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PRSPERNES OF 3-EXMWF8 BS WHECH
ADM? A FREE CYCLiC GRGUP AC‘HON

Thasis for the Degree of Ph. D.
MICHIGAN STATE UM‘JERSETY
JEFFREY LYNN TGLLEFSGN
1968

 

LIBRARY :5
Michigan State E

University r;

THESIS

This is to certify that the

thesis entitled

PROPERTIES OF 3-MANIFOLDS
WHICH ADMIT A FREE CYCLIC GROUP ACTION

presented by

Jeffrey Lynn Tollefson

has been accepted towards fulfillment
of the requirements for

_P_b_.2.__degree infighmtics

B6 1' professor

 

 

 

 

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ABSTRACT

PROPERTIES OF 3-MANIFOLDS
WHICH ADMIT A FREE CYCLIC GROUP ACTION

by Jeffrey Lynn Tollefson

This thesis is a study of the nature of compact 3—mani—
folds which admit a free action by a finite cyclic group
Zk of order k. Some interesting results are obtained on

manifolds M for which there exist a k-sheeted covering

 

map p: M > M.

A proper Zk action is one with the property that a
generator of the action is homotopic to the identity. The
orbit space M/Zk is denoted M*. The main results center

around the following theorems.

Theorem 1: Let M be a compact, connected, orientable,

 

irreducible 3-manifold with Ed M either empty or connected.
If M admits a proper free Zk action, for some prime k 2.2,
such that H1(M*;Z) has no element of order k then M

can be fibered over the circle.

Theorem 2: Let k 3,2 be any integer. A closed, con-

 

nected, non-prime 3-manifold M is a k-sheeted covering of

itself if and only if M = P3 # P3.

Theorem 3: Let k 2:2 be any integer, and let M be

 

a compact connected 3-manifold with connected boundary. If
M covers itself k times then M is a A-prime irreducible

manifold and Ed M is either 81 x 81 or a Klein bottle K.

PROPERTIES OF 3-MANIFOLDS

WHICH ADMIT A FREE CYCLIC GROUP ACTION

BY

Jeffrey Lynn Tollefson

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1968

ACKNOWLEDGMENTS

The author wishes to express his sincere gratitude to

Professor K. W. Kwun for suggesting the problem and for his

stimulating guidance during the research.

ii

To Carolee

iii

ACKNOWLEDGMENTS . .

INTRODUCTION . .

TABLE OF CONTENTS

Notation and Terminology

CHAPTER

I.

II.

III.

3-MANIFOLDS WHICH ADMIT FREE Z ACTIONS
3-MANIFOLDS THAT COVER THEMSELVES

EXAMPLES OF PROPER Z

BIBLIOGRAPHY

k

ACTIONS

iv

k

Page

ii

22
37
48

IN TRODUCTI ON

This thesis is an investigation of compact connected
3-manifolds which admit a free action by a finite cyclic
group. The study was motivated by K. W. Kwun [7] who con-
sidered the class of closed connected orientable 3-mani—
folds (without boundary) which double-cover themselves.
Kwun classified all non-prime manifolds in this class and
showed that the prime manifolds, with certain technical
restrictions, fiber over the circle in the sense of Stal—
lings [18].

3-Manifolds that fiber over the circle have some very
nice properties. Neuwirth [12] has shown that closed ir-
reducible 3-manifolds fibering over the circle are com-
pletely classified by their fundamental group. Recently
Burde and Zieschang [2] have shown a similar result for
irreducible 3—manifolds with boundary. In particular, such
manifolds are completely classified by their fundamental
group and a peripheral system. Moreover, if a 3—manifold
M fibers over the circle then it can be obtained from the
product T x I of a compact surface T and the unit inter—
val I by identifying (T x 0) and (T x 1) by a homeo-

morphism h of T. We write M = T x I/h.

2

Notation and Terminology

A continuous function between two topological spaces
will be called a map. A topological group G is said to

act effectively on a space M if G is a group of homeo-

 

 

morphisms of M onto itself, (g,x) > g(x) is a map

 

G x M > M, and g(x) = x for all x e M implies g = e,

the identity element of G. The orbit of x is the Space

 

G(x) = {g(x)|g e G}. Two orbits are either equal or dis-

joint so that M is partitioned by its orbits. Let M/G

 

denote the set of orbits and W : M > M/G the projec-
tion assigning each point of M to its orbit. The orbit

£2323 of M with respect to G is M/G together with the
quotient topology, i.e. the largest topology such that v is

continuous. The subgroup GX = [g e G|g(x) = x} is the

isotropy group at x. An effective action of G on M is

 

said to be free if Gx = {e} for all x e M.

An n—manifold (n-dimensional manifold, n.fi 3) is a

 

connected separable metric space, each of whose points has

a closed neighborhood homeomorphic to a closed n-cell. We
consider both manifolds with boundary and manifolds without
boundary. A closed n-manifold is a compact n-manifold with-
out boundary. In view of the work of Bing [1] and Moise [11]
we may suppose without any loss of generality that an n-mani-
fold (n fi.3) has a combinatorial triangulation whenever

convenient.

3

We say that a set Q in a manifold M is page if there
is a triangulation of M and a homeomorphism of M onto
itself that throws Q onto a subpolyhedron of M. A 3-mani—
fold M is irreducible if every tame 2-sphere in M bounds
a 3-cell. The connected sum M1 # M2 of two closed 3-mani-
folds is obtained by removing a tame open 3-cell from each
and then matching the boundaries of the resulting spaces by
a homeomorphism (orientation reversing if both manifolds
are orientable). The Sphere S3 serves as identity element.
A 3-manifold is non-trivial if it is not homeomorphic to S3.
A closed non-trivial 3-manifold is pgimg if it cannot be
written as the connected sum of two non-trivial manifolds.

For manifolds with boundary we define a similar opera-
tion. The disk sum M1 A M2 of compact 3-manifolds with
connected boundary is obtained by pasting a tame closed 2-cell
on the boundary of M1 onto a tame closed 2-cell on the
boundary of M2 by a homeomorphism. The closed 3-cell D3
serves as the identity for the A operation. A 3-manifold
with connected boundary is A-prime if it cannot be written

as the A-sum of two manifolds different from D3.

CHAPTER I

3-MANIFOLDS WHICH.ADMIT FREE Z ACTIONS

k

In this chapter we show that under certain conditions
orientable 3-manifolds admitting a free Zk action fiber
over the circle. Zk will denote a cyclic tOpological

group of order k with the discrete topology.

Lemma 1.1: Let M be a compact orientable irreducible

 

3-manifold such that Ed M is either empty or connected.
If either 71(M) is infinite or the genus of Ed M is
positive then M is a K(v,1) space and w = v1(M) has

no elements of finite order.

Proof: If the genus of Ed M is greater than zero,
it is easy to see from the exact homology sequence for the
pair (M, Bd M) that the rank of H1(M;Z) is greater than
zero. Hence 71(M) is infinite under either hypothesis.
But, if W1(M) is infinite, the universal covering space
M, is not compact. Thus by Poincare duality we have
H3(MEZ) = H2(M}Z) = 0. By the sphere theorem of Whitehead
[20], W2(M) = 0 since M is irreducible. The Hurewicz
theorem then implies that M. is contractible. Therefore
M is a K(w,1) space, i.e. W1(M) = W and wi(M) = 0
for i.: 2. It follows from a theorem due to P. A. Smith

[4, page 287] that W1(M) has no elements of finite order.

5
It will be convenient to introduce some basic con—

cepts now that will be used in this chapter. A bundle

 

Q is a triple (E,p,B), where p : E > B is a map, with
the property that there exists a space D such that for
every b e B there is an open neighborhood U of b in

> p-1(U) with

 

B and a homeomorphism ¢U : U x D

p¢U(u,d) = u, u e U, d e D. For each b e B, the Space

p'1(h) is called the fiber of the bundle over b. A

 

bundle map E : C > C', where Q' = (E', p', B'), is a

> B” such that h carries each fiber in E

 

map h : E

 

onto a fiber in E”, thus inducing a map h : B > B'

 

such that p'h = hp. Let f 6 B1 > B be a map. The

induced bundle of C and f, denoted f*(§), is the bundle

 

(E1,p1,B1), where E1 = [(b,x) éiBl x E : f(b) = p(x)} and

 

p1 is the map (b,x) > b.

