problem, for his unceasing patience
and encouragement, and for his sense of humor. I would
like to thank J. G. Hocking, K. W. Kwun, and C. L. Seebeck
for their stimulating discussions. And I would like to

thank Ralph Hautau for starting it all.

iii

TABLE OF CONTENTS

Page
INTRODUCTION 0 O O O O O O O O O O O O O O O O O O 1
Notation and Terminology . . . . . . . . . . 1
CHAPTER
I. NEIGHBORHOODS OF o-DIMENSIONAL COMPACT
SETS IN TOPOLOGICAL MANIFOLDS . . . . . . 3
II. P.L. NEIGHBORHOODS OF ARCS IN TOPOLOGICAL
MANIFOLDS O O O O O O O O O O O O O I O O O 13
III. A CODIMENSION ONE APPROXIMATION FOR
SPHERES O O O O O O O O I O O O O O O O O O 2 l
BIBLIWMPHY O O O O O O O O O O O O O O O 2 7

iv

INTRODUCTION

P. H. Doyle in [7] asks whether there exist small con-
nected neighborhoods of compact o-dimensional sets in topo-
logical n-manifolds which embed in Rn. In Chapter I of
this thesis we answer this question affirmatively and in
Chapter II we show that an arbitrary embedding of an arc in
a t0pological n-manifold has small P.L. neighborhoods.
These results enable one to extend known facts about embed-
dings of compact o-dimensional sets and arcs in P.L. mani-
folds to topological manifolds.

In Chapter III we change direction and consider the
problem of approximating a tOpological embedding of Sn-1
in Sn by one which is P.L. In particular we show that
if f : Sn.1 4 Sn is a topological embedding, N a neigh-

n-l)

borhood of f(S and n # 4, then there exists a P.L.

n—l

embedding g : S 4 Sn such that g(Sn-l) C'N and f

is homotopic to g in N.
Notation and Terminology

The following notation will be used for certain sets

and topological spaces.

Z = {n I n is a positive integer},
Rn = {x I x = (x1,...,xn) an n-tuple of real
numbers}
where Rn is given the topology determined by the euclidian

metric dn‘ The subsets

{x e Rn | dn(O,x) s 1},

Dn
En = [x e Rn I dn(O,x) < 1} and
Sn-l

= {x E Rn I dn(O,x) = l]

of Rn are given the subspace topology.

n n-l

A homeomorphic image of Du, E , or S is called a
closed n-cell, open n-cell, or In-l)-sphere respectively.

A topological n-manifold Mn is a separable metric

 

space in which each point has a neighborhood whose closure
is a closed n-cell. The interior of 'NE, denoted by fin,
is the set of all points which have open n-cell neighbor-
hoods: the boundary of N2, denoted by fin, is defined to
be Mn - fin. The boundary, interior and closure of a sub—
set A of a t0pological space X will be denoted by BXA,
IntXA and Cle respectively, and the subscript 'X' will
be suppressed if there is no possibility of ambiguity.

The basic terminology for t0pological manifold theory
such as locally flat, tame, bicollared, etc., may be found
in [3] and [10] while the terminology, notation, and basic
facts of piecewise linear, P.L., topology may be found in
[11] and [17]. To simplify notation we sometimes use the
same symbol for a polyhedron and its triangulation.

The end of a proof will be denoted by the symbol '0'.

CHAPTER I

NEIGHBORHOODS OF o-DIMENSIONAL COMPACT SETS
IN TOPOLOGICAL MANI FOLDS
P. H. Doyle has shown in [7] that every compact o—dimen-
sional subset of a topological n-manifold has a neighborhood
which embeds in Rn and asked whether there perhaps exist

small connected neighborhoods which embed in Rn. In this

chapter we answer this question affirmatively.

Lemma 1.1: Let on be an n-simplex in Rn and

 

f : on 4 Rn be an embedding such that f(on) is an n-cell
with bicollared boundary. Then there exists a homeomorphism

g : Rn 4 Rn such that gofI = 1 .
On On

Proof: By M. Brown [2] there exists a homeomorphism

n

n such that glof(on) is an n-simplex T . Let

glpRn 4R
92 tRn 4 Rn be a linear homeomorphism determined by any

bijection between the vertices of Tn and on; then

92°910f(on) = on and gzoglof(dn) = dn. If rx denotes

the ray from the barycenter of on meeting on at x then
the homeomorphism g3 : Rn 4 Rn determined by rx liEEES

r is such that g og og of(on) = on and
-l 3 2 l
(gzoglof) (x)

&n = lén.

homeomorphism determined by g4I n = (93°92°gl°fI n)-
o 0

Finally let g4 : Rn 4 Rn be the
l

93°92°91°fl

and

g4IRn °n = an o . Then the required homeomorphism is
—0‘ .-

g = g4°g3°gz°glo D

4
Definition 1.2: The pair (D,f) is an n-euclidean
domain in a topological n-manifold Mn if D chn and
f : D 4 Rn is an embedding such that f(D) is open and

connected in Rn.

Lemma 1.3: Let- Mn be an n-manifold such that

 

(i) (D,f) is an n-euclidean domain in Mn,

(ii) Bn c.Mn is an n-cell with bicollared boundary,

and (iii) C is a subset of D such that Can = Can = q.

n-l

Then there exists an (n-l) cell B c EnnD with bicollared

boundary in En and a homeomorphism g : RD 4 Rn such that
(iv) gofI n_1(Bn-l)=on-l is an (n-l)-simplex in Rn-lxo

B o _
and (v) v*on—1ng°f(C) = gofIQ) 6 On 1 for some v E Rn.

