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mm: :51in STATE UNIVERSWY
MYUNG Ml MYUNG
1970

 
 
 
 

 
 

L I B R‘A R Y
Michigan Sum
- University ‘

 

This is to certify that the
thesis entitled

"PL Involutions of Some 3-Manifolds”

presented by

Myung—Mi Myung

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Mathematics

Major professor

 

Date July 22. 1970

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ABSTRACT

PL INVOLUTIONS OF SOME 3-MANIFOLDS

BY

Myung Mi Myung

Let hl and h be PL involutions of connected,

2
oriented, closed 3-manifolds M1 and M2, respectively.
Let al and a2 be fixed points of hl and h2' re-

spectively, such that near ai the fixed point sets of
hi are of the same dimension. Taking the connected sum

of M1 and M2 along neighborhoods of ai, one can de-

fine a PL involution hl # h2 2

Let M1 and M2 now be irreducible in addition. The

question that under what condition a PL involution h on

of M1 # M induced by hi'

Ml # M2 18 of the form hl # h2

are studied when M1 and M2 are lens spaces (not neces-

sarily having the natural orientations). Henceforth assume

and related questions

that M1 and M2 are lens spaces. Then the main results

are the following:

Theorem 1: Let h be a PL involution of M1 # M2.

If the fixed point set F contains an orientable surface,

 

than F is a 2-sphere and M = - M

2 l’ h being the obvious

involution in this case.

Myung Mi Myung

Theorem 2: Let h be a PL involution of M1 # M2.

 

If the fixed point set F contains a projective plane,
then M1 = M2 is a projective 3-space.
The case M1 = M2 is a projective 3-space P3 is

separately studied.

Theorem 3: Let h be a PL involution of P3 # P3

 

with 2-dimensiona1 fixed point set F. Then F is a 2-
sphere, the disjoint union of two projective planes, or the

disjoint union of a Klein bottle and two points.

Theorem 4: In Theorem 3, if F is the disjoint

 

union of two projective planes, then h is unique and

h = hl # h2 where hi is the unique involution on Mi

with a projective plane and a point ai as the fixed point

set.

PL INVOLUTIONS OF SOME 3-MANIFOLDS

BY

Myung Mi Myung

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1970

To Hyo Chul

ii

AC KNOWLEDGMENT S

The author wishes to express her gratitude to
Professor K. W. Kwun for suggesting the problem and for
his helpful suggestions and guidance during the research.

The last stage of this research was supported in

part by NSF Grant GP-19462.

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 O 1
CHAPTER
I. CODIMENSION ONE EMBEDDINGS OF MANIFOLDS . . 2

II. PL INVOLUTIONS OF SOME 3-MANIFOLDS . . . . . 9

BIBLIOGRAPHY O O I O O O O O O O O O O O O O O O O O 32

iv

INTRODUCTION

Let M be a closed, orientable 3-manifold which

is the connected sum Mi # M2 of two irreducible 3-

manifolds M1 and M2 and let h be a PL involution of

M with a fixed point set F containing a non-orientable
surface F0. Since F0 is one-sided, it would seem that

h cannot interchange Ml-part and M -part and h must

2

be obtained from involutions hl and h2 of M1 2,

respectively, by attaching two involutions along invariant

and M

neighborhoods of fixed points a1 and a of h and h

2 1
respectively, where near ai the fixed point sets of hi

2'

are of the same dimension.

Fremon [2] completely determined all possible fixed
point sets of a PL involution of S1 x 82 and all PL in—
volutions of S1 x 82. Kwun [4] proved that no lens space
except the real projective 3-space P3 admits orientation
reversing PL involutions and in case of P3 there exists
a unique PL involution up to PL equivalences.

Motivated by the fact that S1 x 82 covers
P3 # P3, in this thesis, we consider the possibility of
the above question, when M1 and M2 are isomorphic to
lens spaces.

CHAPTER I
CODIMENSION ONE EMBEDDING OF MANIFOLDS

Let N be a connected n-manifold and M a con-
nected (n - l)-manifold. Corresponding to an embedding

M.::N, we have a double covering p: N +N having the

l
prOperties that (1) each component of p-1(M) is two

1

sided and p- (M) separates N1, (2) p-1(M) is connected

if and only if M is one-sided, (3) N1 is connected if
and only if M does not separate N, and (4) if N - M

is connected, then Nl - p-1(M) has two components each
of which maps homeomorphically onto N - M under p.

Since we use the technique of "cutting along a submanifold"
to obtain N1, we will call p: N1+N the double covering
obtained from N by cutting along M.

In this chapter, we will show that no k + 1 non-
orientable, connected (n - l)-manifolds can be disjointly.
embedded in an orientable, connected n-manifold N whose
2) with coefficient 22 is a
finitely generated group of rank k using the double

homology group H1(N; Z

covering, and examples will follow the theorem.
An embedding will mean an embedding as a closed

subset. We frequently identify an embedding with its

image. Suppose that a connected (n - l)-manifold M is
embedded in a connected n-manifold N. We say that M is
one-sided if M does not separate any connected neighbor-
hood of M. Otherwise, M is two-sided. It can be shown
that every embedding is one-sided if M is non-orientable
and N is orientable, every embedding is two-sided if M
and N are orientable, and if M is one-sided, then

N - M is connected.

