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‘ 800K BINDERY 19.1”

ABSTRACT

INVOLUTIONS OF 3-MANIFOLDS WITH A
2-DIMENSIONAL FIXED POINT SET COMPONENT

BY
Donald K. Showers

Let T1 and T2 be involutions of the 3-manifolds

M1 and M2 respectively and assume T1 and T2 have
2-dimensiona1 fixed point set components F1 and F2.
Taking the connected sum of F1 and F2 in the connected
- sum of ’ M1 and M2 gives a manifold M1 # M2 with an
induced involution T1 5“ I} and a fixed point set
component F1 5? F2. The question studied in Chapter I

of this thesis is the converse of this construction.

It is found that under certain conditions it is
possible to detect that a manifold M.‘with involution T
can be constructed as a connected sum of two other manifolds
with involution by finding a non-zero kernel of the in-
clusion map of a 2-dimensional fixed point set component
of T in homotOpy. Thus the inclusion of 2-dimensional
fixed point set components into an irreducible manifold is
a monomorphism in homotoPy. This allows for classification

of these involutions of S1 x S1 x S1 and of an investigation

1

of S x K where K is the "Klein bottle".

INVOLUTIONS OF 3-MANIFOLDS WITH A
2-DIMENSIONAL FIXED POINT SET COMPONENT

BY

Donald K?'Showers

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1973

Mr.

Dedicated to

and Mrs.

W.F. Showers

. v; -r
l . i

ACKNOWLEDGMENTS

The author wishes to acknowledge his gratitude

to Professor K.W; Kwun for his helpful suggestions and

guidance.

In addition the author wishes to thank Professor

D. E. Blair for his contributions to the authors geometry

background.

The author also thanks Professor L. Sonneborn

and Professor W. Sledd.

ii

TABLE OF CONTENTS

Introduction . . . . . . . . . . . . . . .
Chapter I

BASIC DEFINITIONS AND FIRST THEOREM .
Chapter II

APPLICATIONS . . . . . . . . . . . .

iii

INTRODUCTION

The techniques involved in dividing mathematical
Objects into basic components are of fundamental interest
in mathematics. In this thesis the objects are piecewise
linear 3¥manifolds with an involution having a 2-dimensional
fixed point set component. The result of a general
investigation indicate that under suitable conditions one
can detect whether an involution can be considered as induced
from the connected sum of two involutions by the kernel of
the inclusion of a 2-dimensional component of the fixed
point set in homotopy. The results are applied to
S1 x S1 x S1 and to S1 x K 'where K is the "Klein

bottle". In addition, uniqueness questions are answered.

CHAPTER I

BASIC DEFINITIONS AND FIRST THEOREM

'In this work all tepological manifolds are
assumed to have a piecewise linear structure and all
continuous functions are assumed to be piecewise linear

functions unless stated differently.

Let Mn denote a closed manifold of dimension n.
An involution on M? is a continuous function T not
equal to the identity function which maps Mn to MD
such that T 0 T is the identity function on Mn. The
set F(T) will denote the set of fixed points of T,
F(T) is given the subspace tOpology which makes the
components of F(T) manifolds without boundary. A
n-sphere is said to be the trivial n-manifold since
M9 = M3 ﬁ¥ Sn. A manifold Mn is said to be prime if
it cannot be written as a connected sum of two non-trivial
manifolds. J. Milnor has shown [ 3] that if n = 3
every closed orientable manifold is the connected sum of
prime manifolds and every such decomposition is unique up

to a permutation of factors. Among prime 3-manifolds
-l 2

S x S is unique in that it contains an embedded 2-sphere

 

 

A

which does not bound a 3-cell, the property that every

.embedded 2—sphere bounded a 3-cell is called irreducible.

