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£33? VALENCE 0F TUBULAR
NEJGHBGRHGOES

Thesis for the Degree of Ph. D.
I‘MCHEGAN STATE UféEVERSiTY
JOAN ELIZABETH QUINN
1970

THFC‘

0-169

\\\\\\\ \\\\\\\\\\\\\\\

This is to certify that the

thesis entitled
EQUIVALENCE OF TUBULAR NEIGHBORHOODS

presented by
Joan Elizabeth Quinn

has been accepted towards fulfillment
of the requirements for

Ma degree in Mathemat i C S

K. (W

Major professor
Date / .— 3 / 7o

 

   
  

LIBRARY ’
Michigan State
~ . University

   

allsmlcslls

. 300K BINBERY "1E-

ABSTRACT
EQUIVALENCE OF TUBULAR NEIGHBORHOODS
By
Joan Elizabeth Quinn

Let p: X+X be a connected covering projection. We

"a a... 7,]

say that p is almost regular (AR) if and only if for every

f: X+X,the commutativity of

~ f ~
X-———-—9’X

R/p

X

implies f is a homeomorphism. X is absolutely almost
regular (AAR) if and only if every p is AR. In general,

an AR covering projection may not be a regular covering
projection. Theorem: Among closed surfaces, the 2-sphere,
projective plane, torus, and Klein bottle are the only
(AAR) spaces.

Let M be an (n-l)-manifold locally flatly embedded
in an n-manifold. Then there exists a tubular neighbor-
hood N (a topological l-disk bundle over M) of M. The
pair (N,M) is equivalent to another such pair (N',M) if

.and only if there exists a homeomorphism between the two

Joan Elizabeth Quinn

pairs. There exists a 2-sheeted covering pn:N+M and

(N,M) is homeomorphic to (Mpn, M) (where Mpn is the mapping
cylinder of pn). Theorem: (N1,M) and (N2, M) are equi-
valent if and only if there exists homeomorphisms h and

H such that

'0
5
|-'
z <—,_,z-

is commutative. Let M be a closed surface. Let T be the
set of equivalence classes of pairs (N, M), where N is
obtained by considering all possible locally flat embed-
dings of M into all possible 3-manifolds, except for the
case N = MxI. Let Ki be the subgroups of index 2 of nliM)
and say K1 and K2 are equivalent if and only if there
exists an automorphism of H1(M) that maps Kl onto K2.

Let N be the set of equivalence classes of all Ki‘ Theorem
3: There exists a natural 1-1 correspondence between N

. . _ 2 2
and T. (In particular, if H1(M) - [C1, C2,...CnIC1 C2 ...

an = l] , then T(M)in, and T(projective plane) = 1 and

T(Klein bottle) = 2.

Let (N,M) be defined as above. Theorem: Let hl
and h2 be involutions of N with M as fixed point set.
Then h1 and h2 are equivalent, that is, there exists a
'homeomorphism t of (N,M) onto itself with tIM = 1M and
l

h1 = t h2t.

 

EQUIVALENCE OF TUBULAR NEIGHBORHOODS

BY

Joan Elizabeth Quinn [4

1'3
‘k z

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1970

ACKNOWLEDGMENTS

The author is very grateful to Professor K. W.
Kwun for his suggestion of problems, general guidance,

and many helpful discussions.

ii

INTRODUCTION

Section 1.

Section 2.

TABLE OF CONTENTS

Almost Regular Covering Projections .

Equivalence of Tubular Neighborhoods .

iii

10

INTRODUCTION

Call a covering projection, p, almost regular if

and only if the commutativity of

 

‘51.", M .'_
I

implies f is a homeomorphism. The first section is con-
cerned with what types of Spaces have covering projections
that are not almost regular. Theorem 1.1 says that if a
topological space X has a fundamental group, fll(X,xo),
which has a subgroup H and element t with th-l properly
contained in H and an open covering U with fl1(U,xo) con-
tained in H, then X has a covering projection that is not
almost regular. Then all surfaces are classified as to
whether or not they have non-almost regular covering pro-
jections or not: clearly, a surface with one handle or
crosscap does not, Theorem 1.2 proves all surfaces with 2
or more handles or 3 or more crosscaps do, and Theorem
1.3 proves that the Klein bottle does not. The section is
concluded with an example of a covering projection of the

‘Klein bottle that is almost regular but not regular.