Egggf: Since En is bicollared there exists a neigh-
borhood‘ N of q in D and a homeomorphism h :
Rn‘1x(-1,2) 4 N such that h(0,0) = q, h(Rn-1x0) c 5n and
h(Rnx[0,2)) c Bn. Let on-1 be an (n-1)-simplex in Rn-lxo
with (0,0) 6 3n-l and v = (0,1) 6 Rn-1x(-l,2). Then
foh(V*Cn-l) is an n-cell with bicollared boundary in Rn
hence by Lemma 1.1 there exists a homeomorphism g : Rn 4 Rn

1 Choose 3n.1 = h(on-l). D

v*On-l on-1.

Lemma 1.4: Let Tn = v*on-l be an n-simplex and C

such that gofohI

 

a compact subset of Rn with n g 4 such that

(i) CflTn = v

(ii) Tn — v is contained in the unbounded component

of. Rn - C.

5

Then given 3 > 0 there exists a homeomorphism k : Rn 4 Rn

such that

(111) kITn 1T
and (iv) k(C) c N€(v) where N€(v) is an e-neighborhood

n

of v in Rn.

lggggg: Let ;n = 343” be an n-simplex in Rn con-
centric to Tn such that En-l is parallel to on-1 and
CUTn crfin. Also let xn = :n - Tn be a P.1u manifold with
triangulation T such that IT(1)I contains

(i) a P.L. 1-ball B = 3n*5n (where 'V' denotes

FJH

barycenter),

(ii) a P.L. l-ball a: such that Bins: = agnéi — hi
and sine = ¢ (since Tn - v and In are in the unbounded
component of Rn - C),
and (iii) a P.L. 1-ball a; = (6”‘2*vr**(cn 2*vr" such
that B§OB% = ¢ where Gn-Z is an (n-l)-face of dn-l and

En-Z is the correSponding face of En-l. Next we choose a

fine subdivision T' of T such that N(B3 ,T' )n32 = ¢
1

where N(B§,T') is a regular neighborhood of B3 in T'.

Then Yn = Cl(xn - N(BB,T')) is a P.L. neball, (Yn,Bi) and

(Yn,B%) are (3,1)-ba11 pairs with n g 4 and hence by

E. C. Zeeman [17] there exists a P.L. homeomorphism hi:

i = . Extend h
3:“ lit"

to a P.L. homeomorphism h : Rn 4 Rn by the identity and

Yn 4 YD such that ‘h(B%) = B and hI.

l .
note h(C)nB = ¢ and hIn Now choose a fine
1 Tnu+n= lwnu+n1

subdivision T“ of T? such that N(h(B% ), T")nh(C) = ¢.

6

Then Zn = CIIXn-N(h(B%),T")) is a P.L. n-ball with

h(C) cZn such that h(C)nZn = CflZn = v. Hence there exists

a homeomorphism 6': Zn 4 Zn such that ghh(C) c N€(Tn) and

EI'n = l-n' Finally extend 3' to g : Rn 4 Rn by the iden-
Z Z
tity and hence obtain the required homeomorphism k = g°h :

Rn 4 Rn. D

Definition 1.5: An arc A = pq, is the homeomorphic
l

 

image of D where p and q correspond to the image of

-1 and 1 respectively.

Theorem 1.6: Let Mn be a connected n-manifold such

 

that

O
(i) C and C are connected compact subsets of Mn,

1 2
(ii) (Dl,f1) and (D2,f2) are disjoint subsets of Mn,
(iii) C. c D. for i = 1,2,

1 i

and (iv) U is a connected open neighborhood of C1UC2

. °n

in M .

Then there exists an n-euclidean domain (D,f) in Mn such

that CUCch CU.

1

2529:: Case I For n = l proof is trivial.

Case II For n a 4. Since U is open and con-
nected there exists an n-cell Bn with bicollared boundary
and arcs Ai = piqi for i = 1,2 contained in U such that
annci = ¢,.AiflCi = Pi' Aian = qi and Ai czDi for i = 1,2.
Also by Lemma 1.3 there exists an (n-l) cell Bn-1 ‘with bi—

collared boundary in EnnD, and a homeomorphism g : Rn 4 Rn

7

) = On-l an (n+1)-simp1ex in Rn

-1 _ °n-l n
and v*on figOf1(C1UA1) — gof1(q1) 6 0 for some v E R .

Since 311—1 is bicollared in Bn, C1(Bn-Bn—1)

_1|

n-l
such that g°f1I n_1(B

is an n-cell

by [2] hence we can extend (gohl) to a homeomor-

Gn-l

phism k : v*on-l 4 En such that k(v) = q We now con-

20
struct a homeomorphism h : Rn 4 Rn such that
-1 _ _
hof2(czm2)n[(v*o“ )Ugof2(C1UA1)] — hof2(q2) — v.
For the construction of h we choose an n-simplex Tn =

v*nn-1 C (V*On-1 - on-l) such that (v*on-l,nn-1) is a

n-1 n-1

(n,n-1)—ba11 pair, n is parallel to O and k(Tn) c

D2. We note since Bn is bicollared k can be extended to
a collar of v*on-l, it then follows that fzok(Tn) is an
n-cell with bicollared boundary. Hence bwaemma 1.1 there

exists a homeomorphism h1 : Rn 4 Rn such that h1°f2°kI n
T

= 1 n' W.1.o.g. we may assume fZIAZ) is in the unbounded

T
component of f2(C2) by simply modifying f2 if necessary.
Hence Tn - v is in the unbounded component of Rn -

hlof2(C2UA2). Then by Lemma 1.4 with hlof2(C2UA2) replac1ng

C and setting 3 = min{dist(v,gof1(A1UC2)), dist(v,v*on-l-

Tn)} there exists a homeomorphism h2 : Rn 4 Rn such that

thTn = 1Tn and hzohlof2(C2UA2) c N€(v). Hence

hzohlof.2(c2uh2)rivircn"1 = v and we set h = hzohl.