Theorem 1.1: Suppose that N is an orientable,

 

connected n-manifold such that its homology group
H1(N; 22) is a finitely generated group of rank R. Then
no k + l non-orientable, connected (n - l)-manifolds

can be disjointly embedded in N.

Proof: Let Ai be a non—orientable, connected
(n - l)-manifold, fi: Ai+N an embedding, and pi: Ni+N
the double covering obtained from N by cutting along
f.(A.). Since f.(A.) is non-orientable, f.(A.) is
1 1 1 1 1 1
one-sided, and hence pi-l(fi(Ai» is connected. There-

fore fi cannot be lifted with respect to pi. By the

l1ft1ng theorem fi#nl(Ai) 15 not conta1ned 1n pi#nl(Ni).
Consequently, the composite
f.
l# 91
"1(Ai) -—9 Trl(N) —>1r1(N)/pi#1rl(Ni) = Z2

(nl(N) has different base point for different i's.)

is an epimorphism, where gi is the projection to the

quotient group. Now gi can be factored as

n (N) h H (N) 91 z
1 “F 1 “‘9 2

where h is the Hurewicz homomorphism and gi' is an

epimorphism. Hence we have a commutative diagram:

(A fi# (N 91 2
1T1 1) "“" Tr1 ) "" 2

Hl(Ai) 91

i*

l f l

1*

  

where gi" fi* is an epimorphism.
In order to complete the proof of the theorem, we

need the following lemma:

Lemma 1.2: Suppose that N is an orientable,

 

connected n—manifold such that its homology group

H1(N; 22) is a finitely generated group of rank k, and
suppose that f1(A1)’ f2(A2), ..., ft(At) are mutually
disjoint, non-orientable, connected (n - l)-manifolds
embedded in N. Let Ki be the kernel of 91" where
g." : H1(N; Zz)+z2 is the epimorphism which makes the

1
following diagram commute:

5

f- 9
171(Ai) —££—) «1(N) ——) 22
\L f.* l /
l n
i f J,

i*

 

Then K. n K. n ... nK. , 1 < 11, i2, ..., i < t,

ll l2 1s

s 4 k, and ip # iq for p # q, is of rank k - s.

S

1.

Proof: We use induction on s. Suppose 5
Let Li be the image of fi* and Ki the kernel of
gi . Then Li ¢=Ki and Li # O, for all 1, Since gi fi*
is an epimorphism. On the other hand, since N - fi(Ai)
is connected, Ni - pi—l(fi(Ai)) has two components each
of which maps homeomorphically onto N - fi(Ai)' Hence
all fj, j # i, can be lifted with respect to pi, as

c: _ . .

fj(Aj) N fi(Ai). By the l1ft1ng theorem, Lj czKi

and Ki # Kj for all j # i. Since the exact sequence

O+ker gi"+Hi(N; 22)+ZZ+0

splits, Hl (N; z = ker gi 6-) 22 = Ki (+3 zz. Hence the

2)
rank of K1 is k — l for all 1.

Suppose that Ki1 n Ki2 n ... nKir, 1 < 11, 12,

..., ir s t, r < k, and ip'¥ iq for p # q, is of rank

k - r. Consider the isomorphism

 

where Ki -(K. n K. n ... nK. ) is the smallest sub-
r+l 11 12 1r

group of H (N; Z ) cOntaining K. and

1 2 1r+l
K. n K. n ... nK. . Since L. ¢:K. but
11 12 1r lr+1 lr+1

L. c K. n K. n ...nK. , K. nK. n...nK. ¢K. .
1 12 1r 11 12 1r lr+1

Hence Ki ~(Ki n Ki n... nKi ) must be of rank k.
r+l 1 2 r

Since the rank of Ki is k - l, the quotient group
r+l

K. '(Ki n Ki n ... flKi )

 

is of rank 1. Therefore the above isomorphism and the
fact that the rank of K. n K. n ... HR. is k - r
11 12 1r
n K. n ... nK. is of rank k - (r + l).
l 12 lr+l
This proves the lemma.

imply that Ki

We now return to the proof of the theorem.

Case I: k = 1. Since fl is non-trivial, H1(N; 22)
*

= L1 and gl" is an isomorphism. Hence if there were
another f2(A2) in N disjoint from fl(A1), then

L2 c=Kl = 0, which contradicts L2 # 0. Therefore no
two non-orientable, connected (n - l)-manifolds can be

disjointly embedded in N.