Letting f: Mn—l-tNn be a locally flat embedding as
defined by Brown in [2], a tubular neighborhood T(f) of M
in N corresponding to f is a topological l-disk bundle of
f(M) that is contained in N. Section 2 is concerned with
when there is a homeomorphism h: (T(f),M)——+(T(g),M).

Theorem 2.1 proves that 2 tubular neighborhoods, T(f) and

T(g), of M are equivalent if and only if there is a homeo-

morphism H: T(f)-—)T(g) for which

T(f) e 5(9)

 

 

L

M 9M

:7

 

commutes (where f and 9b are restrictions of the bundle

b
map). For M connected, it is shown that the number of
non-equivalent tubular neighborhoods of M, T(M),: l +
number of non-equivalent subgroups of index 2 of fll(M),
where H1 and H2 (subgroups of "1(M)) are equivalent if and
only if there is an automorphism a: wl(M)—9n1(M) with

a(H1) = H Theorem 2.2 shows that if M is a surface with

2.
n (2 or more) crosscaps, then 3 i T(M):n+l. Also,
T(projective plane) = 2 and T(Klein bottle) = 3. Section
2 concludes with Theorem 2.3 on involutions of tubular

neighborhoods: If hl and h2 are involutions of T(f) with

M as fixed point set, then hl is equivalent to h2‘

Section 1. Almost Regular Covering Projections.

 

Let p: X+X be a covering projection with X con-
nected and locally path-connected and X connected. Note
that this implies X is locally path-connected, since p is
a local homeomorphism.

Definition 1.1. p is almost regular if and only

 

if the commutativity of

f
X———-—)X
P P

X

implies f is a homeomorphism.

Remark 1.1. One can also state Definition 1.1 in
terms of a property of the fundamental group of the
covering space, X. That is,

p is almost regular if and only if p# n1(X,xo) is
not properly contained in any of its conjugates in n1(X,xo)
(where x0 is any point of X and x0 is any preimage of xo
under p, and p# means the map on fundamental groups in-
duced by p).

Proof of Remark 1.1: a. Assume the commutativity of

 

implies f is a homeomorphism.

Suppose p#nl(X,io) were properly contained in one

of its conjugates, say tp#w1(X,xo)t—l. Then since

{P#"l(i,Xo)lXo 6 p-1(xo)} is a conjugacy class in
~ ~ -1 ~ ~l
«1(x,xo) by [11, Thm. 6, p. 73], tp#fl1(x,xo)t = p#fll(X,xo)

1

for some £01 in X for which p(§éo ) = x0. And therefore,

there exists f': X + X such that

 

Xlio $ X'io
P P
X,XO

commutes by'[ll, Thm. 5, p. 76]. Also by [11, Thm. 5, p.
76], there exists no such map from (X,xol) to (X,xo). So
f' is not a homeomorphism which contradicts our assumption.
Therefore, p#nl(X,xo) is not properly contained in any of
its conjugates in n1(X,xo).

b. Assume p#fll(X,xo) is not properly contained in

any of its conjugates. Suppose

f
i,xo———9 mac)
P P

X,X

commutes. (Note that f is a covering projection by [11,
JLemma 1, p. 79]). Suppose f#: nl(X,xO)——9n1(X,f(xo)) is

:not a surjection. Then p#f#: n1(X,xO)—)p#nl(X,f(xo)) is

not a surjection, since f# and p# are monomorphisms by
[11, Thm. 4, p. 72]. But p#f# nl(X,XO) = p#n1(X,Xo) and
by [11, Thm. 6, p. 73], p# nl(X,xo) = p# nl(X,f(Xo)), so
p#f#nl(X,xo) = p#wl(X,f(io)). Therefore, f# is a surjec-
tion. It follows that the multiplicity of p is l by
[11, Thm. 9, p. 73] and thus that f is a homeomorphism.
Examples of almost regular covering projections:
1. All regular covering projections are almost
regular by [11, Thm. 11, p. 74].
2. All covering,projections, p, with finite
multiplicity are almost regular. Since p is n to l, for

any commutative diagram

f
X-——————)X

P\

X

k

pf is n to 1. So f must be 1 to 1 and thus a homeomorphism.