Let M and M be two 3—manifolds on which

1 2
involutions T1 and T2 have 2-dimensional fixed point
set components F1 and F2 respectively. By removing
"small" invariant 3-cells c1 and c2 such that

aci n Fi a s1 i = 1,2 and noting that Tl|ac1 is

isotopic to Tzlac2 through any orientation reversing
identification of acl with acz it is possible to induce
an involution on the connected sum of M1- and M2 which

has a fixed point set component Fl ﬁ¥ F2. This construction

of an involution is denoted by T1 #4 T2 since it

F1#F2
depends on the choice of F1 and F2. Note
F(T1 #P1#F2 T2) = (F(T1)-Fl) u (F(T2)-F2)

u (Fl 7,4! F2).

The purpose of this chapter is to show that under
reasonable conditions involutions on orientable manifolds
induced by the above constructions, can be detected by a
non-trivial kernel of i*Wl(Fi) 4‘n1(M?) where Fi is

any 2-dimensiona1 component of F(T).

Theorem 1. Let M be an orientable closed 3-manifold

with T an involution on M. Suppose S1 x 82 is not a

 

connected sum summand of M and assume further that there

is a 2-dimensional component FC # projective 2-space

such that i*n1(Fc) 4 w1(M) is not a monomorphism, then
M is a connected sum of non-trivial manifolds and T is

induced.

Proof: The proof is divided into several cases:
Case A: PC is two sided:

Case 1: FC is two sided and separates M.

In this case FC separates M into two components

M and M 'with 6M1 ==Fc a 8M2. By the vanKampen theorem,

1 2

i*v1(FC) 4'nl(Ml) i = 1,2 is not a monomorphism. Now
since T‘Ml : M1 4 M2 there can be no other fixed points
and one has that T is a reflection across Fc' However,
using Stallings 100p theorem [‘1] and Dehn's lemma there
exists a disc D in M1 and TD in M2 such that

[5D] # e E wl(FC). Adjoining M1 to M2 along Fc gives
D U TD an invariant 2-sphere S ‘which is non-bounding
and thus since there is no 51 x 52 summand S separates
M into two non-trivial manifolds M1 and M2. "Gluing"

a 3-cell to M along 8' and extending the involution by

1
extending the reflection on S and doing the same to M2
yields two non-trivial manifolds which induce by connected

summing the manifold M and involution T.

Case 2: Fe is two sided but does not separate.

By Stallings [£3] there is a loop a in FC
which bounds a disc D such that FC er = a. By
general position arguments assume D 0 TD = U Si' Si

an embedded l-sphere termed an intersection circle. It
will now be shown how D can be modified to give a disc
D’ such that D’ 0 PC = a and D’ 0 TD' = a. Let B

be an intermost circle in D of D n TD .
Case A: TB n B = ¢L

Near TB choose a "small" regular neighborhood
N of TB in D. The outer rim gives a curve B’. There
is a disc 5 in M "close" to that bounded by B in D
considered embedded in M such that TaD = B’. Form the
disc D’ 'by removing the interior of the disc in D which
is bounded by B' and attaching TD. Then D’ 0 TD’ has

one less intersection circle than D n TD.
Case B: TB n B = B.

Consider a small annulus N of B in D considered
in the zero section of a regular neighborhood of D, embedded
3

in R . TN either lies on one side of N or it does not.

Subcase 1: TN lies on one side of N. Then on
the other side one can put in an annulus N sufficiently

close to N such that N n N = 6N, and B is in the

 

annulus N in N bounded by 5N: Removing the interior
of N and "gluing" in NI gives an annulus N’ 3 TN’ 0 N’ = ¢.

Thus B is removed as an intersection circle.

Subcase 2: TN lies on both sides of N. Then

change the disc by using TD in place of D when DB is

B {3
the disk in D bounded by B. Now use subcase 1.

Thus it is possible to modify the disc D to eliminate
all intersection circles, so assume D has no intersection
circles. Note that D U TD is now an invariant sphere
and that it does not bound a 3-cell since 5D is not null
homotopic in Fc' Proceed as in case 1 to show the involution

is induced.
Case B: Fc is one sided.