Case II: k a 2. Suppose that there are k + l mutually
disjoint, non-orientable, connected (n - l)-manifolds
fl(Al), f2(A2), ..., fk+l(Ak+l) 1n N. S1nce

Li ::K1 n K2“ ... “Ki-l n Ki+ln ... nKk and the rank of
K1 n K2“ ... nKi-l n Ki+ln ... nKk 1s 1, we have

-1 1n ... flKk for all 1

for all i # j. The fact that L1 and

L2 are contained in K3 n K4n ... nKk, the rank of

K3 n... nKk 1s 2, and L

0 # Li = Kl n K20 ... M1
and L. n L. = 0
1 J

n Ki+

l n L2 = 0 imply K3 n K40 ... nK

= Ll @ L2. Repeating this process, we obtain

k

Kk = L1 6-) L2 63 GB Lk-l' S1nce Lk ¢ Kk and the rank

of Lk is l, Lk n K = 0, which gives H1(N; 22)

k

= Kk ® Z2 = Kk GB Lk' S1nce Kk 13 of rank k - 1. There-
fore H1(N; 22) = L1 @L2 6-) ®Lk-l @Lk, and so

L = 0, which contradicts the fact that the rank of

k+l
Lk+l is 1. This completes the proof.

Neuwirth [11] proved a stronger version in case
embeddings are nice, namely, if the disjoint union of k
closed, non-orientable (n - l)-manifolds Mn can be
semilinearly embedded in a closed, orientable n-manifold,

then there exists a homomorphism of w1(Mn) onto the free

product of k c0pies of 22.

Example 1: No two disjoint copies of P2 can be

 

embedded in P3. This will be used later.

Example 2: If H1(N; 22) = 0, then no non-

 

orientable codimension one manifold can be embedded in

N, since H1(N; 22) = 0 implies that N is orientable.

Example 3: No three disjoint copies of P can

2

 

be embedded in P3 # P3.

CHAPTER II
PL INVOLUTIONS OF SOME 3-MANIFOLDS

Let hl and h2 be piecewise linear (PL) involu-
tions of connected, oriented, closed 3-manifolds M1 and
M2, respectively. Let a1 and a2 be fixed points of
hl and h2' respectively, such that near ai the fixed
point sets of hi are of the same dimension. Taking con-
nected sum of M1 and M2 along invariant neighborhoods

of ai, one can define a PL involution h1 # h of

2
M1 # M2 induced by hi' Let h be a PL involution of

a manifold M, where M is isomorphic to the connected
sum Ml # M2 of two connected, oriented, closed, irreduc-
ible 3-manifolds, with a fixed point set F containing a
non-orientable surface F0. Since F0 is one-sided, it
would seem that h cannot interchange Ml-part and M -

2

part and that h must be of the form h1 # h Therefore,

2.
in this chapter, we will study under what conditions a PL

involution h on M = M1 # M2 is of the form hl # h2

and related questions when M and M are isomorphic to

l 2
lens spaces (not necessarily having the natural orientation).
This work was suggested by Kwun [6], who considered
orientation reversing PL involutions of lens spaces, and
proved that no lens space except the projective 3-space P

3
9

10

admits an orientation reversing PL involution and there
exists exactly one orientation reversing PL involution on
P3 up to PL equivalences. In this case, the fixed point

set is a projective plane P2 plus a point.

Definition 2.1: The connected sum Ml # M2 of two

oriented 3-manifolds is obtained by removing the interior

 

of a nice 3-cell from each, and then matching the result-

ing boundaries using an orientation reversing homeomorphism.

Definition 2.2: A manifold M is isomorphic to a

 

Manifold M' if there is a piecewise linear, orientation

preserving homeomorphism between them.

Definition 2.3: A manifold M is non-trivial if

M is not isomorphic to a 3-sphere S3.

 

Definition 2.4: A non-trivial manifold P is

 

prime if there is no decomposition P = M1 # M2 where
M1 and M2 are non-trivial.

Milnor [10] has shown that every connected,
oriented, non-trivial, compact 3-manifold M is isomor-

phic to a sum P # P2 # ... # Pk of prime manifolds and

l
the summands Pi are uniquely determined up to order and

isomorphism.

Definition 2.5: A 3-manifold M is irreducible

 

if every nice 2-sphere in M bounds a 3-cell.

11

Milnor [10] also proved that with the exception of

manifolds isomorphic to S1 x S2 a manifold is prime if

and only if it is irreducible and S1 x 52 is prime, but
it is not irreducible.
From now on, we assume that M1 and M2 are lens

spaces and h is a PL involution on Ml # M2 with a
2-dimensional fixed point set.

Remark: Since the fixed point set F is two-
dimensional, any PL involution h on M has the property
that near each point of F it maps one side of F to the
other side of F. For, if this were not true, one could
find a small invariant 2-sphere S near F such that
h S has a 2-cell as fixed point set. But this is im-
possible. Hence near each point of F, h reverses the

orientation, and therefore h reverses the orientation

globally.

Theorem 2.1: Let h be a PL involution of M

 

= M1 # M If the fixed point set F contains an orien-

2.

table surface, then F is a 2-sphere and M = -M

2 1'h

being the obvious involution in this case.

Proof: We first show that if F contains an
orientable surface, then F is a 2-sphere. Let S be
an orientable surface contained in F. Then by the

Alexander duality theorem [12], over the rationals Q

-0.
H2(M1 # M2 - P: Q) - H (F. Q) .

12

Hence F separates M # M into two parts U and V.

1 2
Since h has to interchange those two parts, Ml # M2 = 25

and we have
Finalttmz-iefi

such that ri is the identity, where i is the inclusion

and r is a retraction defined as follows:

-1

x if x e 6
h (x) if x e h(U).