3. If nl(X,xo) has the ascending or descending
chain condition, then, clearly, any covering projection of
X is almost regular.

Definition 1.2. A group G is said to be not regular

 

 

if there exists a subgroup H of G and an element t of G
for which th-l is properly contained in H. Otherwise, G

is said to be regular.

 

Any subgroup H and element t of a group G that is
not regular, mentioned in the following pages, is under-
stood to behave as in Definition 1.2.

Theorem 1.1. If n1(X,xO) is not regular and there
is an open covering U of X such that wl(u,xo) is contained
in H, then there is a covering projection with base space
X that is not almost regular.

Proof: Let p: (X,io)-—)(X,xo) be the covering projection
with p# fll(X,io) = H constructed as in [11, Thm. 13, p.

1 be the element of

82]. X is clearly connected. Let x0
X which corresponds to a loop w(t) about X0 in X which

corresponds to the element t of fll(X,xo). Consider the

diagram:
fiscal £30
P\\\N ¢///P
X,xo

1

01 ta so) is (w(t)>'

which corresponds to t_1, p#n1(X,xol) = th_1. But th’

p#wl(X,xo) = H. Since p(path from i
l
is properly contained in H, so p is not almost regular.
Remark 1.2. One observes from the construction in
Theorem 1.1 that if M is a connected manifold for which
«1(M) is not regular, then there is a non-almost regular
covering projection of M (because there is a covering of

M by Open sets {U}, for which wl(u,xo) = 0).

Clearly, all covering projections of a surface
with precisely one handle or one crosscap are almost
regular. The following two theorems will complete a
classification of surfaces according to whether or not
they have non-almost regular covering projections or not.

Theorem 1.2. All surfaces with 2 or more handles
or 3 or more crosscaps have non-almost regular covering
projections.

Proof: Fact 1. If a group G is given by the generators
a1, a2, . . . . an and one relation f(a1,a2, . . . an) = l,
and if, further, the element an occurs in this relation
and cannot be removed from it by transformations, then

the subgroup {al’ a2, . . . , an_1} is free and

a1, a2, . . . , an_1 are free generators of the subgroup
by [8, p. 77].

Fact 2. Any free group, G, of 2 or more genera-
tors is not regular (Example by L. M. Sonneborn - Let H
be the subgroup generated by

{a, b‘1 ab, b‘zabz, . . . , b'n abn, . . .}
nl(xn), where Xn is a surface with n (2 or more)
handles, has generators a1, bl’ a2, b2, . . . , an, bn

and the single relation a b a -1b -1 1 -1

1 1 1 1 azbzaz b2 ' ' '
-1 -1

anbnan bn = l by [11, p. 149]. Therefore, nl(Xn) has
a free subgroup with more than one generator by Fact 1.
So by Fact 2, and Remark 1.2, Xn has a non-almost regular

[covering projection.

n1(Yn), where Yn is a surface with n (3 or more)
crosscaps, has generators c1, c2, . . . , cn and the
single relation clzcz2 : 0 ° cn2 = l by [11, p. 149].
Therefore, as in the case for Xn, Yn has a non-almost
regular covering projection.

Theorem 1.3. All covering projections of the
Klein bottle, K, are almost regular.
Proof: Consider the covering projection of multiplicity
2, p: S1 x Sl-—9K. 111(8l x 81) is regular, so, by the
following lemma, nl(K) is regular. Therefore, all
covering projections of K are almost regular.

Lemma 1.1. If G is not regular and N is a normal
subgroup of G for which [G:N]<w, N is not regular.
Proof: Since th-l is properly contained in H, tkHt-k is
properly contained in H for each positive integer k.

There is a k for which tk is an element of N since [G:N]<w.