Subcase 1: FC is the only 2-dimensional fixed point
set component of F(T). Opening M along Fc gives a
manifold M"with boundary N, which double covers FC. The
hypotheses that FC 5? P2 implies that there are no elements
of order 2 in wl(Fc), so by the Van Kampen theorem applied
to an invariant regular neighborhood N and a fattened
complement, one has i*w1(aN) a‘nl(M-N) a non-monomorphism.
Thus i*F1(N) 4‘n1(ﬁl is not a monomorphism. Removing
any isolated fixed points if necessary, one has an

induced involution T : M 4 M which is free. The orbit

space is.a manifold OM. with boundary OB and
1*V1(OB)1:‘F1(OM) is not a monomorphism. Thus one has

a disc D' with boundary aD' which is not null homotopic
in OB. Lifting to M—N gives two discs D and TD.
Using the techniques of case 2 above it can be assumed that
D FITD = O. Now "sew" up M along N’ to give M and a

non-boundary invariant 2-sphere. Proceed again as in case 1

to show that involution is induced.

Subcase 2: PC is not the only two dimensional
fixed point set. Let FK K = l,...,n be the 2-sided
2-dimensional fixed point sets. By case 1 Fi does not
separate. Open M along F1 to obtain lMO alMo = OFl U OF2
where OFl 5 F1 e-OFZ. Form 1MO x Z and let
1M'= 1M0 x z/(x,i) fk§0F1 (Tx,i+l) where T, is the induced
involution on 1M0. .M covers M and has an involution lT'
extending T, and covering the involution T on M. IT, has
one less 2-dimensional fixed point set component and a
neighborhood of PC can be lifted homeomorphically to a
neighborhood of the fixed point set covering FC also
denoted by PC. Note i*W1(FC) 4‘Flﬁ is not a monomorphism.
One could continue this construction removing each Fc .as
a 2—sided fixed point set component of an involution on an

open manifold M ‘with an involution T covering M and T

such that a neighborhood of the one sided fixed point set

components lifts homeomorphically the neighborhood of the

one sided fixed point component sets of M. Now open M

along the one sided components Fd' d = l,...,£ Fd # PC.

This gives a manifold ﬁd' 5M5 kzu aNFd where NFd is

a regular neighborhood of Pd. Consider

= id x Zz/(XxO) +(T(x) x l) where x e U aNFd, M covers

and has an involution T covering T induced by

In?! 3"

N

T x identity on M.d x Z2. T has two Z-dimensional fixed
point sets ch, 2Fc each p.l. homeomorphic to FC and
under the covering projection a neighborhood of ti i = 1,2
is homeomorphic to a neighborhood of B3. Now i*waNch 4
w1(M - “El (Nti)) is not a monomorphism. Hence repeating
the argument in subcase l to get two loops 5’ and T&' which
project to 2-loops in M - NFC a and Ta and two discs D
TD bounding a and Ta. Modify if necessary to get

D n TD = ¢ and a non-boundary invariant separating 2-sphere.

Corollary_l. The inclusion in homotopy of any

 

2-dimensional component # P2 of the fixed point set of an

involution T on a irreducible orientable 3—manifold is a
monomorphism.

Use of this corollary is made in Chapter II to

consider involutions of S1 x S1 x 81.

Corollary. 2. If PC is a 2-dimensional fixed
'point set of an involution of Ml #! M2, M1 and M.2

are orientable 3-manifolds with the conditions of theorem
1 and 1f ‘n1(FC) cannot be a subgroup of W1(Ml) * wl(M2)
then the involution is induced and Fc is induced as a

connected sum between two manifolds not necessarily M1 and M2 .

As an example of the last corollary, consider
P3 a? P3 which can have the "Klein bottle" K has a
fixed point set [4 ]. Since wl(K) is not a free group
and has no elements of order 2, the Kurosch subgroup
theorem says w1(K) cannot be a subgroup of 22 * 22' thus
i* : w1(K) 4‘nl(P3 #! P3) = Z2 * 22 is not a monomorphism
and by Theorem 1 the involution is induced by connected
sum from the unique involution on P3 with 2-dimensional

fixed point set component. This answers the uniqueness

question of K as a fixed point set component in P3 #4 P3 [4-].

One can continue examples with the Kurosch subgroup
theorem in investigating 2-dimensional fixed point sets

in connected sums of two spaces and 3-toruses.