Let U be a neighborhood of gofl(C1UA1) such that

1
C1(U1) c gof1(D10U), C1(Ul)nhof2(A2UC2) = ¢ and

C1(U1)nV*On-l c v*on-1 - Tn. Also let U be a neighbor-

2
hood of hof2(C2UA2) such that Cl(U2) c hof2(D20U),

n-l = n

Cl(U2)nC1(U1) = s and c1(U2)nv*cn'1 c v*n T .

8
We define a homeomorphism
f' : [(v*on-1)U(C1(U1) - (v*on'1)

(v*on_l)°)] 4 Mn

°) U (C1(U2) -

by f' = k,
‘V*On-1
* -1
f'I _ = (get ) I _ I,
C1(U1)-(v*on 1)° 1 C1(Ul)-(v*on 1)°
-1
and f'I = (hof )
c1(U2)-(v*on‘1)° 2 c1(U2)-(v*on‘1)°

l . . ' __ -1
Note f 13 well-defined Since k‘on-l - (9°fl) I0n_1

and k‘1(h.f2)‘1I = k‘lof'loh'l
Tn

-1 -1 -l‘
2 l

0 f2 ohl

-l
othn=k n=
T

lno
T

If we let W be the interior of the component of
C1(Ul)UC1(U2)U(v*dn-1) containing gofl(C1UAl)Uh0f2(C2UA2).

then the desired n-euclidean domain (D,f) is given by

D = fI(V) and f = (fIIV)-1.

Case III For n = 2 or 3. Let g : Dn'lxI 4 U be an

embedding such that g(Dn'1xo) chl - c1, g(Dn‘1x1) cD2 - c2,
and g(Dn_1xI) is an n-cell with bicollared boundary. It
follows that there exists an e > 0 such that

gIDn-lx[0,e)) C'D - C1 and g(Dn-1x[l-e,l)) c D2 - C2 are

1

n-cells with bicollared boundary. Hence homeomorphisms

: R“ 4 Rn exist such that hlofiog(Dn-1x0) = 02-1.

x1) = 03-1, where 02-1 and 02-1 are (n-l)-

hi'hz
-l
h20fzog(Dn

simplexes in R“, hlof1(Dl) c H; and h20f2(D2) c H;
(H; = {x E Rn I x = (x1,...,xn) and xn > 0]), and
hloflog(Dn-1x0)Uh2°f2°g(Dn-1x1) = og‘luog'l is contained in

the unbounded component of Rn - [h10f1(Cl)Uh20f2(C2) ]. Let

9

L : Dn-lxI 4 Rn be an embedding, obtained by "swelling" a

polyhedral are between 01 and 02, such that L(Dn-1x1)

. n-1 n-1
1s a polyhedron, LI _ = o , LI _ = o and
Dn 1x0 1 Dn 1X1 2

xI)fl[h1ofl(C1)Uh20f2(C2)] = ¢. By modifying hl if

necessary we may assume L-lohiOfiogIfin_ 1 : Dn-1x(i-1)
x(i- -l)

4 Dn-lx(i-l) for i = 1,2 both preserve or reverse orien-

L(Dn-l

tation. Then by Fisher [8] there exists an isotopy H" :
'n- 1 ‘n- -1

D XI 4 D xI such that
(I) H"I fin" = L-lohIOfiog‘fin-l for i = 1,2.
x(i- -1) lx(i—l)
. n-l - n-l . . .
Let H : (D x1) 4 (D xI) be a homeomorphism defined
’ -1
by H' I. = H" and H'I = L oh.of.og for
Dn lxI Dn-1x(i-1) 1 1
i = 1,2. H' is well defined by (I). Since Dn-lxI is an
n-cell and H' is defined on (Dn-lxI)' there exists a
homeomorphism H : Dn-lxI 4 Dn-lxI which is an extension of

H'.
Let Ui be a connected neighborhood of hiOfi(Ci) in
hiofi(UnD) such that ¢ ¢ Cl(U.)nL(Dn-1x1) c ;(Bn‘1x1) for

i = 1,2 and note Uan2 C'H+OH- = ¢. We define an embedding

f? : [f(Dn'1x1)U(U1-(L(Dnlx1)°)U(U2-k(Dn-lxl)°)] 4 U

-1 -l I
by f'l _ = 90H at I _ and f'I _
1(n“ 1x1) MDn 1x1) Ui-(£(Dn 1x1))°

= (biofi)‘1' for i = 1,2. f' is well defined since

-1 -1 _ -1o -1 -1 _
9°H °‘ I£(Dn-lx(i-1)) ‘ 9°I‘ hi°fi°9) °‘ IL(Dn 1x
(i-1))
= gog-lofglohgloLoL-II n_1
LID x(i-1))
-1 .
= (h,°f.) I for 1 = 1,2.
1 1 £(D D“ 1x<i-1>)

lO

1

If V is the interior of the component of U1UU2UL(Dn- xI)

containing f(Dn-lxI) then the desired n-euclidean domain

(D,f) is given by D = f'(V) and f = (f')‘1If.(v). D

Corollary 1.7: Let {(Di,fi)]nil=1 be a pairwise dis-
joint collection of n-euclidean domains in Mn, Ci c Di con-
nected and compact for i = l,2,...,m, and U a connected
Open subset of Mn containing C = .3 Ci“ Then there exists
an n-euclidean domain (D,f) such chE C c D c U.