Consider the following diagram:

k -k
————-> tk(HON)t-k——9tkHt_k—-)t Ht 4(HnN)t'k—>O

O
1 111 i 12
0

* > HON ) H ——)H/HflN > 0

 

 

 

i2 is an isomorphism since H/(HnN) is isomorphic to HN/N
and [G:N]<m. So, by the five lemma in [11, p. 185], if
i1 is onto, then i is onto. 'But i is not onto by assump-

tion. Therefore, i1 is not onto and N is not regular.

 

Example of closed PL manifolds of arbitrarily
high dimension that have non-almost regular covering pro-

jections: Let m34. Embed Xn (or Yn) in Sm+l. Take a

1

regular neighborhood, N, of Xn in SI“... . wl(N) = nl(xn).

N is a closed PL manifold of dimension m and by Remark
1.2 has a non-almost regular covering projection.

Example of an almost regular covering projection
that is not regular:
nl(K) has 2 generators, c1 and c2, and the relation
c1 c2 = 1. Consider the covering projection p: (fi,§o»_,
(K,xo) with p#fll(X,xo) = {c2} (where {a} denotes the sub—
group generated by a). c1-1{c2} c1 is not equal to [c2].
Consider w: (I,O)——9(K,xo) where w(I) corresponds to Cl.
There is a lifting of w to w': (I,O%——9(X,Xo) by [11,
Thm. 3, p. 67]. Let w'(l) = ii. Then p(xol) = x0 and
p#nl(X,X;5 = cl-lp#1rl(X,Xo)cl which is not equal to
p#wl(X,Xo). So p is not regular by [11, Thm. 11, p. 74],

and, by Theorem 1.3, p is almost regular.

10

Section 2. Equivalence of Tubular Neighborhoods.

Let M be an n-l manifold and f: M+N be a locally
flat embedding of M into any n—manifold N (where manifold
and locally flat embedding are defined as in [2]).

Definition 2.1. A tubular neighborhood, T(f), of

 

 

M in N corresponding to f is a topological l-disk bundle
of f(M) that is contained in N.

Remark 2.1. Given a locally flat embedding f:
M+N, one can exhibit a tubular neighborhood corresponding
to f in the style of [5] as follows:
Let {Ua} be a basis for the topology on N such that if

Uanf(M)+ ¢ then there is a homeomorphism

h: (En,En-)——)(Ua,Uanf(M)). En’l separates En, say into
E: and EB. Denote h(E$ UEn-l) by U: and b(Equ“'1) by
d' Let {US} = {UIU is an element of {ua}and Unf(M) = ¢

or U = U: or U; for some a}.

Let N1 = {(x,UB)Ix is an element of U8}' Define an
equivalence relation, R, on Nl by (x,UB) is equivalent to
(x1,UBl) if and only if x = x1 and UBnUBInC(f(M)) +¢.
Take N' = Nl/R and denote an equivalence class by [X'UB]°
Let {UB*} be a basis for N' where [X'UY] is an element of
U8* if and only if x is an element of U8' Define n: N'+N
by n[x,UB] = x. n is 2 to l restricted to n-1(f(M)) and
restricted to Cn-l(f(M)) is l to l. N' is an n-manifold
with boundary n-1(f(M)). n-1(f(M)) is collared by [2],

_and its image is a l-disk bundle of f(M).

11

Remark 2.2. Let fb: T(f)+M be the bundle map.
Then fblT(f) is a covering projection of multiplicity 2
and, denoting the mapping cylinder of fblT(f) by Mf,
(Mf,M) is homeomorphic to (T(f),M)-

Definition 2.2. T(f) is equivalent to T(g) if and

 

only if there is a homeomorphism h: (T(f),M)._9(T(g),M).
Theorem 2.1. T(f) is equivalent to T(g) if and

only if there is a homeomorphism H: T(f)__9T(g) for which

 

. r7 .
T(f) > (g)
f 1gb
h
M ; M

 

commutes.