Therefore we obtain the exact sequence
_ 1* rs _
Hi(U; Q) ———9 Himl # M2: Q) -——9 Hi(U; Q)
such that r*i* is the identity. Since Hi(Ml # M2) = 0
for i = l, 2, Hi(fi: Q) = 0 for i = l, 2. Therefore F
must be a 2-sphere.

We now prove that M2==-Ml. Since F is a 2-
sphere, F separates M. Let U and V be the two com-
plementary domains of M - F. Attaching a 3-cell to each
of U and V to eliminate the boundaries, we obtain two
connected, orientable manifolds U' and V', and M
= U' # V'. By the unique decomposition theorem for 3-
manifolds [10], either U' is isomorphic to a 3-sphere
or U' is isomorphic to one of M1 and M2. But, U' cannot

be a 3-sphere. If it were, U would be a 3-sphere minus

a 3-cell and V would be M # M minus a 3-ce11, and

l 2

13

hence h would not be able to interchange U and V.

Hence 0' is isomorphic to M1 or M2. We may assume

that U' is isomorphic to M Similarly, it follows

1.
that V' is isomorphic to M2. Since U and V must
be interchanged by h, V' = -U'. Therefore, M2 = —M1,

and h is the obvious involution. This completes the

proof.

Theorem 2.2: Let M be a connected sum M # M

 

l 2'

where M1 and M2 are isomorphic to lens spaces and h

is a PL involution on M. If a real projective plane P2
is contained in the fixed point set F of h, then

Proof: Suppose that h fixes a real projective
plane A and assume that M has been triangulated so
that h is simplicial and the simplicial neighborhood U

of A is an invariant regular neighborhood of A. More-

over, we may assume that h is fixed point free.

U - A
Since A is 2-dimensiona1, near each point of A h

maps one side of A to the other side. Hence h re—

U
verses orientation. Since U is orientable, but A is

not, A is one-sided in U. Consider the double covering

p: M1+M obtained from M by cutting along A. Then

p-1(A) is connected. Therefore, p-1(A) is isomorphic to

l

a 2-sphere. p—1(A) separates M1 and M1 - p- (A) has

two components each of which maps homeomorphically onto

14

M - A. Therefore, p (U) is a two-side collar neighbor-
hood of p-1(A), and each component of its boundary, which
is isomorphic to p-1(A) = 82, maps homeomorphically

onto the boundary of U. Hence (U, A) is homeomorphic to
(N, A) where N is the mapping cylinder of a double
covering 82+A.

Let U' and (N - U)' be the connected manifolds
obtained from U and N - U by attaching a 3-cell to
each. Then M = U' # (N - U)'. By the unique decomposi-
tion theorem for 3-manifolds [10], U' is isomorphic to
33, M1’ or M2. But U' cannot be isomorphic to a
3-sphere, since the fundamental group nl(U') of U' is
Z2, but wl(S3) is trivial. Therefore we may assume
that U' is isomorphic to M1’ Now hIU can be extended
to an orientation reversing PL involution h' of U' 3 M1'
since U' - U is a 3-cell and Iqu(U) : Bd(U)+Bd(U) is

fixed point free, and hence h'lU. can be defined by

- U

the cone over h Bd(U)' Since no lens space except the

projective 3-space P3 admits an orientation reversing

PL involution, U' = M1 must be P3.

From U' = M1 and the unique decomposition theorem,
we get (N - U)' = M2. Since (N - U)' - (N - U) is a
3-cell and h Bd(N _ U) 15 f1xed po1nt free, lett1ng
h"|(N _ U)' _ (N _ U) be the cone over h Bd(N _ U)’

th _ U can be extended to a PL involution h" of (N - U)',

and moreover h" is orientation reversing, since

15

h reverses orientation. Therefore (N - U)'

Bd(N - U)
= M2 must be P3. Hence M1 = M2 = a real projective
3-space P3. This proves the theorem.

Henceforth assume that Mi , i = l, 2, is a
manifold isomorphic to a projective 3-space P3 and h
3 # P3 with a 2-dimensional fixed
point set. Since the case where a 2-dimensional component

is a PL involution on P

is an orientable surface has been taken care of, we have
only to consider the case where each Z-dimensional com-

ponent is non-orientable.

Lemma 2.3: Let h be a PL involution of P3 # P

Then there exists a PL involution h': S1 x SZ+S1 x S2

3.

 

such that the following diagram commutes

I
S1 x 52 h S1 x 82
p' p'
h
P3 # P3 P3 # P3
where p': S1 x SZ+P3 # P3 is a 4-to-1 covering projection.
Proof: Consider the covering space S1 x S2 of

P3 # P3 and the usual 2-to-l covering map p: S1 x 32

_ 1 2 ..
+P3 # P Let H — (S x S ) and G — 1r1(P3 # P3).