Definition 1.8: A topological Space E..l§ o-dimensional
if for every x E X there exists arbitrarily small neigh-

borhoods of x 'with empty boundary.

Theorem 1.9: Let U be an open connected neighborhood

 

of a o-dimensional compact set C in the interior of a
t0pologica1 n-manifold Mn. Then there exists an n-euclidean

domain (D,f) such that C c D c U.

Proof: For each x 6 C there exists an open n-cell neigh-
borhood Ox of Mn and an open and closed ‘Gc in (2 such
that x E Vx c 0x c U. Since C is compact there exists a cov-

j-l
ering {V ]IP_ of C. We define U. = V -~U V for j = 1,2,
x. 1—1 3 x. ._ .
1 3 1—1 1
m

j=1 is a pairwise disjoint o-dimensional

...,m and note [Uj]
compact covering of (2 and hence by theorem 3 of Osborne [13]
there exists a collection of pairwise disjoint arcs {A1]T%l

such that Ui C.Ai c Ox flU. Since Mn is normal there exists
i

a pairwise disjoint collection of Open subsets of Mn,

m

I

{W i=1

. such that A. cW.flU CW. for i= l,2,...,m;
1 1 1 1

11

hence {winox nU}?_1 is a pairwise disjoint collection of
i _

Open subsets of U such that winox nU embeds in Rn and
i
U. c:A. ch.nO nU for i = l,...,m. Letting D. denote
1 1 1 xi 1
the component of Ai in Winox nU for i = l,2,...,m it
i

follows from Corollary 1.7 there exists an n-euclidean do-

main (D,f) such that C c D C U. CI

Corollary 1.10: Any two disjoint compact O-dimensional
subsets in the interior of a connected topological n-manifold

Mn lie in disjoint homeomorphic n-euclidean,domains.

Proof: Let C C be the o-dimensional subsets and

1' 2

U1 and U2 disjoint open connected neighborhoods of Cl

0

and C2 respectively in MI. Then by Theorem 1.9 there
exists disjoint n-euclidean domains (Dl,f1) and (D2,f2)
such that Ci c Di c Ui for i = 1,2. By D. Tondra [16]
there exists an Open connected subset U of Rn such that

O O
fl(Dl) =igéxi and f2(D2) = IgfiYi where the un1ons are

monotone, and Xi' Yi and II are homeomorphic. Since f(Ci)
for i = 1,2 are compact there exists an N E Z such that

f(Cl) cIX and f(CZ) C'Y . Hence the required n-euclidean

N N
. . -1
domains are (D§,f§) for 1 = 1,2 where Di = f1 (XN),

and f*=f .CI

-1
* =
D f2 (YN) ' f* 2 2I

=f|
2 1 1 f-1(X

-l

N) f2 (YN)
Corollary 1.11: Any o-dimensional compact subset in

the interior of a connected tOpological n-manifold Mn lies

on the boundary of a k-cell in in for k = l,2,...,n.

Proof: Theorem 1.9 and Osborne [13]. D

12
Definition 1.12: A o-dimensional subset Of a topolo-
gical n-manifold is called locally flat if it lies on a

locally flat closed l-cell.

Corollary 1.13: Any compact o-dimensional set C in
the interior of a connected tOpological n-manifold Mn lies
in an Open n-cell On except for perhaps a locally flat

o-dimensional subset C' of C.

Prggf: Let (D,f) be an n-euclidean domain given by
Theorem 1.9 and Nn a connected P.L. compact n-manifold
neighborhood of f(C) in f(D). Then by M. Brown [4] Nn =
PnUR where Pn is an open n-cell, R is an (n-1)-complex
in some triangulation of Nn, and R is disjoint from Pn.

But then f(C)nR = Un-l (CflUn-l) is the finite union of

a 6R
tame o-dimensional compact sets and hence by Osborne [13] is
tame. Therefore there exists a locally flat arc containing
0
f(C)nR in Nn. But then we are finished by setting 0n =

f'1(o“) and c' = f-l(f(C)nR). [:1

Corollary 1.14: Any compact o-dimensional set C in

 

a topological n-manifold Mn has an open neighborhood which

embeds in Rn.

‘ggggf: Let Nn = MnUanx[O,1) where f : Mn 4.Mnx0
is defined by f(x) = xxO for all x 6 Mn. Then Nn is a
tOpological n-manifold without boundary, hence by Theorem
ulég there exists an n-euclidean domain (D,f) such that
C c:D c Nn. But then D - (Nn—Mn) is an Open subset of Mn

which contains C and embeds in Rn.I3

CHAPTER II

P.L. NEIGHBORHOODS OF ARCS IN TOPOLOGICAL MANIFOLDS

This chapter is concerned with the construction of P.L.
manifold neighborhoods of arcs in topological manifolds. We

begin with the following definition.

Definition 2.1: Mn is a P.L. n-manifold if there

 

exists a homeomorphism t : IKI 4 Mn where K is a finite
simplicial complex in which the star of each vertex is P.L.

homeomorphic to an n-simplex. In this case we may write

n

 

M = (K,t) and call IKI a combinatorial n-manifold.
The next two lemmas are well known facts in P.L.
topology.

Lemma 2.2: If two combinatorial n-manifolds M2, M2

 

are attached by a P.L. homeomorphism h along P.L. (n-l)-mani-
folds in their boundaries then the adjunction space MIUhMg

is a combinatorial n-manifold.