Proof: Assume T(f) is equivalent to T(g). Then there is
a homeomorphism h: (Mf,M)-—9(Mg,M). Let 91 be the cover-
ing projection, 91: T(g)xI—1,MxI, induced by gb and let
i: M—)Mxl—-)MxI be the natural inclusion. Then, by the
diagram:

, id
T(g)xI + M > M

 

91+i 9'

 

\v
MxI

g' is continuous by [4, Thm. 3.2, p. 123]. (id will al-

,waysindicate the obvious identification map.)

 

12

Case 1. Assume T(f), T(g), and M are connected.

We have

h*: T(f)xI——l-d—> Mf—fl—mg—L—nyI

where h*(x,l) is in Mxl. Consider the diagram:

T(g)xI

)7
/'

IL" 91
/

o / h*

T(f)xI ———> MxI
There exists h' with h'IT(f)x0 = h by [11, Thm. 3, p. 67].
Note that h'(T(f)x1) is contained in T(g)xl. Also

4':

h*|T(f)xl is a covering projection.

Now consider the diagram:

T(g)xl
h' 91

o h*
T(f)xl >Mxl

h'IT(f)xl is a covering projection, since T(f), T(g), and
M are connected and locally path-conntected by [11, Lemma
1, p. 79] and therefore h'lT(f)x1 is open and onto.
h'|T(f)xl must clearly than be 1 to 1, so h'IT(f)xl
induces the desired H.

Case 2. Assume T(f) and T(g) are not connected

and that M is connected.

13

Then there are T1 and T2 (each connected) for
which fblTi: Ti-—+M is a homeomorphism. As in Case 1,

h*: T(f)xI-—9MxI. For this case, consider the diagram:

T(g)xI
‘3
//
h.l / g1
l/ R
/
/ h*
T x1 44; MxI

 

There exists hil that makes the diagram commute and

 

hil(TixI) is connected. So, we get a l to 1, onto map
h':T(f)—->T(g) that clearly must be Open.

Case 3. Assume M is not connected. The argument
is the same as the preceding ones applied to the components
of M and their preimages.

Assume there is h: T(f)—+T(g) for which
0 H O

T(f) ————-> T(g)

b 96

M hfixM

 

commutes for some homeomorphism h: M-—9M.

Consideration of the following diagrams completes

the proof:

14

    
 

T(f)xI + M £6 Mf T(g)xI + M——1‘9—;Mg
hxl+1 lel'l'i
T(g)xI + M T(f)xI + M -1
h h
id id
Mg Mf

Remark 2.3. Given a commutative diagram as in
Theorem 2.1, one can alter h to be base-point preserving
by an isotopy and H can be altered accordingly. So we
will assume h is base-point preserving.

Remark 2.4. Up to equivalence, there is one
tubular neighborhood, T(f), of M with T(f) not connected,
namely, MxI.

Remark 2.5. Given any subgroup H of n1(M) of
index 2, there is a corresponding tubular neighborhood,
T(p'), of M. Let p: E-—9M be a connected covering pro-
jection of M with p#nl(E) = H. And take N to be the
mapping cylinder of p and p': M-—)N to be the natural
map.

In this paragraph, assume T(f), T(g), and M are

connected. Consider the diagram:

on 33? , ’

15

By Theorem 2.1, if T(f) is equivalent to T(g), then there
exists 5 that makes the diagram commute, and therefore
h#: wimp—”104) with h#fb#1rlT(f) = gb#1r1'l‘(g) by [11,
Thm. 5, p. 76]. Suppose there is no automorphism
a: “(m—”1m with afb#1r1T(f) = gb#1r1'l'(g). Then T(f)
is not equivalent to T(g). One concludes that the number
of non-equivalent tubular neighborhoods, T(f), of M with
T(f) connected 1 the number of non-equivalent subgroups
of n1(M) of index 2, where if H1 and H2 are subgroups of
G, H1 is equivalent to H2 if and only if there is an
automorphism a: G—9G with a(H1) = H2.

When M is a surface, every automorphism of rl(M)
is induced by a homeomorphism of M by [10, Thm. 2, p. 542].
Therefore, if M is a surface, T(M) = the number of non-

equivalent subgroups of n1(M)+1. Let n1(M) = G. Define

a subgroup of G of index 2 by a homeomorphism from G to

22 and let H1 and H2 by 2 such subgroups.
Case 1. Suppose G has generators a1, b1, . . . ,
. -1 -l -1 -1 _
an, bn and the relation alblal bl anbnan bn — l.