Given a topological group G, a free right Ge§pace is

 

 

a space M together with a free group action M X G > M.

 

defined by (x,s) > xs.

Definition 1.2: A principal G—bundle Q is a bundle

 

 

(E,p,B) satisfying the following conditions.

(1) E is a free right G-Space with a continuous trans-

 

lation function t : E* > G, where E* is the set
[(x,xs) e E X E : x e E, s e G} and t is the function
with the prOperty that x t(x,x') = x'.

(2) There is an equivariant bundle isomorphism

 

 

h : C > (E,w,E/G) such that h : B > E/G is an

homeomorphism.

6
It follows that each fiber p—1(b) of a principal
G-bundle is homeomorphic to G. The induced bundle of a
principal G—bundle is also a principal G-bundle. If M
admits a free G action, denote the orbit Space M/G
by M*. The bundle (M,p,M*)~ is a principal G-bundle, where
for convenience we now let p denote the projection map.

Further properties of bundles are presented in [5] and [19].

Lemma 1.3: Let M be a compact 3—manifold admitting

 

a free Zk yaction, where k 3.2 is prime. If H1(M*;Z)

has no k torsion then there is a bundle map

9 : (MapIM*)

 

> (sl.p£sl),
where p' is the standard k to 1 covering projection

of 81.

Proof: Since C = (M,p,M*) is a principal Zk bundle,

 

there is a map f : M* > Loo’l‘ a classifying Space for

CD
Zk' such that C is the induced bundle of the universal

bundle (Sc), poo’ Loo) and f [5]. We will take S

CD

and Loo as the following. Consider the commutative dia-
gram
81C 33 C S5
P11 Pa! 95!

L1C L3 c L5

where (Sl,pi.Li) is the standard k to 1 covering of

the generalized lens space Li - L(k:1,---,l) by S1 [17,

page 88]. S1+2 is to be considered as the double suspension

7

of 81. Define SCD and LCD to be the unions Us1 and

ULi respectively, with the weak topology. 800 is contrac—
tible so LCD is a K(Zk,1) space. q = (Soo’pco’Loo) is
a principal Zk bundle and hence by [19, page 101] is a
universal Zk bundle.

Since Loo is a K(Zk’1) space, the set of homotopy
classes [M*.La; is in a one to one correspondence with the
hom omorphisms of v1(M5 into Zk [4, page 198]. The homo-
topy class of f is completely determined by a hom omorphism
¢ : 71(M*) -—> Zk' o is onto since Q is not the trivial
is!

bundle. ¢ must factor through H1(M*;Z) since Zk

an abelian group. We get the commutative diagram

7T1 (M*) q) > Z

a\ /

H1(M*7Z)

 

k

where a is the projection to the abelianization of w1(M*).
H1(M*:Z) is finitely generated_and has no k torsion by
hypothesis. It follows that H1(M*;Z) has a free part
mapping onto Zk' Thus a can be factored through Z and

we get the commutative diagram

771(M*) > Z

a\ /y

Z

'Because 81 and LCD are K(v,1) spaces there are

 

 

maps c : 81 > Loo and g : M* > 81 corresponding

8
to y and B_ respectively. We may suppose that f was

chosen so that f = cyg. Consider the following diagram.

 

E(f*(n)) =M >s°°

LQI

 

 

The induced Zk bundle c§(q) is equivalent to the

standard k -sheeted covering of 81. There is a bundle

 

map 6 : M > 81 since Q = f*(q) - g*(c;(q)) [5, page
19]. Hence we have shown the existence of the required

commutative diagram

 

M g > 81
p p'
M* g > S1

 

where p' is the standard k to 1 covering of S1 and

g is a bundle map.

Corollary 1.4: Let M be a compact 3-manifold admit-
ting a free Zk action, k 2.2 prime. If H1(M*;Z) has

no k torsion then H1(M*;Z) has a nontrivial free part.

9
Definition 1.5: Suppose M admits a free Zk action
generated by the homeomorphism h. Call (U,T) an equivar-

iant hepartition of M if T is the disjoint union of

 

connected two-sided 2-manifolds regularly embedded in M
(i.e. Bd T = T 0 Bd M) such that the following conditions
are satisfied:

(1) hi(T) n hj(T) = e if i i j (mod k),

(2) U is open in M.

(3) hi(U) n hj(U) = e if i i j (mod k),

(4) Fr U T U h(T),

(5) M = (hi(U) u hi(T)).

u<:W

Lemma 1.6: Let M be a compact orientable 3-manifold

 

admitting a free Zk action and such that if Bd M # ¢

then Bd M is connected. If there is a bundle map from
(M,p,M*) to (Sl,p',Sl), where p' is the standard k to
1 covering projection of 81, then there is a compact 29p—
nected orientable polyhedral 2-manifold T such that T
determines an equivariant h-partition (U,T) of M with
the properties that U is connected, and if Bd M # ¢ then

Bd T is connected and does not separate Bd M.

Proof: Consider the diagram below whose existence is

given by hypothesis.

 

M g > 31
p p'
g 1

 

10
We may suppose there is a point a e 83 such that g-1(a)
is the disjoint union of polyhedral two-Sided orientable
compact 2-manifolds regularly embedded in M*. Let p'-1(a)
[ai} k C SI. By suitable choice of labeling we may let A

i=1
be the arc in $1 with endpoints a1 and a2 such that

no other ai lies on A. Let U = g'1(A - {a1,a2]) and
T = g'1(a1). Since 6 is a bundle map, (U,T) is an
equivariant h-partition. The difficulty comes in trying

to adjust (U,T) to satisfy the conclusions of the lemma.
First we need to adjust the frontiers of the components
of U. Suppose a component U1 of U has at least two
components T1 and T2 of T on its frontier. Let C
be a polyhedral arc in C1 U1 with one end on T1 and the
other end on T2 but otherwise lying in Int U1. Let N(C)
be a closed tubular neighborhood of C in C1 U1 such that
N(C) n T1 and N(C) 0 T2 are closed disks. Take as a new
U the set

(U - N(c)) U h[IntM N(C) U IntT((T1 U T2) 0 N(C))l

and as a new T one obtained from the old one by replacing
T1 U T2 by C1[T1 U T2 U Bd N(C) - Int(N(C) 0 (T1 U T2))].
The number of components of T decreases and the number of
components of U does not increase. Repeating this process
and a similar one for h(T) we obtain an equivariant h-
partition (U,T) such that each component of U has at
most one component of T and one component of h(T) on

its frontier.

11

We adjust (U,T) further. Suppose a component U1
of U is such that Fr U1 = T1 C,T, where T1 is a com-
ponent of T. Then crossing T1 from U1, one enters
hk-1(U). Whenever this case occurs take as a new U the
set (U - U1) u h(Cl U1). Notice that Fr h(U1) czc1 U.
Obtain a new T by dropping T1 from the old one. Re-
peating this process and a similar one for h(T), i.e. re-
place U by (U - Ui) U h-1(Cl U1) where necessary, we
obtain a new (U,T) partition where each component of U
has exactly one component of T and one component of h(T)
on its frontier.

Let T1 be a component of T. T1 lies on the fron-
tier of a unique component U1 of U. Let T1(1) C h(T)
be the component of h(T) on the frontier of U1. T1(1)
lies on the frontier of an unique component U1(1) of h(U).
Continuing the construction of this chain, let T1(2) C h2(T)
be the component of h2(T) on Fr U1(1). As before, let
U1(2) be the unique component of h2(U) with T1(2) on
its frontier. Repetition of this process yields a sequence

of components

1 2 _ _
T11 U11 T1( )I U1(1)l T1( )1 ...I T1(k 1): U1(k 1)!
— —1
T21 U21 T2(1)I U2(1)I T2(2)I 000: T2(k 1): U2(k )I
(j) (j) (k-l) (k-l)
... , Ti , Ul ,, . , Tn' . Un .
where Ti c T, Ui c U, Ti(3) c h3(T), Ui(3) c h3(U).

j+1>

Fr Ui(3) = Ti(3) U Ti( and Ti(k) is identified with

12

th

T This sequence must return to T1, say at the n

i+1'
stage, and we identify Tn(k) with T1 completing the
cycle.