Lemma 2.3: Let" U be an open neighborhood of a com-

 

pact set C in a combinatorial n-manifold M“. Then there
exists a combinatorial n-submanifold Kn c U such that

(i) C C’Kn

(ii) Kth is a combinatorial (n-l)-manifold containing

COM in its interior.

lemma 2.4: Let A be an arc in the interior of a

 

13

14
topological n-manifold Mn. Then .A c (D1,f1)U(D2,f2)
where (Di'fi) for i = 1,2 are n-euclidean domains

in Mn.

Proof: Let C be a countable dense subset of A, then
C lies in an open n-cell On by [6]. But A - On is a
compact o-dimensional set and hence lies in an n-euclidean

domain by Theorem 1.9. D

We now introduce special notation to be used for the
rest of this chapter. Let A = pq be an arc in the interior
of an n-manifold Mn. If B“ is a c1osed n-ce11 neighbor-

hood of x €.A then denote by p'n, the "first" point in A
B

that meets Bn and: denote by q'n, the last point of A
B
that meets Bn and note that A = pp'nxq'nq. Also let
B B
° 0 0
Sin = p'nx - Bn and an = xq'n - Bn.
B B B

Lemma 2.5: Let

(1) Mn = D1UD2 be an n-manifold without boundary such
that D1 and D2

(ii) A = pq an arc in Mn,

. n
are open in M ,

(iii) x 6.AnDlnD2,

(iv) C a compact set in Mn such that Cn{x] = ¢.
Then there exists an e > 0 so that for any closed n—cell
neighborhood Bn of x such that diam (Bn) < e we have

n i f
(V) B USBnUSBn clenD2

and (vi) CfiIBnUSinUan) = ¢.
B B

15
nggf: Since A is locally connected there exists a
connected neighborhood U of x in A such that U c
Afl[(DlnD2)-C] and we note that U is a subarc of A. Hence
there exists an Open set V in [(DlnD2)-C] such that
ADV = U. If we let e = dist (x'Mn—V) then for any closed
n-cell neighborhood Bn of x such that diameter (Bn) < e

'we have Bn C'V, BnnA c U, p'nxq'n c [(DlfiD2)-C],
B B

BnUSl USf c [(D 0D )-C], hence conditions (v) and (vi).[3
Bn Bn 1 2

Theorem 2.6: Let A be an arc in the interior of a
topological n-manifold Mn. Then there exists a P.L. n-mani-

fold 1»:n such that A c 131“ c it“.

Proof: By Lemma 2.4 we may assume Mn = D1UD2 where

. . . _ 1+1
(Di'fi) are n-eucl1dean doma1ns for 1 — 1.2. Let {xi}i=0

be an increasing ordered set of points in A (with respect

to a parameterization of A) from x = p to x

0 1+1 = q

such that the subarcs xixi+1 c Dimod2+l for 1 = O,1,....L

and xi 6 DlnDZ i = l,2,...,t. We may also assume q = x“1

E DlnD2 by extending A if necessary.
We will construct inductively a sequence of P.L. mani-

. . n _ . =
fold pa1rs 1n M , (N*'LI)"(Kl'Hl't1)'""(NI+1'LI+1)

(Kb+1’HL+17tL+l) such that for i = 1,...,L+l

(1) Ki is a combinatorial n-manifold,

(2) Hi is a combinatorial n-cell in Ki'
(3) L: is a bicollared n-cell in DlnD2 such that

O
x. L?
1 6 1'

(4) ppfiusi. c N
1 1

l6
(5) (SIE Uq' q)nN* = (sf uq' quv = (sf Uq' q)nf.*
L; L; i L; L; i L: L: i'

and (6) {xi+1,...,x£+l] c qizq.

Step I Construction of (Ni,Li) = (K1,H1:tl).
Choose a combinatorial n-manifold IJEI c fl(Dl) con-
taining fl(xoxl) in its interior with mesh so fine that
. n . .
(1) f1(x1) 6 01 where a? E J? 13 an n-S1mplex

and (ii) diam (fI1(o?)) < 31 where is given by Lemma

61
2.5 with Mn = D1UD2,
A = A,
x = x1,
and C = x Oszq.

Hence (iii) f11(dn)USif_1 n US: _1 n c (DlnD2)-(xOUx2q).

f1 (01) f1 (01)
Next we Obtain a combinatorial n-manifold IREI c IJRI by

Lemma 2.3 with

M" = I914?“
C = f1(pp'f_1(on) M:1(:1))'
and U = (IJnI- on)nf1[D1- (S:f11(on)m Wfi(o?)q)].
Hence (iv) f l(pp' f_1(dn)w wi f(on)) clifll.

(v) IKnI c (IJnI- -o:)n£l[nl- (s: q)].

f-l n
f1 l(0"1‘)LJq f1 (01)

and (vi) IRPInOn is a combinatorial (n-l) manifold with
f 1(pp _ _ mon in its interior.

£11m“ Tc'llto")
By Lemma 2.2 with h = l and (vi) we have

time?

 

 

17

 

(vii) |K1“Q2£4-‘E?‘Uo$ is a combinatorial n-manifold.
Now we set
_ -1
t1 ’ f1 ‘0?
H = on
(viii) l 1
Ni = 11(‘K1‘)
Li = 11(02)

Condition (1) is satisfied by (vii); condition (2) by (viii)
and (vii); condition (3) is satisfied since £11 is defined
on a neighborhood of 0?, (iii), and (i); condition (4) is
satisfied by (iv) and (vi): condition (5) is satisfied by

(v) and (vii); and condition (6) is satisfied by (iii) and

(V).