A. If for both H1 and H2, one and only one
generator (aj for H1 and a1 for H2) is mapped to l in 22,
then there is clearly an automorphism a of G with
a(Hl) = H2.
B. If H1 is the subgroup of G defined by
sending a1 and only al to l and H2 is the subgroup of G

defined by sending bl to 1, then there is an automorphism

16

a of G for which a(Hl) = H2. Namely, define a(a1)

and a(b1) = al-l. Otherwise, a(ai) = ai

all
and a(bi) = bi'
C. If H1 is the subgroup of G defined by
sending ai and only ai to 1 and H2 is the subgroup of G
defined by sending ai and bi to 1, then there is an auto-

morphism a of G with a(H1) = H2. Namely, a(aj) = a. and

J
a(bj) = bj for 3 not equal to 1 and a(ai) = aibi and
a(bi) = ai-l. One concludes when M is the torus, T(M) = 2.
Case 2. Suppose G has generators c1, c2, . . . , cn
and the single relation c12c22 ° ° - on2 = 1.

A. Let Hs denote the subgroup of index 2
defined by mapping C8 to 1, then there is an automorphism

s of G with s(HS)=H (where without loss of generality

t

assume s<t).

Define: s(ci) for i<s, i>t

ll
0

s(cs) = c c
S(cs+l) = C3 Ct cs+1ct Cs

3(cs+2) = C3 Ct cs+2ct Cs

s(ct_l) = c ct Ct-lct c
s(ct) = c
B. Let Hi denote the subgroup of G of index
2 defined by mapping c. , c. , . . . , c. to 1. Then
11 12 1n

there is an automorphism a of G with a(Hi) = Hj' Namely,

17

define a = hlh2 ° ° - hn where hk = 1k if ik<jk and hk = 3k

if ij1k and hk = identity if 1k = jk (where 1k and 3k are

defined as above).
C. Suppose Hi is defined by sending ci and

Q o O ' C I

only ci to l and Hj is defined by sending c. j
s

Ic-I
31 32
to l where s is even. Then there does not exist an auto-
morphism a of G for which a(Hi) = Hj' Abelianize G

giving Hi and H3, corresponding to Hi and Hj' In Hj',

there is a non-identity element, namely clc2 . . . cn,

whose square is equal to 1. So H3 is not free abelian,

 

but Hi is clearly free abelian. So Hi is not isomorphic
to HE.

3
One concludes:

Theorem 2.2. T(surface with n crosscaps) is‘: 3
and i.n+1 (where n 3 2). T(projective plane) = 2 and
T(Klein bottle) = 3.

Let M be any surface and f: M-—,N a locally flat

embedding with corresponding tubular neighborhood T(f).

Let h1 and h2 be involutions on (T(f),f(M)) with f(M) as
fixed point set. As is standard, h1 is equivalent to h2
if there is a t: T(f)-—sT%f) with thlt”1 = h2'

Theorem 2.3. hl and h2 are equivalent.
Proof: Case 1. f(M) separates T(f), say into N1 and N2.

Claim: Either hi(Nl) = N and hi(N2) = N or hi(Nj) = N..

2 1 3
Suppose not and there are n1 and n1* which are elements of

. Nl for which hl(nl) = ni which is in N1 and h1(n1*) = n2

18

which is in N2. Then there is a path 1 in N1 from n1 to
n1* and hl(i) is homeomorphic to I with hl (n1) = ni and
hl(n1*) = n2. So there must be an n in i for which hl(n)
is an element of f(M). This is a contradiction so the
claim holds. But we know hi(Nj) is not equal to Nj'

= N Therefore, the

Thus, we get hi(N and hi(N2) = N

1’ 2 1°
orbit spaces of h1 and h2 [acting on T(f)-f(M)] are clearly
homeomorphic, say by f. (Oh is the orbit space of h: N——>N
and h* is the projection from N to Oh.) Consider the

diagram:

 

There exists t which makes the diagram commute for each of

N1 and N2. Now,

hgthlt’1(n) = fhihi t 1(n) = fhit—l(n) = h5(n>.