All of the components of hi(U), i = 1, ..., k, must
have appeared as M is connected. From the construction

th

it is clear that h(Tl) will appear 1/k of the way

through the cycle and from there on the sequence is nothing

th

more than the first 1/k under repeated application of h.

Let K be the union of everything appearing in the first
l/kth of the sequence. Then the new U is to be chosen
as the set K - (T1 U h(T1)) and the new T as T1. This
version of (U,T) is an equivariant h-partition with

U and T connected, U open, and T regularly embedded
in M. So if Bd M = ¢ we are done.

Finally suppose Bd M # e. We must adjust (U,T) so
that Ed T is connected. Bd T consists of disjoint simple
closed curves {Ci}i:1 lying on Bd M. If n 2.2 a poly-
gonal arc A in Ed M can be found with endpoints a and
b lying on different curves Ci and Cj such that
A - (a, b] lies entirely in either U or h-1(U). Suppose
A - (a, b] C‘U. (If it lies in h—1(U) a similar operation
with the obvious modifications is used). Let N(A) be a
closed tubular neighborhood lying in C1 U, containing A
in its boundary, and meeting T in two closed disks Di
and Dj which contain arcs of Ci and Cj in their re-

spective boundaries. In particular, Di n Bd M C Ci and

Dj n Bd M C Cj. N(A) is chosen so that there is a

13
homeomorphism f mapping A x [6,1] onto N(A) 0 Bd M
with f(A x 1/2) ='A. For our new U we take the set

(U - N(A)) U h[Int N(A) U f(A x (0,1)) U Int (Di U Dj)].
Replace T by the set
T U Bd N(A) — [f(A x (0,1)) U Int (Di U Dj)].

This Operation decreases the number of components of Ed T
by one but does not disturb the connectedness of U and T.
Eventually this gives rise to an equivariant h-partition
such that Ed T, U, and T are all connected.

Cl U 0 Bd M is an oriented 2-manifold with boundary
Bd T U h(Bd T). It is easy to see that Ed M - Bd T is

connected already without further adjustment.

Remark 1.7: If Bd M # ¢, then (U,T) in the above

 

lemma is such that U 0 Bd M is connected.

Definition 1.8: Let M admit a free Zk action and

suppose that the action is generated by the homeomorphism h.

The Z action is said to be proper if h is homotopic to

k
1M.

Theorem 1.9: If M is an irreducible closed orientable

 

3-manifold such that for some prime k 3.2 M admits a
proper free Zk action and H1(M*;Z) has no element of

order k, then M can be fibered over the circle.

Proof: From Lemma 1.3 and Lemma 1.6 we get the existence

of an equivariant h-partition (U,T) of M with T a

14
connected orientable closed polyhedral 2-manifold. Let A
be an arc with one endpoint x0 6 T and the other endpoint
h(xo) e h(T) but otherwise lying in U.
There is a retraction r0 of Cl U onto A such that

r;1(xo) = T and r;1(h(xo)) = h(T). Define a retraction

 

ri of h1(Cl u) onto hl(A) by ri = hlr0h_l = hri_1h‘1
for i = 1, ..., k-1. The ri's define a retraction
k . o .
r : M > P, where F = U hl(A), such that rh1 = hlr
i=1

for all i. Hence r is equivariant and induces a retrac-

 

tion r0 : M* _> p(F). Let f be r0 followed by a
homeomorphism of p(F) onto 81. Then there exists a bundle

map f such that the following diagram commutes

 

M f >51

p I
f

M* > 83

 

where p' is the standard k to 1 covering of 51.

Since the Zk

erator h of Zk is homotopic to the identity homeomorphism

action is proper we have that the gen-

on M. Restricting the homotopy we get a map

 

Go : T )<[0,1] > M such that Go(x,0) = x and

 

Go(x,1) = h(x). Define the maps Gi : T x [i, i+1] > M

by Gi(x,t) = hlG0(x,t—i), for i = 1, ..., k-1. Then
i+1
(

Gi(x,i+1) = Gi+1(x,i+1) = h x) and Gk_1(X,k) =

Go(x,0) = hk(x). Hence the Gi's define a map

G : T x sl-——> M, where s1 is considered as [p,k]/[O,k}.

15

We Show that the degree of G

 

deg G = d. Let i : T

Look at the exact homology sequence of the pair

 

 

 

is not zero. Let

> M be the inclusion map.

(M,T).

 

Since H2(M,T;Z) is free, it follows that the sequence
*
o > H2(T;Z) > H2(M;Z) is split exact. From this
'*
we see that H2(M;Z) l > H2(T;Z) > O is exact using

the universal coefficient theorem for cohomology which is

functorial. In the sequel,

with integer coefficients will be used exclusively.

a e H3(T x 51) and a e H3(M)
G*Q = dB.

mutative diagram:

 

 

 

 

H2 (T X X0)
iI-X—
*
H2 (T x sl) G >
Q0
H1(T X SI) >
where i' : T X x0 > T x 81

 

singular homology and cohomology

LEt

be generators such that

Using Poincare duality we get the following com-

H2(T)
1*
H2(M)
I den
H1(Ml *

H'tl

 

> H1(Sl)

is the inclusion for

Using the universal coefficient theorems [17], write

H2(T x 51) =

H1(T x 81)
where P = H2(T) e H°(sl). Q
R = H0(T) e H1(sl),
is an epimorphism, for each
that u

+ u' e Im G*.

phism P maps onto R.

and S = H1(T) e Ho(Sl).

u e P

Since

P $ Q.

R O S,

H1(T) n H1(sl),
Since i*

c Q such

there is a u'

Under the Poincaré duality isomor—

f*G*(S) = 0 we have that

16

E*G*(R) = f*G*(a n p) = E*G*(d n G*(H2(M)) = E*(de n H2(M)).
Hence f*G*(R) = f*(dH1(M)).

We can identify R with W1(Xo x 81) for a fixed
x0 e T. Choose the point so that {x0} = F n T. R is gen-
erated by an element represented by a loop u going around
x0 x 81 exactly once. Gu is a loop starting from x0 6 T
and passing through h(xo) e h(T) l/kth of the way around,
and thereafter repeating itself k-1 times under the action
of hi, for i = 1, ..., k-l. Hence qu will start from
some point so in 51, wrap around some number of times,
and then reach so + ei(2Fj/k) when Gu is j/kth of the
way around, for j = 1,2, ..., k. Therefore under fG, the
first 1/kth part of u is wrapped around 81 m + 1/k
times. Hence under fG , LL is wrapped around 51
k(m + 1/k) = km + 1 times. Thus f*G*([u]) e F1(Sl) is
an integer congruent to 1 modulo k. Since f*G*(R) =
f*(dH1(M)), after identifying H1(sl) and w1(sl), it
follows that km + 1 is a multiple of d = deg G. There-
fore deg G i 0 (mod k) and in particular deg G # 0.

Let M'-E-—> M be a covering corresponding to the sub-

 

G
group w' of W1(M), where W' = Im(71(T x Sl)‘-—#—> w1(M)).
If G' is a lifting of G we get the commutative diagram
fi'
GI
P
T x 51 ___£i____> M ____£___? 31

p is a finite to one covering projection since deg G is

17

N

finite and deg G = (deg G*)(deg 5). Hence v' = W1(M)

 

has a finite index in v1(M). G# : v1(T x 81) > W1(M)
is an epimorphism, so every loop in M, will circle around
31 under the map EE' some n number of times, where n
is a multiple of km + 1.

Let the circle C be the component of p-1(F) that
contains the basepoint (we may assume that basepoints have
been chosen nicely). Let c be a non-singular loop on C.
t = deg p divides d = deg G and 5c circles around F
at most t times (in absolute value). Thus EEc circles
around 81 at most t times. It follows from these re-
marks that t :.d :.km + 1 i.n :.t, and hence t = d. In
other words, deg G = deg 5' and [71(M), F'] = d. So ‘5
is a d to 1 covering projection.