§E£E_ll Suppose (Ni:Li) = (K1,H1:tl),...,(N;,L;) =
(Kr,Hr:tr) for r 5 I have been chosen to satisfy condi-
tions (1) through (6). Since L; is a bicollared n-cell in

DlnD2 there exists a homeomorphism hr : Rn » Rn such that

n. ' .—
hr°frmod2+l°tr . Hr a R is a P.L. embedding. Let gr —
hrofrmod2+1 and choose a combinatorial n-manifold |J§+1| c
gr1Drmod2+11 with mesh so fine that

' * H"!
(l) gr1xrxr+lULr) c r+l ’
.. 0 n . .
(11) gr(xr+1) 6 02+1 where 0r+l E Jg+l 15 an n-s1m-
plex,

and (iii) diam[g;1(o§+1)] < er+l where er+1 is given by

Lemma 2.5 with

n—
M - DlUDZ'

A = A,

18

x = xr+1'

= *
and C Narr+2q.

Note N;n[x:+1] = o by condition (5) and (6). Hence
. f-
(1v);1(0n )USi
+11£1102+11f162+11C

(DlnD2)-(N;er+2q)o
We now obtain a combinatorial n-manifold 1 @111 C 1 fi+11

by Lemma 2.3 with

M“ = IJE+1I- <3”+1Ug(L;)>.
C=gr (Sf Ln P _ 115:1):
L* 1*w1<0?+1>r1<°?+1>
and U = 11fi+11'1ymlug (L*))]ng r1Drmod2+1‘ [C1(N*-L*)U

f

s ' q]].
-1 -1

gr 102+1) gr (02+1)

f . 1

Hence (v) 9

US -1(Gn 1 C 1 E+11'
(Vi) 1 r+11 c 11 r+l1-3n1r+rlug (Lr11 n

-[c1(N*-L*)usf Uq' Q]].
1 g r1or+11 g;l(dn

r+1

gr1Drmod2+

Kn+11nan r+1
with g r1SL*UqL*pg

(vii) 1 is a combinatorial (n-1)-manifold
1
US -
1 0:11) 9_1(an
1ngr (L*)' =1 E+1|ngr (N*) is a combinatorial
i

us
9;l(on+1)

in its interior,

and (viii) 1f r+l

ngr(L;))'

in its interior.

Then by Lemma 2. 2 with h = 11? 100n+1 we have
r+1

def.

 

 

(ix) 1 M111 1 r+l1uf r+1 is a combinatorial n-mani-
fold and by Lemma 2. 2 we_ have
def. -—- .
(x) 1Kr+11— 1E+11U119rtr1_11— is a

I “+1109 (L*r )

l9
combinatorial n-manifold.

Finally we set

_ -l
tr+1 ‘ trugr

_ n
Hr+l — Or+1

N:41 = tr+1lKr+l1
L{5+1 = tr+1(o:+1).

Condition (1) is satisfied by (x): condition (2) is satis-

fied by (xi), (x) and (ix): condition (3) is satisfied since

-1

r is defined on a neighborhood of 0211, (iv) and (ii):

g
condition (4) is satisfied by (v), (vii) and (viii): condi-
tion (5) is satisfied by (vi), (ix) and (x): and condition
(6) is satisfied by (iv) and (vi).

The induction is now complete and to finish the proof

'we note A c:ppL* US:* ULi c:N* by condition (4) and

1+1 1+1 “'1
(2) with i = 3+1. [3

Corollary 2.7: Let A be an arc in the interior of a
topological n-manifold Mn. Then A lies in an open n-cell On

except for perhaps a locally flat compact o-dimensional set C.

Egggfz By Theorem 2.6 we may assume that Mn is a com-
binatorial n-manifold with a triangulation T. By M. Brown
[4] 1T1 = RUOn where On is an Open n-cell and R c T(n-1)
is disjoint from On. we may also assume by adjusting A
slightly if necessary that AnR is a compact o-dimensional

set or empty. Hence A lies in On except for perhaps a

compact o-dimensional set C =.AnR.[3

20
Corollary 2.8: An arc A in the interior of a tOpo-

logical n-manifold lies in the union of two open n-cells.

Proof: This follows immediately from Corollary 2.7
and the fact that a locally flat compact o-dimensional set

lies in an open n-cell. D

CHAPTER III

A CODIMENSION ONE APPROXIMATION FOR SPHERES

In 1957 Bing [1] proved that a Z-Sphere embedded in S3

can be pointwise approximated by a flat 2-sphere. A genera-
lization of Bing's theorem to higher dimensions seems diffi-

cult. But in this chapter we prove the following: Let

n-l 1

f : s 4 sn be an embedding, N a neighborhood of f(sn' )

Sn-l n

and n #-4, then there exists a P.L. embedding g : 4 S

such that g(Sn-1) c:N and [f] ~ [g] E nn_1(N). We begin
by first proving the following theorem which approximates a

sphere separation of two compact sets in Sn by a P.L. one.

n-l

Theorem 3.1. Let f S 4 Sn be a topological em-

n-l)

bedding with n > 4 such that f(S separates two com-

pact subsets A and B. Then there exists a P.L. embedding

g : Sn"1 4 Sn such that g(Sn-l) separates A and B.