3' P#"1
Then the index [G: H] of G modulo H is 2, since p
is 2-to-l. Suppose h#H # H. Since [G: H] = [G: h#H]

= 2, neither h#H contains H nor H contains h#H, and

l6

moreover H and h#H are normal subgroup of G. Let

L = H n h#H. Then L is a normal subgroup of G, since

L is the intersection of two normal subgroups, and

[G: L] = [G: H][H: L]. We now show that [G: L] = 4. Let
H' = h#H. Clearly HH' = G and H/L = H/H n H' = HH'/H'
G/H' = Z2, which shows [H: L] = 2. Consequently [G: L]

= 4. Furthermore, h#L = h#(H n H') = H' n H = L. Hence

by the lifting theorem, there is a PL involution h' on

S1 x S2 such that p'h' = hp', where p': S1 x SZ+P3 # P3

is a 4-to-1 covering projection. This proves the lemma.

We now show that the possible 2-dimensional non-
orientable surfaces which can be fixed under a PL involu-
t1on of P3 # P3

. n .
Cons1der S as a suspen81on of S

are a projective plane and a Klein bottle.

“'1 and let kn be the

simplicial involution of Sn that leaves Sn-l pointwise
fixed and interchanges the suspension vertices, and define
1 2

two involutions hl and h2 of S x S by

hl(X. y) = (kl(x), y), h2(x, y) = (x, k2(y)) .

Kwun [5] considered PL involutions of S1 x S2 and proved

the following two theorems:

Theorem: Let h be a PL involution of S1 x 82

with homogeneously two dimensional fixed point set F. If

F is not connected, then h is PL equivalent to hl'

17

Theorem: Let h be a PL involution of S1 x S2

with 2-dimensiona1 connected fixed point set F and

orientable orbit space. Then h is PL equivalent to h2'

Fremon [2] completed the work and proved that all

possible fixed point sets of a PL involution of S1 x 82

are 82 U 82, S1 x 51, 82 plus two points, and a Klein

bottle. Hence by the Lemma 2.3 we obtain:

 

Lemma 2.4: Let N be the connected sum P3 # P3

and h a PL involution of P3 # P3. Then all possible
2-dimensional, non-orientable components of a fixed point
set F of h are a Projective plane and a Klein bottle.
We shall first consider the case that a projective
plane is fixed and show that, in this case, the fixed

point set F is a disjoint union of two projective planes

and h is uniquely determined.

Theorem 2.5: Let N be the connected sum P3 # P3

 

and h a PL involution of N. If a projective plane is
contained in the fixed point set F, then F is the dis-

joint union of two projective planes.

Proof: Suppose that a projective plane A is con-
tained in F and assume that the simplicial triangulation
of N is such that h is simplicial, the simplicial
neighborhood U of h is an invariant regular neighbor-

hood of A, and h U is fixed point free. Then we

-A

18

have seen that U is homeomorphic to P3 minus a 3-cell

and N - U is homeomorphic to P minus the interior of

3
a 3-cell in Theorem 2.2. Moreover, we have seen that

h N _ U can be extended to an orientation reversing PL
involution h' of P3 obtained from N - U by attaching
a 3-cell. Let F' be the fixed point set of h'. Then
by the parity theorem and the Lefschetz fixed point for-
mula, the dimension of F' is either 0 or 2. But by
work of Kwun [6], the dimension of F' cannot be 0, and
F' is the disjoint union of a projective plane and a point
p. By the way we extended h to h', the cone vertex must
be an isolated fixed point. Hence p is the cone vertex,
and p lies in the 3-cell attached to N - U to obtain

a P Therefore h has a projective plane as the

3' N - U

fixed point set. Using the same argument for h as

U

above, we obtain that F is the disjoint union of two

projective planes. This completes the proof.

Theorem 2.6: Let N be the connected sum P3 # P

and h a PL involution of N. If the fixed point set F

3

 

of h contains a projective plane, then h is unique and
h is of the form h1 # hz, where hi is the unique PL
involution on P3 with a projective plane and a point as
the fixed point set.

Proof: By Theorem 2.5, F is the disjoint union

of two projective planes. Let A and A' be two projective

l9

planes whose union is F. We may assume that h is simpli—
cial and simplicial neighborhood U and U' are invariant
regular neighborhoods of A and A', respectively. Fur-
thermore, assume that U is disjoint from U'. Then U

and U' are isomorphic to P3 minus a 3-cell, and by

the unique decomposition theorem for 3-manifolds,

N - (U U U') is isomorphic to $3 - two Open 3-cells

= 82 x [0, 1] such that h(S2 x i) = 82 x i for

i = 0, l, h N _ is fixed point free, and

(U U U')
h(N - (U U U')) = N - (U U U'). Work of Livesay [7] shows

that if f is any fixed point free involution on

82 x [0, l] satisfying f(S2 x i) = 82 x i for i = 0, 1,
then there exists a homeomorphism t: 82 x [0, l]->S2 x [0, 1]
such that tft"l = 9, where g is the involution on

2 2

S x [0, 1] defined by g(x, t) = (-x, t) for x e S ,

t 6 [0, l] and -x denotes the antipode of x. That is,

there is a unique involution g on 82 x [0, 1] up to PL

equivalences. Hence it suffices to analyse h U and h U"

We now analyse h U as in Kwun [6]. Let O be
the orbit space and let f: 640 be the orbit map. Then
0 is a compact 3-manifold and the boundary components
are f(Bd(U)) and f(A). Let V be a regular neighbor-
hood of f(A) in O disjoint from f(Bd(U)). Let W be

f-l

(V). Then W is a neighborhood of A in U and f
is 2-to-l except on f(A), since h is fixed point free

on U'- A. We triangulate W in such a way that for each

20

simplex s in V, there corresponds two copies of s in
W. By collapsing two corresponding simplices as we collapse
5, we can collapse W to A. Hence W is a regular

neighborhood of A in U, which is disjoint from Bd(U).