Let y be the element of 71(M) represented by a
loop circling around F exactly once. Identify F1(Sl)
with the additive group of integers Z. Consider the cosets
W', yv', ..., y w'. These cosets are distinct since
f#(yiv') E i (mod d), so that their disjoint union is all
of W1(M). Moreover W' is the kernel of a homomorphism
from W1(M) onto Zd and therefore is a normal subgroup.

We have v1(T x 51) = v1(T) x W1(Sl) naturally ;

split, so w1(fi) = G# (W1(T)) x G'#(W1(Sl)). If we let

 

K - Ker(f# : 7r1(M) > 7r1(sl)), it is clear that K c Tr'.

Since 5% is a monomorphism, we have that K

 

5%,(Ket((f5)# : W1(M) > V1(Sl))) But considering the

18
effect of (f5)# and G# it is clear that K = E#G#(W1(T)).
Hence K is finitely generated and 71(M)/K = Z. Moreover,
by Lemma 1.1, v1(M) has no elements of finite order.
Therefore we can apply the following thoerem due to Stallings

[18] to complete the proof of the theorem.

Theorem 1.10: If M is a compact irreducible 3—mani—
fold, and if W1(M) has a finitely generated normal sub-
group K different from 22, whose quotient group is Z,
then M is the total space of a fiber space with base space
a circle and with fiber a 2—manifold T embedded in E

whose fundamental group is K.

 

Remark 1.11: In Theorem 1.9 the hypothesis that Zk
be a proper action may be replaced by the weaker condition
that for some equivarient h-partition (U,T) such as in

Lemma 1.6 and some non-trivial h e Zk we have that th

 

is homotopic to the inclusion i : T > M.

Remark 1.12: Let M be a closed orientable 3-manifold
satsifying the hypothesis of Theorem 1.9. It follows from
the proof that M is covered by M, in a finite to one
manner. The multiplicity of the covering is relatively
prime to k. By Neuwirth [12], M'= T x 81, where T is
a closed orientable 2-manifold. Moreover, 71(T) =
G#W1(T), which is isomorphic to the fundamental group of

the fiber of a fibration of M over 81 [18].

19

Theorem 1.13: If M is an irreducible compact orient-

 

able 3-manifold with connected nonempty boundary such that

for some prime k 3.2 M admits a prOper free Z action

k
and H1(M*;Z) has no element of order k, then M can be

fibered over the circle and Ed M = 81 x 51.

nggfi: The proof of this theorem follows that of
Theorem 1.9 very closely, so the obvious overlap will be
omitted here. There exists an equivariant h- partition
(U,T) of M with T a connected orientable compact poly-
hedral 2-manifold such that Ed T = T n Bd M is connected
and does not separate Bd M.

Let A be an arc with endpoints x0 6 T and h(xo) e

h(T) such that A *'{xo, h(xo)} C U. As before there is an

 

equivariant retraction r : M > F, where

k .

F : U h1(A), which induces a retraction r0 : M*
i=1

Let f be r0 followed by a homeomorphism of p(F) onto

 

> p(F).

81. We get the commutative diagram

f

 

 

 

M 31
p p'

f
M* s:

 

where f is a bundle map and p' is the standard k to 1

covering of 81. In View of h g 1M we get a map

 

G : T X S1 > M.
We require a slightly different diagram this time to

show that deg G = d is not zero.

20
H2(Bd M, Bd T) = Z since Bd T is a nonseparating
simple closed curve in Ed M. Consider the following com-

mutative diagram:

 

 

> 0

 

H1(Bd M) ii—eH1(Bd T)

l l

' *
H2(M,Bd M) l >H2(T,Bd T)

l

0

>H2(Bd M,Bd T) >H2(Bd M)

 

where i and j are inclusion maps. It follows from the

fact that j* is an epimorphism that i* is an epimorphism.
Let a e H3(T x S1,Bd T x 31) and a e H3(M,Bd M) be

generators such that G*Q = dB. Consider the following com—

mutative diagram:

 

 

 

 

 

 

 

H2(T x x ,Bd T x x0) H2(T.Bd T)

. * _*

1' 1

G*

H2(T x sl,Bd T x 51) > H2(M.Bd M)

an dBfl _

G f*

H1(T x 81) * > H1(M) > H1(Sl)

where i' : (T x X0. Bd T x X0)

 

> (T x $1,Bd T x 31) is
the inclusion map for a fixed xo 6 81. Write

H2 (T x sl,Bd T x 51) = P 0 Q, where P = H2(T,Bd T) s H0 (51)
and Q = H1(T,Bd T) e H1(sl). Let H1(T x 31) = R e s as
before. Then f*G*(R) = f*(dH1(M)). The remainder of the
proof to show that M fibers over the circle is the same

as that of Theorem 1.9.

21

Theorem 1.14: If a closed compact 3-manifold M fibers

 

over the circle, then M is a prime manifold.

Proof: Denote the fiber of a fibering of M over 81
by T (we may assume that T is connected). From previous
remarks it follows that there is a homeomorphism h : T ->T
such that M = T x I/h. If T = $2 or P2, then M is
either an 32 or P2 bundle over $1, hence prime (in
fact irreducible in the second instance). Otherwise, the
universal covering space of T is R2, and hence the uni-
versal covering space of M is R3. Therefore M is ir-

reducible in this case, and prime in either case.

CHAPTER II
3-MANIFOLDS THAT COVER THEMSELVES

In this chapter we consider 3-manifolds M that admit
a free Zk action such that the orbit space is homeomorphic
to M. Kwun's results [7] on closed orientable 3-manifolds
(without boundary) which double—cover themselves are con-
tained here as special cases of Theorem 2.8 and Theorem
2.16.

A bundle (E,p,B) is called a covering Space of B if
every b e B has an open neighborhood U such that p-1(U)
is a disjoint union of open sets in E each of which is
mapped homeomorphically onto U by p. The map p is
called the covering projection. Let E be path-connected.

The number of sheets of p is the cardinal number of the

 

discrete set p-1(b), which is independent of b e B. The
covering projection is regular if for some x0 6 E,
p#v1(E,xo) is a normal subgroup of w1(B,p(xo)). The

group of covering transformations G(EIB) of p is the

 

group of homeomorphisms f : E > E such that pf = p.

Let M be an n—manifold. If (M,p,B) is a regular

covering space, then G G(M|B) acts freely on M and
the orbit space M* = M/G is homeomorphic to B. On the
other hand, if a finite group G acts freely on M, the

bundle (M,p,M*) is a regular covering Space, where p

denotes the projection. In either case

22

23
G = W1(M*,p(xo))/p#(w1(M,xo)), and the number of sheets of
p equals the order of G. -For more details on covering
spaces see [17, Chapter 2].
In the introduction we defined the concept of a con-
nected sum. Closely related to this is the operation of

adding a handle to a 3-manifold M. Remove the interiors

 

of two disjoint tame 3—cells in M and match the resulting
boundaries by a homeomorphism. If M is orientable, the
result is homeomorphic to M # 81 x 82 when the attaching
homeomorphism is orientation reversing, and homeomorphic to
M # N when it is orientation preserving. By N we mean
the non-orientable locally trivial $2 bundle over 51.

If M is non-orientable to start with, the resulting space
is homeomorphic in either case to M # 51 x 82 since
m#slxsz=M#N.

Milnor [10] has shown that every closed orientable
3—manifold M is homeomorphic to a sum P1 # P2 # ...# Pk
of prime manifolds, where the summands Pi are uniquely
determined up to order and homeomorphism. It has been ob-
served by Raymond [16] that Kneser [6] actually proved,
modulo the truth of Dehn's lemma, a unique decomposition
theorem for closed 3-manifolds, orientable or not. Kneser's
theorem states that every closed 3-manifold can be written
uniquely in "normal form" as the sum of prime manifolds.

In normal form means as the sum of irreducible manifolds
and handles, where the number of non-orientable handles is

minimal (i.e. 1 or 0, depending on whether the irreducible

24
summands are all orientable or not).