So, t lh1 t 1(n) = n or h2 (n).

If t 1h1t1(n) = n, then hlt1(n) = t-1(n) which is a contra-
diction. Extend t to T(f) by t(x) = x. Then thlt-1 = h2

and hl is equivalent to h2.
Case 2. f(M) does not separate T(f).
A. Assume M is not the projective plane.

Let x be an element of f(M) and let U be a neighborhood of

19

x in N such that f(M) separates U, say into U1 and U2.
Then, as 1n Case 1, hi(Ul) = U2 and hi(UZ) = 01'
Let N' be defined as in Remark 2.1. Let
M = n-1(f(M)) and T = n-1(T(f)). Let 51 and 52 be the
involutions on T induced by h1 and h2' Then if x is an
element of M for which n(x) = n(x'), then fil(x) = fi2(x) = x'.
T is homeomorphic to MxI so without loss of generality
~ . ~ _ ~ ' = ~ , ,
assume T 18 MxI. Let B - Ohl and M Ofil|MxO Wthh is
homeomorphic to f(M) and therefore not the projective
. . . , = ~
plane. Let h. ML-—9B be the 1nc1u31on. M1 OHl'Mxl
and "1(Mi) = nl(M') = n1(B). B is clearly compact, con-
nected and a Poincare 3-manifold. Therefore, 0a is
1
homeomorphic to M'xI by [1, Thm. 3.1, p. 485]. Similarly
for O . So 0 is homeomorphic to O , say by f.
H2 H1 52

Consider the diagram:

 

 

~ t1 ~
MxI - e;.MxI
”* *
hl fi2
f
O s; 0
F11 52

Since f can be chosen so that f 5* n (MxI) = h n (MxI)
# l# l 2 1

#
and hi and 55 are covering projections by [11, Thm. 7,

p. 87], there is a t1 which makes the above diagram com-

-1

mute. As in Case 1, tlfilt1
l

by t1, we get thlt- = h2.

= hz. Letting t be induced

20

B. Let M be the projective plane.

By Theorem 2.2, M has 2 non-equivalent tubular neighbor-
hoods. One is equivalent to MxI and is taken care of in
Case 1. The other tubular neighborhood can be defined
by the 2 to 1 covering projection p: S2->M as (Mp,M).
hi: (Mp,M)-—9(MP,M) induces an involution hi on 82x1.
By [9, Thm. 1, p. 582], there is a t': Ssz->SZxI for
which t'hit'-1 = hi. t' induces t: (Mp,M)-—->(MP,M)

with th t-1 = h 2

l 2' E

 

BIBLIOGRAPHY

10.

11.

BIBLIOGRAPHY

Brown, E. M., Unknotting in szI, Trans. Amer. Math.

 

Brown, Morton, Locally flat embeddings of topologi-
cal manifolds, Ann. Math. (2) 75 (1962), 331-341.

 

Curtis, M. L. and Kwun, K. W. Infinite sums of
manifolds, Topology, 3 (1965), 31-42.

Dugundji, Topology, Allyn and Bacon, Inc., 1966.

Gluskina, E. D., Locally flat embeddings of codimen-
sion 1, Sibirsk Mat. Z, 7 (1966), 217-220.

 

Kneser, Hellmuth, Die kleinste bedeckungszahl
innerhalb einer klasse von flachenabbildung,
Math Annalen, 103 (1930), 347-358.

 

Kneser, Hellmuth, Glattung von flachenabbildung,
Math Annalen, 100 (1928), 609-617.

 

Kurosh, K. A., Theory of Grogps, Vol. 2, Chelsea
Publishing Company, 1960.

 

Livesay, George R., Involutions with two fixed points
on the three-sphere, Ann. Math., (2) 78 (1963),
582-593.

 

Mangler, W., Die klassen von topologischen abbildungen

einer geschlossen fléche auf sich, Math, g’44
(1938), 541-554.

Spanier, Edwin H., Algebraic Topology, McGraw-Hill,
Inc., 1966.

 

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