It has been recently shown by J. Tollefson that
Theorem 1 generalizes to non-orientable manifolds. Such
manifolds may have a one—dimensional component in F(T) as
well as a two-dimensional component. The proof of Theorem 1
above uses orientability to conclude the non existence of 1-

dimensional fixed point set components.

CHAPTER II

APPLICATIONS

In this chapter the results of Theorem 1 will
be used in considering involutions of S1 x S1 x S1
which have a 2-dimensional component in the fixed

point set. By P. Conner [2 ] the fixed point sets of

such involutions are S1 x S1 and S1 x S1 U S1 x 81.

Examples of these are h : S1 x S1 x S1 4 S1 x S1 x S1

'l l l

. = . x
given by h(zl,z .23) (22.21.23) and K . S S x S 4

2
l l l . — .
S x S x S given by K(zl.z2,z3) — (21.22.23) respectively
where of course the zi are complex numbers such that
|zi| = 1. It will be shown that these are the only

examples up to conjugation.

Lemma. If S1 x S1 is a fixed point set component

of an involution T : S1 x S1 x S1 4 S1 x S1 x S1 then

S1 x S1 is a retract of S1 x S1 x 81.

Proof: First note that S1 X S1 does not separate

S1 x S1 x 81 since from Theorem 1, i*v1(Slel) 4 nl(Slelxsl)

is a monomorphism. but w1(Slxsl) = H1(Slxsl), wl(Slxslel) =

H1(Slx51xsl) letting S1 X S1 = T2 S1 x S1 x S1 = T3 so

 

H1(T3,T3) has rank 1. From the universal coefficient

10

\I

ll

theorem rank H1(T3,T2) = l and since T3—T2 is open in
T3 it is orientable thus from Lepschetz duality the

rank of H2(T3-T2) is 1, but it is shown in Theorem 1
that if T2 separates T3,~ it separates T3 into two

homeomorphic components thus rank H2(T3-T2) is even.

This contradiction shows T2 cannot separate.

Now open T3 along T2 by removing a small
invariant regular neighborhood of T2. This gives a
manifold Mr‘with two boundary components 51,52 each
homeomorphic to T2 and i*v1(ai) 4 n1(M) is a monomor-

phism for i = 1,2. Also M, has an involution T’ such

that
T
M 4 M
U
p135;

notice that T, carries al to 52. Preform the following

construction.

First form .M x Z, then using the relation

x x 1 ‘ﬂ- T(x) x (i+1) with x e 81 the quotient space

ﬁggzg is a connected manifold denoted by M. M covers

T3 and as such w1(M) is an abelian group of rank O,l

Or 20 Let Zn = {-n,-n+l,...,-l,0.1,..o,n-1 ] then

froming M'x zn with the relation x x i-xe Tkx) x (i+1)

12

ﬁxZ
with x e 81 and considering the quotient space n

as Mn one has 'nl(M) a lam w1(Mh). By the Van Kampen
i n

N * ~
theorem, wl(Mn) 4 71(M

n+1) is a monomorphism and vl(M1)

is non-abelian unless the inclusion of the boundary and

a1
52 into M, is an epimorphism but w1(Mi) survives in

W1(M) which is abelian thus the inclusion of 31,52 into

~

M is an isomorphism in homotopy. So now by Brown [1.]

M =- T2 x I and one has
p.l.
T2 xI '3 T2 )(I
v P
Ki ’5. Si

consider the relation x x Oc§ T(x) x l and form the
2
quotient to give K = E—ll .

8?

Lemma. K =- T2 x S1 iff T#: W1(T2xI) 4 1T1(T2xI)

is the identity.

Proof: More generally let M be any manifold with
T a p.l. homeomorphism from M to itself. Consider ‘Ell

where x x O T(x) x 1. Without loss of generality assume
X‘xI
O

 

xv = T(x0) and let

0 = a. One has now

’\l

i
O 4Trl(M) 4# Wl(%£) 47Tl(Sl) 40

the claim is i#T#B = [a]—li#B[a] for every B e W1(M).