Milnor [10] also proved that, with the exception of S3
and 81 x 82, an orientable closed manifold is prime if and
only if it is irreducible. In light of Raymond's observa-
tion, the proof given by Milnor can easily be extended to
the non-orientable case if N is also excluded.

We are now ready to initiate our investigation of com-
pact manifolds covering themselves. First we need to define
a property of covering spaces closely related to proper

group actions.

Definition 2.1: Let (M,p,B) be a regular k-Sheeted

 

covering space, k 2.2 prime. We say that M properly
covers B if the action of the group of covering trans-

formations G(M|B) = Zk on M is proper.

The next two examples will serve to motivate the dis-
cussion in this chapter and to exhibit the general character

of manifolds covering themselves.

Example 2.2: Let T be a compact 2-manifold and

k 3,2 any integer. Then the map

> szll

 

I . 1
1T x p . T x S

where p' is the standard k-sheeted covering projection

of the circle, is a proper k-sheeted covering projection.

Example 2.3: Consider the connected sum P3 # P3 of

 

two real projective 3-spaces. P3 # P3 is homeomorphic to

25
the sum P(k) = P3 # S3 # --° # $3 # P3, where P(k) con-
tains k - 1 summands of S3 (note that orientation need

not be specified since P3 admits an orientation reversing

 

homeomorphism). Let p : P(k) > P3 # P3 be the covering
projection by which the sphere summands S3 of P(k) al-
ternately double cover the P3 summands of the base space,
the first P3 of P(k) covers the left half of P3 # P3,
and finally the last P3 of P(k) covers the left (right)

half of P3 # P3 if k is even (odd).

 

Remark 2.4: Let p : P3 # P3 > P3 # P3 be the
k-sheeted covering just described. Notice that if k = 2,
then p# v1(P3 # P3) is a normal subgroup of r1(P3 # P3),
and hence p is a regular covering projection in this case.
This is the only case in which p is a regular covering
projection. Moreover, none of these coverings are proper.

Theorem 2.8 shows that Example 2.3 is the only non-
prime closed 3-manifold to cover itself in a non-trivial
way. The next three lemmas lead up to this theorem.

Let Ho = {S3} and let H1 denote the collection of
non-trivial prime closed 3-manifolds. For j 3.2 let Hj
denote the collection of closed 3-manifolds which are homeo-
morphic to connected sums of exactly j elements of H1.

For convenience of notation we will let N* denote a handle

of either type, i.e. either 81 x S2 or N.

Lemma 2.5: If M 6 H2 then M covers itself k

 

times, k 3.2, if and only if M = P3 # P3.

26

 

Proof: Suppose p : M > M is a k to 1 covering
projection. Let M = A # B, where A, B 6 H1. Write

A # B = A' U B', where A' (B?) is obtained from A (B) by
deleting a tame open 3-cell. A' H B' = S is a 2-sphere

_ k
and p 1(S) is a disjoint collection of 2—spheres {Si}i=1’

Case 1. Each Si separates A # B, for i = 1, ..., k.
Therefore M - p-1(S) has k + 1 components with the closure
of each component covering either A' or B'. The closure

of at least two components, say U1 and U2, have con-
nected boundary, say 81 and 82 respectively. Let C (D)
be obtained from C1 U1 (Cl U2) by sewing a 3-cell along

81 (82). C (D) covers either A or B exactly once, so

C and D belong to H1. Since M 6 H2, A # B = C # D

and each of the remaining components must have disconnected
boundary of exactly two componern:s lying in p-1(S) and

be homeomorphic to S3 minus two tame open 3-cells. An
analysis of the situation reveals that either both A and

B are double-covered by $3, or A = B and one of them

is doubleocovered by S3. We need now to invoke the follow—

ing theorem due to Livesay [8].

 

Theorem: If T : S3 > S3 is any fixed-point-free

homeomorphism of period 2 on the 3—sphere, then there

> S3 such that hTh-1

 

exists a homeomorphism h : S3
is the antipodal map.
It follows that the orbit Space of the action on 83

by any free involution must be P3. In particular, this

proves that A B = P3.

27

Case 2. A # B — Si is connected for some i, say
i = 1. Since A # B - 81 is connected, either A or B
must be N*. Suppose B = N*. The closure of each com—
ponent of M - p—1(S) covers either A' or B'. If the
closure of a component U covers B', Cl U plus some tame
Open 3-cells sewn along its boundary components is homeo-
morphic to N*. This is because W1(N*) = w1(B') = Z has
exactly one subgroup of a given finite index, and hence has
a unique connected m-sheeted covering for any given m.
It follows that the closure of only one component of
M - p-1(S) covers B'. Otherwise we would have at least
three handles N* upstairs which is in violation of M
being in H2. Therefore only Cl U covers B' and hence
does so in a k to 1 fashion. The boundary of Cl U must
be p-1(S). Since M - $1 is connected there must be
another copy of N* upstairs, in addition to Cl U with k
open 3-cells attached. Hence A = N* and B = N*. By the
uniqueness of coverings for N*, the closure of some com-
ponent, say Cl V, covers A' k times and is such that
Cl v n c1 U = p'1(s). But this implies that there are
k - 2 handles upstairs besides C1 U and Cl V (with
appropriate 3-cells attached). Therefore we must have k = 2
to avoid contradicthmythat M 6 H2. This case is ruled out
by the following argument which is essentially that of
Kwun's [Proposition 3.1, Case 2] with a slight modification

to extend it to the non-orientable case.

28

We have then that neither 81 nor $2 separates
A # B, and so A # B - 51 U 82 has two components P and
Q. Since p(P U Q) = A # B - S is disconnected, p(P) =
A' - B' and p(Q) = B' — A‘ (by prOper choice of labeling).
Let C and D be manifolds obtained from C1 P and Cl Q
reSpectively by attaching 3-cells. Then A # B = C # N* # D.
By the uniqueness of the decomposition, either A or B,
say A, is a handle. But this implies that either C or
D is N*, which in turn implies that B = N*, and finally
that both C and D are handles. Hence the connected sum
of three copies of N* would be homeomorphic to that of
two copies of N*. This contradiction rules out this case

and thus completes the proof of the lemma.

Lemma 2.6: Let M1 6 Hm and M2 6 Hn' and suppose

that M1 contains no handles. If there exists a k to 1

 

covering projection p : M1 > M2, then m 3.n (iF nitll),

Proof: The proof is by induction on m. Let m = 1.
Write M2 = A # B, where A 6 H1 and B e Hn_1. As
usual write M2 = A' U B', with A' n B' = S, a 2-Sphere.

Since M1 has no handles, each component Si of p-1(s) =

31 U --- U Sk must separate M1. There are at least two
components of M1 - p-1(S), each of which covers either A'
or B' exactly once. But this is impossible unless n = 1

since M1 is prime.
Now suppose that the lemma is true for q i.m and

let M1 e Hm+1’ M2 e Hn We suppose that n+1 > m+1

+1'

29
and Show this leads to a contradiction. Write M2 = A # B =
A' U B', where A 6 H1, B e Hn, and A' n B' = S, a
2-Sphere. Again every component Si of p_1(S) separates
M1. Hence there must be at least two components of
M1 - P-1(S) such that the closure of each must cover either
A' or B' in a one to one manner. Let C' be any com-
ponent of p-1(B), and let C be the manifold obtained by
capping the 2-sphere boundary components of Cl C' with
3-cells. Then >C e Hq, where1sq.fi.m. However, p induces
a covering of B by C, which violates our induction

hypothesis since q < n. This completes the induction.

Lemma 2.7: If M e Hn' n > 2, then M does not

cover itself k times for any k 3.2.

Proof: Let M = A1 #'A2 # ... # An’ where each

Ai 6 H1.