Proof: Let D denote the closure of a complementary

n-l)

domain of f(S containing A. Since D is a compact

absolute retract there exists compact P.L. n-manifolds with
connected boundaries N1, N2 and N3 which are neighbor-

hoods of D such that Ni c I3i+l (i = 1,2,3), any polyhedron

of dimension é n-3 in N2 is contained in a P.L. n-cell in

N3. N1 C'N2 and N2 C’N3 are null homotopic, and for i =

1,2,3 Ni c Sn-B (the proof of this statement may be found

in section 2 of M. L. Curtis and D. R. McMillan's paper [5]).

n-1)

Since f(S is a l-connected absolute neighborhood retract

21

22

n-1) n-1)

there exists a neighborhood V of f(S such that f(S

c V induces the trivial map on 1'11 and V c N -A. Also let

1
N be a P.L. n-manifold neighborhood with connected boundary

in DUV. Hence by Lemma 3.2 of [9] we can do surgery on N in
V to obtain a closed P.L. (n-l) manifold Id in \I which is
l-connected and we let Dd denote the P.L. n-manifold which

' O
M bounds in N and note that .A c.M.

1
We now show the pair x_= (N2,§24M) is 2-connected and

thus obtain a homeomorphism h : N 4 N such that h(M)nN2(2)=

2 2

o and h‘fi = lfi by Stalling's engulfing theorem [14].
2 2

Let 'k : (Al,51) 4 1 be a P.L. representative of a

class in n1(x). By the van Kampen theorem M is simply
connected and hence it easily follows that k represents
the trivial class in n1(x).

Now let L : (A2,A2) 4 A be a representative of a class
in H2(k). W.1.o.g. we may assume L is a P.L. homeomor-
phism and that L = n-1(L(A2)nM) is a compact P.L. 2-mani-
fold in the interior of A2 which is contained in some sub-
division of A2. Choose a regular neighborhood E of L in

.32 such that L(E-L) is contained in an "outer" collar C

1 and 1(52)nC = ¢. Since L(E) c MUC

0
of M where CUM c N
and MUC is l-connected we can extend L to L' :

(AZXO U E(1)xI) 4 N such that L'(b,t) = L(b) for all b e

2
' . (1) . (1)

B and t e I, L (E xI) c:MUC, and z (E x1) c.C. By
the van Kampen theorem C is l-connected; hence we can ex-
tend L' to L" : (AZXO U E(1)XI U Exl) 4 N2 such that

£"(EX1) =:E. Since L"(Ex0 U E(l)xI U Exl) c MUC c N1 and

23

NlCN2

(AZXO U EXI) 4 N2. Finally since Lm(e,t) = L(e) for all

is null homotopic we can extend L" to L” :

e E B and t E I we can extend L” to L(lv) 2

: A xI 4 N2
. . (iv) _ 2
by defining L (x,t) — L(x) for all x 6 A -E and t E
I. Therefore L(lv) is a homotopy re1 52 between L and
L(lv)‘ 2 where L(lV)‘ 2 (Ale) c NZ-M, hence L repre-
A x1 (A x1)

sents the trivial class in U2(l) and l is 2-connected.

Now extend h : N2 4 N2 to a P.L. homeomorphism h' :

N3 4 N3

sisting of the 2-ske1eton of N2 together with all the sim-

by the identity and let F denote the complex con—

plexes in N -N2. Then we have h'(M)nF = ¢. If G denotes

3
the complementary (n-3)-skeleton to F in the first derived

subdivision of N3 then there exists a P.L. n-ce11 On in

N3 containing G since dimension G a n-3 and G C’NZ. We
note that 6“ is a neighborhood of G and that h'(M)nF =
¢ hence by Theorem 8.1 of [15] there exists a P.L. homeomor-

phism h" : N 4 N such that h" is fixed on F and G

3 3
and h'(M) c h"(0n). But then A c'M = (h')-1(h'M) c
(h')-1oh"(0n) C'Sn-B and (h')‘1.h"(6“) is a P.L. (n-l)-

sphere which separates A and B.[3

We now prove two lemmas which are used in the main

theorem.

n-l

Lemma 3.2: Let f : S 4 Sn-{p,q} be an embedding

such that f(Sn-l)

Un_1(Sn-{p.q}).

separates p and q, then [f] generates

24

n-l) separates p and q there

Proof: Since f(S
exists a strong deformation retract of Sn-{p,q} onto
f(Sn-l). If r : Sn-{p,q} 4 f(Sn-l) denotes the retraction

n-l) c

in the last stage of the deformation and i : f(S
n . . - . n
S -{p,q} 18 the inclusion map then r* : n*(s -{p,q]) 4

H*(f(Sn_l)) is an isomorphism and 1* : n*(f(Sn-1)) 4

H*(Sn-{p,q}) is the inverse of r*. Since f' : Sn-1 4
Sn—{p,q], defined by f'(x) = f(x) for all x e sn‘l, is a
representative of a generator of nn_1(f'Sn_1)) we have

i*[f'] = [iof'] = [f] E Hn_1(Sn-{p,q]) is a generator of
nn_1(s“-{p.q}). [:1

Lemma 3.3: Let f : Sn"1 4 Sn-{p,q} be an embedding .

such that f(Sn-l)

separates p and q, and N a neighbor-
_‘|

hood of f(Sn ‘) in Sn. Then there exists a neighborhood

w of f(sn'l) in N such that if c is any compact set

in W there is a map h : Sn-{p,q] 4 N such that
h - = -
|f(sn 1)uc 1f(sn 1)UC.