Therefore Cl(U - W) is homeomorphic to 82 x [0, l] on

which h is fixed point free. By Livesay [8], the orbit

space of h is homeomorphic to P2 x [0, 1] with

Cl (U - W)

P2 x 1 as f(Bd(U)). Since V is a collar of f(A) = P

V is homeomorphic to P2 x [0, 1]. Therefore O is

2'

homeomorphic to P x [0, 1] such that P2 x 0 and

2

P2 x 1 correspond to f(A) and f(Bd(U)), respectively.

We will construct a PL involution on U and show that for

any two PL involutions h and h2 on U, there exists

1
a PL homeomorphism t: U+U such that h1 = t-lhzt. Con-

sider a PL double covering j: Bd(U) = 82+A. Then there

exists only one non-trivial covering transformation 9 of

j. Let h be the covering transformation 9. Then

Bd(U)
U is a mapping cylinder of j. Hence g can induce a PL
involution on this mapping cylinder in the obvious way.

Let h1 and h2 be any two such PL involutions on U and
let g1 and g2 be orbit maps: U+P2 x [0, 1] correspond-
ing to h1 and h2,
ponding to the fixed point set A. Since

respectively, with P2 x 0 corres-
giU-Aisa
universal covering, there exists a PL homeomorphism t

from U - A to U - A such that gl = g2t. Then t can

be uniquely extended to a PL homeomorphism t: U+U such

21

that g1 = g2t. But th = h t as t respects covering

1 2
translation. This is true on U - A and by continuity

this is also true on U. Hence h1 = t_lh2t and there—
fore there exists a unique PL involution of U up to PL
equivalences.

The same argument applied to h shows that

U'
there exists exactly one PL involution on U' up to PL
equivalences and hlU' is exactly the same type of in-
volution as h'U' Therefore there exists a unique PL
involution on N with a projective plane in the fixed
point set.

Now h = hl # h2’ where hi is the unique in-
volution on P3 with a projective plane and a point as
the fixed point set, since hllU = hlU' hZIU' = hlU" and
hl|P3 _ U (h2 P3 _ U,) is the cone over hl|Bd(U)

(hled(U'))' Th1s completes the proof.

From now on we shall consider the case that F
contains a Klein bottle and shall show that F is the
disjoint union of a Klein bottle and two points. To prove

this we need following lemmas:

Lemma 2.7: Let N be the connected sum P3 # P3

 

and let K be a Klein bottle contained in N and U a

regular neighborhood of K in N. Then n1(N - U) = Z.

22

Proof: Suppose that U is a regular neighborhood

of K in P3 # P3. Consider the double covering p1: NI+N

1

obtained from N by cutting along K. Then p1- (K) is

connected and separates Nl into two components. Since

1

N1 is orientable, pl- (K) is also orientable, otherwise

pl (K) cannot separate N -1 -1(K)+K is

1' Pl p1 (K): Pl
2-to-l. Hence p1-1(K) must be homeomorphic to S1 x 81.

Now consider p1-1(U). pl-1(U) is a regular neighborhood
of p1-1(K), since each component of N1 - p1-1(K) maps
homeomorphically onto N - K, and hence pl-1(U) is a

collar of pl—1

1

(K). Hence each component of the boundary

1

of pl- (U) is homeomorphic to S x S1 and p1 maps

81 x S1 homeomorphically onto the boundary of U. There—
fore Bd(U) is homeomorphic to S1 x 81. By the Meyer-

Vietoris sequence
1 l f — _ g
H2(N) + H1(S x S) H1(U) ®H1(N U) H1(N) + O
we obtain

0+Z®Z£(Z+Z2)®Hl(N-U)g22+22+0

where f is one-to-one and the kernel of g is f(Z C)Z).
Therefore H1(N — U) is a group of rank 1.

n1(N - U) is abelian. For, since S1 x 82 covers
N in 2-to-l fashion, S1 x 82 - a Klein bottle, which is

homeomorphic to R2 x 51, covers N - U 2-to—l. Hence we

obtain an exact sequence

23

0 + Z i fll(N - U) E Z2 + 0

Hence if we choose a e nl(N - U) such that g(a) # 0 and

a generator b in Z, then nl(N - U) is generated by a

and b. Suppose aba-1 = b-l. We abelianize it. Then

1 = aha-lb.l = (aba-1)b—l = bnlb-1 = (b-1)2. Hence

b2 = 1. And a is not of infinite order either, since
g(a2) = g(a)g(a) = 1 implies that azéi Z, and therefore
a2 = l or a4 = l in H1(N - U). Therefore H1(N - U)

is finite. This contradicts the fact that the rank of
H1(N - U) is 1. Hence nl(N — U) is abelian, which

implies

nl(N - U) = Z + Torsion part.