Case 1. At least one Ai = N*. Suppose that Ai = N*

<

for 1 :.i :-m and that Ai # N* for i > m (1 j_m _.n)

Write M = Ai U A; U --- u AA, where A] and A; are
obtained from A1 and An respectively by deleting a

tame open 3-cell, and A; is obtained from Ai by deleting
two tame open 3-cells for 1 < i < n. Since M is con—
nected, A; n Ai+1 = Si’ a 2-sphere. A connected t-
sheeted covering of Ai, for

1 < i < m (and for i = m if m # 1, n),

must be homeomorphic to N* minus 2t tame open 3-cells

30

(or t 3-cells if i = 1 or i = m = n). Hence each com—
ponent of M - p-1(S) covering Ai , i j,m, is of this
form. To cover A1 #-A2 # ... #'Am k times requires at

least k(m-1) copies of N* upstairs. So M must have
at least k(m-1) handles. But M has exactly m handles

< m holds only when k = m = 2. An analysis

and k(m - 1)
of this special situation reveals that it takes more than
two copies of N* upstairs to double—cover M. Since M
has only two handles such an M cannot double-cover itself.

This rules out Case 1.

Case 2. No Ai = N*. Write M = A # B, where A = A1
and B = A2 # °°~ # An. As before write M = A' U B'
such that A' n B' = S, a 2-sphere. Since M contains
no handles, each component of p-1(S) must separate M.
Hence the closure of at least two components of M - p‘1(S)
must have connected 2-Sphere boundary and cover in a one to
one manner. Call two of these components U and V. We
must have that Cl U = Cl V = A', for otherwise we would
have too many summands upstairs (i.e. at least 2(n—1) > n).
Let W' be a component of Cl(p—1(B')). If we cap the 2-
sphere boundary components of W' and B' with 3-cells we
get the closed 3-manifolds W and B, W e Hq and B e Hn_1.
where q < n-l. But p|W' induces a covering of B by W

which is impossible in view of Lemma 2.6. This rules out

Case 2, thus completing the proof.

31

The following theorem is an immediate consequence of

the above three lemmas.

Theorem 2.8: Let k 3.2 be any integer. A closed

 

non-prime 3-manifold M is a k-sheeted covering of itself

if and only if M = P3 # P3.

We digress for a moment to organize some of the prob-
lems that have arisen in the above discussion. In Lemma 2.6
we have shown that no irreducible 3-manifold can cover a
non-prime one. However, it is possible for a prime 3-mani-
fold to cover a non-prime one. For example 51 x S2 double-
covers P3 # P3. We ask if the opposite situation can occur,

that is, can a non-prime 3-manifold cover a prime 3-manifold?

Conjecture 2.9: A closed covering Space of a prime

 

closed 3-manifold is itself prime.

We now would like to obtain a result for manifolds

with connected boundary Similar to Theorem 2.8.

Lemma 2.10: Let k :_2 be an integer. A closed 2-

 

manifold T is a k-sheeted covering of itself if and only
if either T = 51 x 81 or T = K and k is odd (K de—

notes the Klein bottle).

Proof: Let x(T) denote the Euler characteristic of
T. In view of the fact that x(T) = kx(T) [4, page 277],
we must have x(T) = 0. Therefore T must be either a

torus or a Klein bottle K. Since K can only cover itself

32

an odd number of times, the lemma is proved.

Lemma 2.11: If a manifold M covers itself k times

for some k :_2, then w1(M) is infinite.

Proof: It follows that M covers itself kn times
for all integers n.3 1. Hence W1(M) contains subgroups

of index kn for all n :_1.

Theorem 2.12: Let k 3.2 and let M. be a compact
3-manifold with nonempty connected boundary. If M covers
itself k times, then M is a prime irreducible 3-mani-

fold and Ed M is either the torus or the Klein bottle.

 

Proof: Suppose p : M > M is a k to 1 covering

> Bd M is also a

 

projection. Then p|Bd M : Bd M
k to 1 covering projection. By Lemma 2.10, Bd M is
either 81 x S1 or K.

Now suppose that M = A A B, where A and B are not
closed 3-cells. By proper choice of notation we may suppose
that Ed A = Bd M and Ed B = 82. Consider 2M, the double
of M, obtained by sewing two copies of M. together along
their boundaries by the identity map. It is clear that
2M = 2A # 2B, where 2A and 2B are non-trivial. p in-
duces a k to 1 covering of 2M by itself. According to
Theorem 2.8, 2M = P3 # P3. By the uniqueness of the con-
nected sum decomposition we must have that 2B = P3. But
this would imply that 22 = W1(B)*Wfl(B), where * denotes
the free product. This is a contradiction. Thus M must

be prime.

33

We want to show that M is also irreducible. Since
2M covers itself k times there are only three cases to
consider, namely 2M = P3 # P3, 2M "N* (i.e. 51 x 52 or
N), and 2M irreducible. For clarity in notation we sup-
pose that 2M = M U M', where M = M‘ and M n M' =
Bd M = Bd M'.

First suppose that 2M - P3 # P3. Let S C Int M C22M
be a tamely embedded 2—Sphere. If S does not bound a
3-cell in M, then S must bound P3 less a tame open
3-cell. By the symmetry of 2M, 3 corresponding tame 2-
Sphere S' in Int M' also bounds P3 less a tame open
3-cell. If we let M and M' denote M and M' respec-
tively with the P3's removed and replaced by 3-cells, we
get 2M = S3. Since there is a retraction of 83 onto M,
W1(M) = 0. It follows that W1(M) = 22. But this is im-
possible by Lemma 2.11. Hence this case is ruled out.

Suppose 2M is irreducible. Let S C Int M C.2M be
a tamely embedded 2-sphere. Let 2M -'S = A U B and sup-
pose that A C Int M. Then Cl A must be a 3-cell. For
if A were not a 3-cell, Cl B would have to be. But
there is a 2-Sphere S' in Int M' corresponding to S.
S' C Int M' C Cl B and S' bounds a homeomorphic c0py of
A in C1 B. This is a contradiction Since Cl B is ir-
reducible. Hence 2M irreducible implies M is irreducible.

The last case, when 2M = N*, is taken care of by the

next lemma.

34
Lemma 2.13: Let M be a compact 3-manifold with con-

nected boundary. If 2M = N* then M is irreducible.

ggpgfi: (Let S C Int M be a tamely embedded 2-sphere.
If S separates N*, then S must bound a 3-cell in N*
and this 3-cell would lie in Int M. Hence it is sufficient
to show that every 2-sphere tamely embedded in Int M CiN*

must separate N*.

 

Let i : Bd M > 2M be the inclusion map. Suppose
that S C Int M C 2M - Bd M is a non-separating tame 2-

sphere. Then i(Bd M) CI2M - S and the induced map

 

i# : F1(Bd M) > w1(2M) is trivial. This is true Since
N* - S = 82 x (0,1). Consider the commutative diagram ob—

tained using the Van Kampen theorem.

 

W1(Bd M)
e X
G ' 7T1(M) i G' ‘ 7T1(M)
.#
L/
71(2M)

If we let F1(Bd M) = (Z : t), G = (i : f), and G' 3
(§ : S), then we can write

W1(2M) = (§,§ : E,S,{el(zk)ez(zk)- : 2k 6 2}).
Since i# = 0 and M is a retract of 2M, 61 and 62 are
trivial. Therefore w1(2M) = G*G', where G = G'. But
this is a contradiction since W1(2M) = Z and Z is not

the free product of two isomorphic groups. Therefore every

35

tame 2-sphere S in Int M must separate 2M and hence
bound a 3—cell in M.

Relating to Lemma 2.13, Kwun observed that if 2M 3
81 x 32 and Ed M connected, then M = D2 x 81 (the

solid torus). This follows from the above mentioned lemma.

Corollary 2.14: Let M be a compact 3-manifold with

 

connected boundary. If 2M = 81 x 82 then M is
a solid torus. If 2M = N then M is the product of the

Mobius band with the interval.

ggppfi: Suppose 2M - N*. Then there is a retraction
of N* onto M. Since v1(N*) = Z, it follows that F1(M)
is either Z or 0. But v1(N*) is isomorphic to an
amalgamated free product of two copies of W1(M), hence
71(M) = Z. M is irreducible by Lemma 2.13. By Theorem
1.10 (Stallings), M fibers over the circle with the closed
2—disk as fiber. There are only two locally-trivial 2-disk
bundles over SI, the solid urns and the non-trivial one.
If 2M = 81 x 52 then M must be the solid torus since a
non-orientable 3—manifold cannot be embedded in 81 x 82.
On the other hand, the double of the non-trivial 2-disk

bundle over 81 is N.