Proof Since N is an absolute neighborhood retract
there exists an e > 0 such that if f,g : X 4 N (X arbi-

trary) are such that dist (f(x),g(x)) < e for all x E X

n-l) be a retrac-

then f..g in N. Let r : Sn-{p,q] 4 f(S
tion, V = {x E N ‘ dist (x,r(x)) < e] be an Open neighbor-

hood of f(Sn-l) in N, M a connected compact P.L. n-mani-
fold neighborhood of f(Sn-l) in v and ‘w = i. If we let

f = 1M : M 4 M and g = r : M 4 M ‘we have dist (1M(x),

r(x)) < e for all x E M hence lM ~ r in N, say

25

H

MxI 4 N such that H‘Mxo = l and H‘Mxl = r. Now let

M

M 4 I be a Urysohn function such that k(Cuf(Sn-1))

k

= O
and k(M) = 1, and L : M 4 Graph k chxI a map defined by

L(x) = (x,k(x)) for all x 6 M. Finally define h : Sn-{p,q]

4 N by h(x) ={r(x) if x E Cl(Sn-(MU{p’q})). We first
H01,(X) if X 6 M

note h is continuous for if x E M, HoL(x) = H(x,k(X)) =
H(x.1) = r(x): h<sn-{p.q}> c N since r(s“-<Mu{p.q}>> =
f(Sn-l) c N’ and HoL(M) c H(MxI) C'N: and if x E f(Sn-1)UC
then h(x) = HoL(x) = H(x,k(x)) = H(x,0) = 1(x). Therefore

h is the required map.[3

We now prove the main theorem.

n-l n

Theorem 3.4: Let f : S 4 S be a topological em-

bedding of the (n-l)-sphere in the n-sphere with n # 4 and

n-1)

N a neighborhood of f(S in Sn. Then there exists a

P.L. embedding g : Sn"1 4 Sn such that g(Sn-1) c-N and

[f] = [9] 6 nn_l(N).

Proof: Let C and D be the complementary domains of

f(Sn-l) in Sn, A = cn(sn-N) and B = Dn(Sn-N). Note if .A

n-l

(or B) is empty then any P.L. embedding of S into Cl(C)

(or Cl(D)) suffices since Cl(C) and Cl(D) are contractible.

Now apply Theorem 3.1 to A and B to obtain a P.L. embed-

n-l

ding g : S 4 Sn such that g(Sn-1)

separates A and

n-1 4 Sn (note

B, and f and g are pointed maps of S
that Theorem 3.1 could have been stated for pointed maps and

Spaces by slightly adjusting the image of g in Sn-(AUB)).

26
Now by Lemma 3.2 we have [f] = t[g] E Hn_l(Sn-{p,q}) and

‘we may assume, by preceeding g with an orientation reversing

n-l

homeomorphism of S if necessary, that [f] = [g]. If

Sn-l

H : x1 4 Sn-{p,q} is the homotoPy between f and g in

Sn-{p,q} then letting C = g(Sn-1

) in Lemma 3.3 we obtain
an h : Sn-l-[p,q] 4 N such that hoH : (Sn-{qu})x1 4
(Sn-{p,q]) is a homotopy between f and g in N. Hence

g is the required P.L. embedding. E]

BIBLIOGRAPHY

10.

ll.

12.

l3.

l4.

BIBLIOGRAPHY

H. Bing, "Approximating surfaces with polyhedral
ones", Ann. of Math., 68 (1958), 17-37.

Brown, "A proof of the generalized Schoenflies
theorem", Bull. Amer. Math. Soc., 68 (1960), 74-76.

Brown, "Locally flat embeddings of tOpological
manifolds", TOpology_ of 3-Manifolds Lnd Related
Topics, Prentice-Ha11,Eng1ewood Cliffs, N. J., 1962,
83- 91.

Brown, "A mapping theorem for untriangulated mani-
folds", Topology of 3-Manifolds Lnd Related TOpics,
Prentice-Hall, Englewood Cliffs, N. J., 1962, 92-94.

L. Curtis and D. R. McMillan, "Cellularity of sets
in products", Mich. Math. J., 9 (1962), 299-302.

H. Doyle and J. G. Hocking, "A decomposition theorem
for n-dimensional manifolds", Proc. Amer. Math. Soc.
13 (1962), 469-471.

H. Doyle, Summer Seminar pp_T0polpgy (mimeographed
notes), Michigan State University, 1967.

Fisher, "On the group of all homeomorphisms of a
manifold", Trans. Amer. Math. Soc., 97 (1960), 193—212.

F. Duvall, Jr., "Neighborhoods of l-connected.ANR's
in high dimensional piecewise linear manifolds",
Doctoral thesis, University of Georgia, 1965.

K. Fort, Jr., Topolpgy of 3-Manifolds Lnd Related
Topics, Prentice-Ha11,Englewood Cliffs, N. J., 1962.

F. P. Hudson, Piecewise Linear Topology, Benjamin,
New York, N.Y., 1969.

R. McMillan, Jr., "A criterion for cellularity in a
manifold", Ann. of Math., 79 (1964), 327-337.

P. Osborne, "Embedding Cantor Sets in a manifold",
Mich. Math. J., 13 (1966), 57-63.

R. Stallings, "The piecewise-linear structure of

euclidean space", Proc. Camb. Phil. Soc., 58 (1962),
481-488.

27

28

15. J. R. Stallings, "0n topologically unknotted spheres",
Ann. of Math., 77 (1963), 490-503.

16. R. J. Tondra, "The set of generating domains for certain
manifolds", Doctoral thesis, Michigan State University,
1968.

17. E. C. Zeeman, Seminar pp_Combinatorial Topology, (mimeo-
graphed notes), Inst. Hautes Etudes Sci., Paris, 1963.