13

Let DS be the deformation retraction of M x I
to M x [1} given by DS(m,t) = (m,T-S(l-t)) and let

B be a map of I into M such that B(O) = x0 = B(l).

Consider the following homotopy h(T,S) : I x I 4

M.x I given by

. 1
h('r,s) — (x0,3st) 1f 3 s 2 t
h(t s) = (Tong-13:3) s) if 1s < t and - l 5+1 > t
' 3-2s ' 3 — 3 —
and h(t,s) = (xo,3s(l-—t)) if - 31- s+1 _<_ t

Now T°B(3t-l,l) is identified with B(3t-l,a) so
i#T#B = [a]-1° i#B o [a].

In the case of the lemma if K e-Tz x S1 then

1T1(K) lS abelian so 1#T#B = 1#B => T#B = identity and
if T# is the identity then [a] does not act on 1r1 (K)
so 0 4 2+2 4 nl(K) 4 Z 4,0 splits and W1(K) = Z+Z+Z so

K must be T2 x S1 using Stallings fibration theorem [9 ].

Thus the lemma allows classification of the involution

of T2 x S1 with a 2-dimensional fixed point set component

by considering actions T on T2 x I such that T : T2 x O 4

T2 x 1 and T#: 1r1(T2xI) 4 Trl (szI) is the identity.

Now suppose that T2 is the only fixed point set,

this implies that for classification of involutions with

T2 as fixed point one need only consider orientation

14

~ ~

reversing free involutions T of T2 x I with g¥ the
identity and considered as 22 action on wl(T2), T; is
the trivial action. Let Q be the quotient space of

T1 x I under Ti since T; is trivial one has 0 4 ZxZ 4
wl(Q) 4 z2 4.0 with w1(Q) = Z+Z+Z2 or Z+Z. The
universal covering of Q is a contractable space and

there are no finite fixed point free actions on a contractable

space. Thus W1(Q) = Z+Z.

Thus Q is a 3-manifold with one boundary component
50 which is included monomorphically into Q. By Stallings
[ 9] Q must be S1 x M. where M is the closed Mobius

strip.

Now, if there is another free 22 action T2 on
S1 x S1 x I which carries al to 52 with Tzﬁg = identity,

then letting the orbit map be P2, one has the following

diagram:
SlxslxI p154 SxM é—p—2-— SlelxI
i1} i3} i2?
51 ————> S1 x M <———-- 51
pl/a1 pz/al

11* = identity = 12*

15

pl/al* and pz/al* are isomorphisms.

1

1 .
p2*(S x S x I) and thus p1 and p2

the covering translation theorem. Thus

T2 as a fixed point set are conjugate.

Now assume that T has T2 as

2 2

Thus pl*(Sl x S1

x I)
are conjugate by

any Z2 action with

a fixed point set

of T': T x I 4 T x I. i* : wl(T2) 4 n1(T2xI) is a

monomorphism and hence by a similar argument to the above,

2

2 . . ~
T separates T x I into Ml,M2 ‘w1th T . M1 4 M2 a

homeomorphism. By the Van Kampen theorem, i* : F(Tz) 4‘F(Ml)

. 2 . .
and 11* . wl(T ) 4‘n1(M2) is a monomorphism. But

2
1r1('1‘ XI) " 1riml) *z+z

1T1(M2) and T*i*

= 1* so 1* must

be an epimorphism. Thus i* 1T(T2) 4 1r1Ml is an isomorphism

It is possible to switch the ordering of selection of the

two fixed point set components to get that the other boundary

component of M1 induces by the inclusion an isomorphism

in homootpy. Thus by Brown [ 1], again

M1 = T2 x I and

M: = T2 x I and T is conjugate to a reflection.