Lemma 2.15: If a compact 3-manifold M properly

 

covers itself k times for some k 3.2, then M is

either N* or an irreducible K(v,1) space.

36
Proof: This follows directly from Lemma 1.1, Theorem

2.8, Lemma 2.11, and Theorem 2.12.

Theorem 2.16: Let k 3.2 be a prime integer. Suppose
M is a compact orientable 3-manifold such that H1(M;Z)
has no element of order k and Ed M is either empty or
connected. If M properly covers itself k times, then

M can be fibered over the circle.

.ggppfe M admits a proper free action by G(MIM) = Zk’
with M* - M. If M is closed, then by Theorem 2.8 and
Remark 2.4, M must be a prime closed manifold. Since

81 x 82 satisfies the conclusion of the theorem we may as-
sume that M is irreducible. Application of Theorem 1.9
shows that M fibers over the circle. If Bd M # ¢, then
by Theorem 2.12 M is irreducible. Application of Theorem

1.13 completes the proof.

In the next chapter examples are given to show that in
general M need not be a product of a 2—manifold with the

circle.

CHAPTER III

EXAMPLES OF PROPER Z ACTIONS

k

In this chapter we present examples of 3-manifolds with
free Zk actions that can be extended to effective 30(2)
actions. Such Zk actions will clearly be proper. Closed
3-manifolds admitting effective 80(2) actions and the
actions have been classified by Orlik and Raymond [14]. We
follow their notation which seems convenient for the pre-
sentation of our examples.

First we need to describe some elementary SO(2)
actions on solid toniwhich will serve as "building blodks"
for more complicated actions on arbitrary 3-manifolds ad-
mitting an 80(2) action. Parameterize the solid torus
D2 x 81 by (peie,eiw), where 0.: p.: 1, 0.: 6,¢ < 2v.

Define an ordinary action on D2 x S1 by

 

Z X (peiQZeiw) > (peie,zei¢),

where z ranges over the complex number of norm 1.
Let u and v be relatively prime integers with
0 < v < u. Define the standard linear action (u,v) of
80(2) by
z x (pele,elw) -————> (zvpele,zuel¢).
The center circle (0,e1¢) will be called an exceptional

orbit of type (u,v). The isotropy group of any point on
such an orbit is Zu C 80(2). The principal orbits wind

37

38

around an exceptional orbit of type (u,v) u times and
around a bounding curve m - (eie,eiwo), wo fixed, v times.

We may choose a closed curve q on the boundary torus
such that q is a cross-section to the standard linear
action, i.e. q cuts each orbit on the boundary torus in
a Single point. The group 80(2) is naturally oriented
and by choosing an orientation for the solid torus a com—
patible orientation of q is determined. Orient m =
(eie,eiwo) such that the homology relation m ~tnq + Sh,
where 5v E 1 (mod n), holds. Notice that we can assume
g was chosen so that 0 < B < u, since q can be modi-
fied by q ~Iq' + sh, where s is an arbitrary integer
and h denotes a principal orbit on the boundary torus. In
this case we refer to (u,B) as being in normal form.

Using these building blocks, Orlik and Raymond [14]

defined the symbol

[b3'(0rg:0:00;(a1151): ...! (anIBn)}I

which describes both a closed orientable 3-manifold and a
standard SO(2) action on this manifold. For completeness,
a brief description of this notation follows.

Let the integers g 3,0 and b be given and let
“ai'Bi)]iE1 be pairs of relatively prime integers such
that 0 < Bi < oi. Let M+ be a closed orientable 2-mani—

, +
fold of genus 9. Define an SO(2) action on M X 81

by e19 X (m+ x elw) > (m+ x eigelw)

 

. The orbit Space
(M+ X Sl)/SO(2) is homeomorphic to M+. There is a cross-

section to this action which can be identified with (M+ x 1).

39

Choose n points x1+, °'°, xn+ in M+ and remove
the interior of a closed disk neighborhood Dj+ around each
of the points. The resulting manifold Mo = M+ - Int(Dj+ X SI)
is a compact 3-manifold with n toral boundary components,
Tj. Since we removed only invariant tubular neighborhoods
of principal orbits, 50(2) still operates freely on Mo.
On Tj, let qj = Bd(Dj+) x 1 and let h be any one of the
principal orbits. qj and h form an orthogonal curve
system on Tj'

Now sew a solid torus jV with standard linear action
(Oj,Vj) on Tj by matching the principal orbits on the
boundary of jV to the principal orbits on Tj' and match-
ing a cross-section jq (to the action on jT) to the
cross-section qj. This is done in an orientation reversing

manner. Moreover the cross-section jq is chosen such

'1
th t v. E . d a. .
a 3 B3 (mo 3)

If b # 0 remove one more invariant tubular neighbor—
hood Int(D; x SI) of a principal orbit over xo+. On To
we have the cross-section curve qo = Bd(D:) x 1. Equi-
variantly sew in a solid torus 0V on which we have an
ordinary action by matching the principal orbits on the cor-
responding torus boundaries, but sending qo to a cross-
section curve oq on the boundary of 0V satisfying the
homology relation 0m ~voq + bh, where am is a bounding
curve on OT and h a principal orbit on OT(OT = Bd 0V).

We send —qo to Oq to retain orientation.

40
The resulting closed 3-manifold M is denoted by the
symbol {b:(o;g,0,0);(a1,61), ---, (an,Bn)}. Observe that
this symbol also describes an effective 80(2) action on M
without any fixed point sets and with exactly n isotropy
groups ZQ, a = ai, for i - 1, ---, n, corresponding to

the exceptional orbits of the n solid tori which were

sewn in.

Theorem [14, Theorem 2]: Let 80(2) act effectively
on a closed, connected, orientable 3-manifold M with no
fixed point sets. Then there exists a 3-manifold with

standard action M1 = [b:(0,9.0.0)7(al.61).°°°.(anlfinll

 

and a homeomorphism H : M1 > M together with an auto-

 

morphism a : 80(2) > 80(2) such that for all m 6 M1

and g 6 50(2), H(g(m)) = a(g)H(m)-

Notice that any subgroup Z C 80(2) with (k,ai) = 1

k
for i = 1,---, n, acts on M with a proper free action.
If we apply Van Kampen's theorem n + 1 times to M1, keep-

ing in mind how the solid tori were sewn in, the fundamental

group of M1 can be calculated. The calculation yields:

_b a. B.
W1(M1) = (ai,bi.qj,h: w*h ,[ai,h],[bi,h],[qj,h],q 3h 3),

where W* = q1 °°° qn[a1,b1] --- [ag,bg].
One final observation is needed before we are in a
position to consider specific examples. Suppose we have

the 3-manifold M = [b;(o,g,0,0)7(a1:51)r ..., (an,Bn)].

Let k 3,2 be an integer such that (k,ai) = 1 for i = 1,...,n.

41
There is a proper free Zk action induced on M by the sub-
group Zk of the given standard 80(2) action. The orbit
space M* - M/Zk admits an effective 80(2) action induced
by the original one on M. We wish to determine a set of
invariants for M* and this induced 80(2) action.

Let T be a solid torus with the standard linear
action (a,v), where so + tk = 1 and 0 < s < k. If we
let T* denote the orbit space T/Zk' T* has an induced
standard linear 80(2) action (a,tv).

Recall the construction of M from the system of in-
variants. Mo = M+ X 81 — Int(n solid tori). We let

M3 = Mo/Zk, which is just M+ x (sl/zk) less the interior

 

of n solid tori. Under the projection w : M > M*.

a principal orbit h is wrapped around its image, h*, k

times. A cross-section q on one of the toral boundaries

of M0 is mapped by v bijectively to a cross-section q*.
Hence the orbit space M* is obtained from M0 by

sewing in the solid tori jV* by matching principal orbits

with principal orbits as before. The only difference in

the construction is that the cross-section jq* on the

boundary jT* of jV* which is matched to the meridian

on the boundary of Mo now satisfies the homology relation

jm o'ajjq* + kth*. We do not require Qj and Bj to

satisfy the normalization conditions 0 < Bj < aj as before.