Consider the covering space S1 x 82 of N. Since

S1 x 82 minus a Klein bottle is homeomorphic to R2 x 81,

we have a universal covering

R2 x R1 + R2 x S1 + N - U.

But no non-trivial finite group can act freely on a finite
dimensional, contractible space. Hence nl(N - U) = Z.

This proves the lemma.

Lemma 2.8: Let N, K, and U be the same as in

 

the Lemma 2.7. Then N - U is irreducible and is homeo-

morphic to D2 x 81.

24

Proof: Suppose N - U is not irreducible. Then
there exists a nice 2-sphere which does not bound a 3-
cell. But every nice 2-sphere in N - U is a bounding
2-sphere, and actually every bounding 2-sphere bounds a
3-cell. Let S be any nice 2-sphere in N - U, and

consider the following diagram:

R2xSl<=R3c=S3

7)
.— P
/
l/ l

/ i
S———)N-U

Since w1(s) = O, by the lifting theorem there exists

I: S+R2 x S1 such that i = pf. S separates N - U

into two parts W and V, and

W1(W) * wl(V) wl(W) * fll(V) _

Ill-(WUV) = “1(an) =——-fi-(§-y———1T1(W) * 1(V).

 

But by Lemma 2.7, nl(W U V) = nl(N - U) = Z. Hence n1(W)
= 0, which implies that 1(8) consists of disjoint copies
of S in R2 x 81. Then it can be embedded in 83. By
the theorem of Alexander [1], it bounds a 3-cell and
therefore S bounds a 3-cell. Hence N - U is
irreducible.

Since N - U is an orientable, irreducible, com-
pact 3-manifold with the fundamental group isomorphic to

Z and the boundary is homeomorphic to S1 x 81, N - U is

homeomorphic to D2 x 81. This completes the proof.

25

Lemma 2.9: Consider S1 x S1 as the boundary of

D2 x 81. Let f: S1 x Sl+Sl x S1 be a homeomorphism

1

 

such that f*: nl(Sl x Sl)+1r1(s1 x S ) is presented by

1 q
o 1

using canonical generators for 1T1(Sl x 81). Then f may

be extended to a homeomorphism f: D2 x Sl+D2 x 81.

Proof: By [9], isotOpy classes of homeomorphisms

of S1 x S1 are precisely isomorphism classes of

nl(S1 x Sl,*) (We disregard base points as 1r1(S1 x 81)

is abelian.). Since extendability is an isotopy invariant,

we may suppose that f(eznlt, e2flls) = (e2m'(t + gs),
eZW1s) 2n1t’ 082n1s) = (€2n1(t + gs), pe21ris

).

. Define f(e

This proves the lemma.

Theorem 2.10: Let N be the connected sum P3 # P3

 

and let h be a PL involution of N. If a Klein bottle
K is contained in the fixed point set F of h, then F

is the disjoint union of a Klein bottle and two points.

Proof: Let U be an invariant regular neighbor-

hood of K in N. Then by Lemma 2.8, N - U is homeo-

morphic to D2 x 81. Denote h D2 x S1 = h'. Then either

the fixed point set Fh, of h' is of dimension 0 or 2,

or h' is fixed point free.

26

We shall show that, in the case that dimension of
Fh' is O, the number of fixed points is 2 and shall

rule out the case the dimension of F is 2 and the

hl
case h' is fixed point free.
Case I): dim Fh' = 0. Suppose that h' fixes
x1 , x2 , ..., xk in D2 x S1 and no other point. We

may assume that h' is simplicial with xi as vertices
and that closed stars of xi are mutually disjoint. Let
M be obtained from D2 x S1 by removing open stars of

xi. Then h" = h M is a free involution on M reversing

orientation of each boundary component of M. Then the
Lefschetz number of h" is l + l - k + 0 = 0. Hence

k = 2.

Case II): dim F = 2. Consider the double covering

h!
p: S1 x SZ+P3 # P3. h' n Bd(Sl x D2) = o, Fh' is
1

contained in the interior of S x D2, and hence P—1(Fh,)

Since F

is disjoint union of two 2-dimensional components and

p-1(Fh.) must be a fixed point set of a PL involution of

S1 x 82. By Fremon [2] p-1(F is the disjoint union

h')
of two 2-spheres, which implies that Fh' is a 2-sphere.

Since S1 x 02 is irreducible by Lemma 2.9, Fh' bounds

a 3-cell. But the other side of F cannot be a 3-cell,

h!
and therefore h' will not be able to interchange those

two parts. Hence the dimension of F cannot be 2.

h!

27

Now in order to rule out the case Fh, is fixed

point free, we need to see how the boundary S1 x S1 of

U is attached to the boundary S1 x S1 of 81 x02 = N - U.