It has thus been shown

1 l 1

Theorem 2. If T is an involution of S x S x S

 

with one 2-dimensional component it is conjugate to

1 l l 1 1 l

h : S X S x S 4 S x S x S given by h(Z1oZ2oZ3) =

(z2,zl.z3) and if T has two 2-dimensional components

then T is conjugate to K : S1 x S1

l 1 l l

x S 4 S x S x S

16
The classification of the involutions of S1 x S1 X S1
‘with 2-dimensional fixed point set components allows for an
investigation of S1 x K where K is the “Klein bottle".
Suppose K is a fixed point set component of an involution

- 1 1* 1 . .
T on S x K. Now wl(K) 4 'nl(S xKﬂ is a monomorphism

thus lifting to the S1 x S1 x S1 covering space gives a
torus T2 covering K. T2 is 2-sided and the covering
translation on T2 reverses orientation since K is not.
orientable. Consider a two-sided invariant saturated regular
neighborhood of T2, say T2 x [-l,l]. With the fixed

point set T2 x O. The claim is that under the covering

2 x [0.1] goes to T2 x [0.1]. If not

translation h T
the regular neighborhood N of K is S x K ‘would be
orientable. However, the image of all Open connected
saturated sets must be all nonorientable or orientable and
since S X K is nonorientable, N must be nonorientable,
this contradiction yields that h(T2 x [0,1]) = T2 x [0,1].
Hence K is two-sided in S x K. Open S x K along K

to Obtain a manifold M: Note

T2 4 T2x I
1 1
i 4 if

yields immediately by Brown [11] that M’ =- K X I.

17

The position of the investigation is similar to Theorem 2.

Theorem 3. If K is a component of the fixed
point set of an involution S x K then T is conjugate
to either

i) a reflection x identity on S x K;

ii) the induced map on S x K gotten by using
the covering translation T of the double
covering of K by itself. Forming
T x 1-t : K x I 4 K x I and identifying
(x,0) with (E'(x),1). Since 5 is
fixed point free Ta“ = identity, so the

identification gives S x K.

Proof: It has already been noted that K is
2-sided and the complement of a regular neighborhood of
K is homeomorphic to K x I which is covered by
S x S x I. The involution T on K x I generates an
involution T' on S x S x I. The claim is that

1%3 = identity and thus T' is induced from an invOlution

on S x S x S. Having

N

T#
WIS x S x I 4 WIS x S x I
{Pig 1P2;
T
1T1(K x I) 47‘! TrlK x I
identity

substantiates the claim.

18

Now in Theorem 2 it is shown that S x S x I has

2 involutions which are candidates for TﬁZ' One has
T2 as a fixed point set and T, is a reflection. T2
covers a Klein bottle in K x I and T is a reflection
giving case i. The other case is where T, and hence T
is a free involution. Now as in the proof of Theorem 2,
only the orbit space need by unique. In this case calling
the orbit space Q. By Scott [ 7] Q is a line bundle
over a Klein bottle. However, by Quinn [ 6] and the fact
that Q has K as a boundary Q is the unique mapping

cylinder of the double covering of K by itself.

Unfortunately there are involutions of S1 x K with
T2 as a component of the fixed point set so a complete
classification of involutions on S1 x K cannot be claimed

at this time.

 

BIBLIOGRAPHY

 

 

+WMK‘IA.HIQL _-_.: ~°"'-. ' -

BIBLIOGRAPHY
E.M. Brown, "Unknotting in M2 x I", Trans. Amer. Math.
Soc., 1966, p.480-505.

P.E. Conner, "Transformation groups on a K(w,l) II",
Mich. Math. Journal, Vol. 6, 1959, p.413-417.

J. Milnor, "A unique decomposition theorem for 3-manifolds",
American Journal of Math., Vol. 84, 1962, p.l-7.

Myung M. Myung, Thesis, M.S.U., 1970.

C.D. PapakyriakOpoulos, "On Dehn's Lemma and asphericity
of knots", Annals. of Math., Vol. 66, 1957, p.l-26.

Joan Quinn, Thesis, M.S.U., 1970.

P. Scott, "On sufficiently large 3-manifolds", Quarterly
Journal of Math., Vol. 23, 1972, p.159—172.

J. Stallings, "On the Loop Theorem", Annals. of Math.,
Vol. 72, 1960, p.12-19.

"On fibering certain 3-manifolds", Topology

 

of 3-manifolds and related topics, Prentice-Hall
1962, p.95-100.

l9