The solid torus 0V corresponding to b will come down to

one, 0V* in M*, which is sewn in the same as before ex—

cept with b replaced by kb. Therefore

42
M* = (kb:(o.g,o.0);(a1,k51), ---. (an.k6n)]N,

where the N indicates that the invariants are not neces-
sarily in normal form (i.e. we do not require 0 < kBi < ai).
This set of invariants can be put in normal form by reducing
the kBi terms modulo ai for each i and adjusting kb
to compensate for the "untwisting" of the solid tori cor—
responding to the exceptional orbits. This normalization
seems very complicated in general, but is is computable in
our special cases.

We first consider a class of examples which indicates
that we cannot conclude in Theorem 2.16 that the 3-manifolds

which satisfy the hypothesis are products of the form

wxsl.

Example 3.1: Let M be the closed, orientable, ir—

 

reducible 3-manifold {—1;(o,g,0,0);(h+1,1),(A+1,x)], where
A,g > 0. Then M is a proper k-Sheeted covering of itself
for every k E 1 (mod A+l)-Moreover, H1(M;Z) is a free
abelian group of rank 29 + 1, but M is not a product

of the form W X 81.

‘ggpgfi: From the above discussion the orbit space M*
of the induced free Zk action is homeomorphic to
{-k;(o,g,0,0);(h+1,k),(h+1,kk)}N. Computation of
H1(M*;Z) from this Set of invariants reveals that it is a
free abelian group of rank 29 + 1. Using this information

we can determine the normal form for M*. It is easy to

43
see what all the invariants will be except the "b" term.
Hence we have M* = {b':(o,g,0,0);(h+1,1),(h+1,x)} with

b' undetermined. It follows that

-b' h+1 7x A+1‘
H1(M*:Z) = [26 Z] 6 (q1,q2,h =A qrqzh .q1 h .qz h)-
g .
The group given by this presentation is free if and only if
b' = -1. Hence M = M*, i.e. M is a prOper k—sheeted

1 (mod A+l).

covering of itself for k
The given standard 80(2) action is the only one which
can be defined on M [14, Theorem 4]. Therefore M cannot
be homeomorphic to the product of a 2-manifold and 81.
It is interesting to note that if we were to let 9 = 0

in this class of manifolds we would get a handle. That is
31 x 32 = {-1;(0.0.0.0);(>\+1,1),(hi-1AM.

It is natural to ask at this point what conditions of
this type are necessary to characterize products Of'the

form 1W X 81. Kwun has posed the following question.

Question: If a closed orientable 3—manifold M covers
itself properly k times, for every prime k, is M a

product of a 2-manifold and 81?

If we alter the question by only requiring M to
properly cover itself k times for every pgd_prime k, we
can answer in the negative with an example. Let M be a
manifold from the class given in Example 3.1 with A = 1.
Then M properly covers itself k times for every odd

integer k, H1(M;Z) is free, and M is not a product.

44
Another description of M when 9 = 1 may be illustrative.

Let T2 be a closed surface of genus 2. Consider the

 

homeomorphism h : T2 > T2 obtained by interchanging
the holes of T2 in such a way that h has exactly two

fixed points and h2 = 1. Then M = T2 x I/h.

Example 3.2: Let k 3.2 be given and let M be the
closed, orientable, irreducible 3-manifold described by
{b;(o,g,0,0);(a1,61), °-°,(an,Bn)], where b(k+1) + n = 0,
ai = (k+1)Bi for all i, and g,n > 0. Then M is a
prOper k—sheeted covering of itself and is not a product

of a 2-manifold and 81.

nggf: The orbit space M* = M/Zk is homeomorphic to
[kb;(o,g,0,0);(a1,kfil); ...; (an,an)}N. Reversing the
orientation on M* changes the invariants by replacing kb
with -n-kb = b and by replacing kBi with ai-kBi = Bi

[14]. Again M is not a product by [14, Theorem 4].

We now wish to consider some examples related to
Theorem 1.9. In particular we show that we cannot drop
from the hypothesis either that the Zk action be proper

or the condition that H1(M*;Z) have no k torsion.

Example 3.3: Let M be the closed, orientable, ir-
reducible 3—manifold [b;(o,g,0,0);(a,B)}. The given 80(2)
action has the single isotropy group Za' If (k.d) = 1

and k 3.2, there is a proper free Zk action on M such

45
that H1(M;Z) has k torsion. Moreover, no M in this

class fibers over the circle.

Proof: The orbit space corresponding to Z C 80(2)

k
is homeomorphic to {kb;(o,g,0,0);(a,k5))N. From this we

note that H1(M*;Z) = [6} Z] 6 (q,h : A qh-kb,qath). A
29
simple reduction yields H1 (M :Z) = [0 Z] 6 Z

29

kp’ where

p=lba+B

 

To Show that M does not fiber over the circle for

any 9, notice that H1(M;Z) = [0 Z] O (q,h : A qh-b.qah6).
29

where h is the element generated by a principal orbit of
the 80(2) action on M. The order of h in H1(M:Z).
o(h), is always finite. In particular, o(h) = |ba + bl.
But by the next theorem, manifolds of this type do not

fiber over the circle unless o(h) is infinite.

Theorem [15, Theorem 4]: Let M be a closed, orient-
able 3-manifold such that M = {b;(o,g,0,0);(a1,51),...,(an;5n)].
Then either M is a T-bundle over S1 (T denotes a torus
of genus 1) or M fibers over S1 if and only if the

order of the element in H1(M;Z) generated by a principal

orbit h is infinite.

When 9 = 0, the manifolds in Example 3.3 are lens
spaces. In particular, L(B,-a(mod 3)) = {0:(o,0,0,0);(qfi)).
Therefore, if (k,d) = 1, the lens Space L(B,—a(mod 5))

is a pr0per k-sheeted covering of L(k5,-a(mod k6)).

46

Example 3.4: Let M be the closed, orientable, ir—

 

reducible 3-manifold {0;(o,1,0,0);(2,1),(2,1)). There is
a free 22 action on M that is not proper and is such
that the orbit Space M* = (0:(o,1,0,0);(2,1)}. M does

not fiber over the circle, although H1(M*:Z) = Z e Z.

Proof: As in the construction of M, let M0 denote
M less two invariant tubular neighborhoods of its excep-
tional orbits. Then M0 = T+ X 81, where T+ denotes a
torus of genus 1 less the interiors of two disjoint closed
disks. These disks can be chosen such that there is a
free involution of T+ which interchanges its two circle
boundary components. This involution induces a free 22
action on M0 which can be extended to M. It is clear
the M/Z2 is the manifold M* given above. To see that
M does not fiber over the circle, all we need observe is
that the order of the element generated by a principal or-
bit in H1(M;Z) = Z 0 Z O Z4 is two.

In light of [15, Theorem 4], the next theorem indicates
the close connection between the properties of fibering
over the circle and not having k torsion in H1(M/Zk;Z)
when in the class of closed orientable 3-manifolds with

effective 80(2) actions.

Theorem 3.5: Let M = [b;(o,g,0,0):(a1,B1),---,(an,5 )].

n

Let k 3.2 be such that (k,ai) = 1 for i = 1, ..., n.

Denote the elements of H1(M:Z) and H1(M/Zk:Z) generated

47
by a principal 80(2) orbit by h and h respectively.

Then o(h) = k o(h), where 0 denotes the order.

Proof: M/zk = [kb;(o,g,0,0);(a1,k51),°°°,(an,an)].
The calculation of the first integral homology groups of

M and M/Zk yields:

-b OL- B-
H1<M:z> =[2021 6» (qlwqulhm q1---qnh ’qi 1“ 1*
g
_ -b a' 5' -k
H1(M/Zk:Z)=[g>Z] 0 (q1,...,qn,h,x:A q1...qnx ,qi lx l,x=h )
9

It follows by inspection that o(h) = ko(h).

Corollary 3.6: Let M = {b;(o,g,0,0):(d1.61),---(dn.6n

If there is an integer k 3,2 such that (k,ai) = 1 for

i - 1, ..., n, and such that M/Zk =‘M, where Z C 80(2),

k

then M can be fibered over the circle.

[1]

[2]

[3]

[4]

[5]

[5]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

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