1 1 l 1

Hence let f: S x S +S x S be the attaching map and let

a and b be the canonical generators of 1rl(S1 x 81)

of S1 x S1 covering K and a and B the canonical
generators of n1(Sl x 81) of S1 x S1 contained in
S1 x D2. Let p: S1 x Sl+K be the covering projection

and i: S1 x Sl+S1 x D2 the inclusion. Suppose f#(a)

ra + 38. We may assume that the

= pa + qB and f#(b)

determinant

II
P

Since the covering projection p takes a to a

and b to b2 with relation bah-1a = 1, and the inclu-

sion i takes a to c and B to l, we have

—1
22 * 22 = n10?3 # P3) = {a, b, c | bab a = 1, a = cp,
b2 = Cr}
with generators a, b, c and relations bab-la a l, a = cp,
and b2 = cr. Since bcpb-lcp = l and b2p = crp' crp

= h(bZPm'l = 1:>(<:P)rb'1 = (bcpb‘l)r = (c'P)r = c'rp. Hence

c2rp = l.

28

Case (1): The order of c is finite. Since a = cp, the
order of a is also finite. But actually the order of a

is infinite. For, consider the subgroup {a} generated

by a. Since bab-1 = a-1 and a = cp commutes with c,

{a} is a normal subgroup of Z * Z Therefore we obtain

2 2°

0+{a}§Z*Z g2

2 2 * Zz/{a} + O

2

which is exact. Hence 22 * 22/{a} is presented by

{a, b, c I bah-1a = 1, a = cp, b2 = cr , a = l}

{b, cllob’l = 1, cp = 1, b = c

{b, chP = 1, b2 = cr} .

Since ps - rq = l, cpsc-rq = c. On the other hand,
cpsc-rq = (cp)sc-rq = c-rq = (cr)-q = (b2)-q = b-Zq.
Hence b-2q = c. Therefore 22 * Zz/{a} is generated by
'5, where '5 is the image of c under the projection.

Since 22 s 22 3 5' = 22 * ZZ/{a} can be factored

through Z2 + 22, we obtain

* ZZ/{a} = {E}

Z2
%

where g' is onto. Hence {5} = 0 or Z

22 s 22 -—Ji——9
22 + 22

2. But {0}
cannot be trivial, since Z2 * 22 # {a}, which implies

{E} = 22 is finite. Consequently, {a} must be infinite,

29

which shows that the order of a is infinite. Therefore

c cannot be of finite order.

Case (ii): The order of c is infinite. Since c2rp = l,
rp = 0, which implies either- r = 0 or p = 0. But p

cannot be 0. For, if p = 0, then we would have

Hence we could define a homomorphism y: Z2 * ZZ+Z as

follows:
y(b) = r and y(c) = 2,

which is impossible. Therefore r = 0, and consequently
we obtain that p = s = 11. We may further assume that

p = s = l, and have that f# is presented by

1 q

0 1

l as the double covering space

Considering S1 x S
of K and t as the non-trivial covering transformation
of the covering projection, t#: fll(Sl x Sl)+1rl(S1 x 81)

is presented by

30
and we have the following commutative diagram

-1 0

0 1
“1(31 x 51) ———-> n1(sl x 51)

l q 1 q
0 l 0 l
nl(Sl X Sl) -—j;r———?'"l(sl X 51)
where h': S1 x Sl+S1 x S1 is a PL involution of S1 x S1
l 2

considered as the boundary of S x D . Consequently, h'#

is presented by

and by Lemma 2.9, h' can be extended to a PL involution

h' on S1 x D2, and h'#: n1(Sl x D2)+1r1(Sl x D2) sends
l to -l, which implies that the Lefschetz number is not
0. Therefore h' cannot be fixed point free.

Hence F is the disjoint union of a Klein bottle

and two points. This proves the theorem.

Remark: The uniqueness question for h in case F
is the disjoint union of a Klein bottle and two points is

not settled. If h is unique, then h = h1 # h2°

31

For, let Fh be the disjoint union of a projective plane
1

Ai and a point pi which is the fixed point set of hi'

Taking the connected sum P3 # P along invariant neighbor-

3
hood of ai, where ai 6 A1, we obtain a PL involution

h1 # h2 whose fixed point set F is the disjoint

hl # h2
union of a Klein bottle and two points (A1 # A2 U {p1, p2}).

BIBLIOGRAPHY

[l]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

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Thesis, Michigan State University, 1969.

M. Hall, Jr., The theory of groups, Macmillan, New
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K. W. Kwun, Non existence of orientation reversing

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SOC o p — o

, Piecewise linear involutions of S1 x 82,

 

Midfi. Math. J. 16(1969), 93- 96.

, Scarcity of orientation reversing PL in-
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G. R. Livesay, Fixed point free involutions on the

 

3-sphere, Ann. of Math. 72(1960), 6034611.

, Involutions with two fixed points on the

 

three sphere, Ann. ofiMath. 78(1963), 582-593.

 

W. Mangler, Die Klassen von topologischen Abbildungen

 

einer geschlossenen FlaEhe auf sich, Math. Z.
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J. Milnor, A unique decomposition theorem for 3-
manifolds, Amer. J. of‘Math. 84(I962), l- 7.

 

44

L. Neuwirth, In groups and imbeddings of n—compleces

 

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' O

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York, 1966.

32

 

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