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This is to certify that the

thesis entitled

SOME FIBER PRESERVING INVOLUTIONS
OF ORIENTABLE 3-DIMENSIONAL
HANDLEBODIES

presented by

Roger Bruce Nelson

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in WCS

GAO jo AOprofessor

Date July 20, 1976

 

0-7639

 

    
      

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ABSTRACT

SOME FIBER PRESERVING INVOLUTIONS
OF ORIENTABLE 3-DIMENSIONAL HANDLEBODIES

BY

Roger B. Nelson

An involution h on an orientable handlebody M is
a product involution if M admits a product fibering,
M 24A x I, where A is a compact, bordered surface
and I = [-1.1], and if h is equivalent to either
(i) a x idI, where a is an involution on A, or (ii)
idA x r, where r(t) = -t for all t E I. we ask which
involutions of M. *with homogeneously 2—dimensiona1, tame
fixed point sets, are product involutions. It is shown
that the obvious, necessary conditions are also sufficient.
Namely,

l) h is a product involution of type (i) if and only
if the fixed point set of h consists only of
disks and annuli, and.

2) h is a product involution of type (ii) if and
only if the fixed point set F is connected.
orientable, and X(F) = X(M).

Orientable handlebodies may also be viewed as twisted line

bundles over non—orientable, compact. bordered surfaces.

Roger B. Nelson

Necessary and sufficient conditions are also proved for h
to be equivalent to a fiber preserving involution for such

a fibering.

A handlebody M generally admits several different
product fiberings. Accordingly, it is natural to ask when
involutions a1 x idI and a2 x idI, ‘with al not equiva-
lent to a2, are equivalent in M. we prove that two product
involutions are equivalent if and only if they have homeomorphic
fixed point sets which both separate or both fail to separate
M. These results, together with a complete classification of
involutions having l—dimensional fixed point sets. on compact,
orientable, bordered surfaces, provide for a complete classi-
fication of product involutions on orientable. 3-dimensional

handlebodies. The classification on surfaces is provided.

Finally, any compact, orientable, 3-manifold X contains
an orientable handlebody H. such that H and i7:7§ are
homeomorphic. we show that H may be chosen to be h-
invariant if X is endowed with an involution h, having
a tame 2-dimensional fixed point set. Hence, the objects
studied are building blocks of involutions of a much larger

class.

SOME FIBER PRESERVING INVOLUTIONS
OF ORIENTABLE 3€DIMENSIONAL HANDLEBODIES

BY

Roger Bruce Nelson

A DISSERTATION

submitted to
Midhigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1976

ACKNOWLEDGMENTS

I wish to express my gratitude to Professor K.w; Kwun

for his helpful suggestions, guidance and advice.

I must also thank my wife, Fran, for her patience,

encouragement and love.

ii

TABLE OF CONTENTS

INTRODUCTION............. ...... ....... ..... ........ ..... ..1

SECTION
I. INVOLUTIONS ON SURFACES........................4
II. INVOLUTIONS ON 3-DIMENSIONAL HANDLEBODIES......12

III. RELATIONSHIP TO A LARGER PROBLEM........ ....... 34

BIBLIOGMMOOOIO 0000000 .0 ..... .OOOOOOOOOOOOOOOOO ...... .0038

iii

INTRODUCTION

Let X be a topological space. An involution on X
is a period two homeomorphism ‘h:x 4 X. The orbit space
of the involution h is x; = X/o, where xdy if and
only if y = xoh(x): the projection map is denoted by
ph, though the subscript is deleted when no confusion can

arise. The fixed point set of h is F(h) = {x 6 th(x) = x].

Involutions h and g on X and X’ respectively,
are said to be equivalent, denoted by h ~ 9, if and only
if there exist a homeomorphism f:X‘+ X' such that
h = f.1 o g o f. Equivalently, there exist homeomorphisms

f and f such that the diagram
f

x 4 x’
Pht 1P9
* ,*

"h " x
f 9

commutes. In what follows, it is often convenient to find
f, thereby establishing h ..g, by constructing homeo-

morphism E:x; 4 X§* which can be lifted.

In this study, all spaces are compact, bordered n-

manifolds, n = 2 or 3, and are assumed to carry a piecewise

linear (PL) structure. According to Bing [1], this is no
restriction. Attention is restricted solely to involutions
h where F(h) is tame and homogeneously of codimension one
in X. A result of Kwun [7], applied to the double of the
manifold X, guarantees that X may be triangulated so
that h is PL. Therefore, we will assume all maps and

spaces to be in the PL category.

Kim and Tollefson [6] have shown that if A is a compact
surface and h is a PL involution of A x I such that
h(A x BI) = A x a1 (I = [-l,1]). there exists an involu-
tion 9 of A such that h is equivalent to the involution
h' of A x I defined by h'(x,t) = (g(X)ol(t)) for
(x,t) 6 A x I and X(t) = t or -t. If A is a closed
surface, this condition is always satisfied. If A has
boundary, then A x I is a handlebody M. It is natural to
ask for which involutions M admits a product fibering such

that the condition holds.

Section II deals with this question for involutions
with homogeneously 2-dimensional, tame fixed point sets on
orientable handlebodies. Theorems 2.1 and 2.2 provide nec-
essary and sufficient conditions that such an involution be
equivalent to a fiber preserving involution for some product
fibering of M. Theorems 2.1' and 2.2' generalize these
results to M as a line bundle (possibly twisted) over a

compact surface. Theorem 2.3 gives criteria for deciding if

two product involutions are equivalent to each other. The
proofs of Theorems 2.1 and 2.3 are essentially by induction
on the "complexity" of the involution. Two technical lemmas
are necessary in order to reduce this complexity and apply

the induction hypothesis.

Section I provides a complete classification of involu—
tions with l-dimensional fixed point sets on compact, orientable.
bordered surfaces. Together with the results of Section II,
this provides, theoretically, for a complete classification
of product involutions with 2-dimensional fixed points sets

on orientable, 3-dimensiona1 handlebodies.

Section III is a short discussion of a limited relation-
ship between the study of involutions on orientable handlebodies
and the same prdblem on arbitrary, compact, orientable 3—

manifolds.

SECTION I

INVOLUTIONS ON SURFACES

In this section, the involutions with 1-dimensiona1
fixed point sets on orientable, compact, boardered surfaces
are classified. The first step is to classify all such
involutions on closed surfaces. The orbit space MI ‘will
play an important role in this and later sections. we
need to know that M? is a manifold: it is useful to know
what the orientability or non-orientability of M? says
about the involution. The following two lemmas address

these concerns.

Lemma 1.1. Let (M be a compact n-manifold (possibly
with boundary), and h an involution of M.*with homogeneously
codimension-l fixed point set F. Then M; is a compact,

bordered n-manifold with 8(M;) = ph(BM) U ph(F).

Proof: If x t F, then x has a regular neighborhood

NK 2 1Rn (or IR: if x 6 6M) which is mapped homeomorphically

onto ph(Nx), a neighborhood of p(x) in Mi. If x e F - BM,
n

then x has a small invariant neighborhood Nx a 1!. Since
F n Nx separates Nx into two pieces homeomorphic to

*
1R2, p(Nx) 2 JR: and is a neighborhood of p(x) in M .

If x E F n BM. then x has an invariant neighborhood
Nx a- ]R: which is separated by NK n F into two pieces,
each homeomorphic to IRE. p(Nx) 9: IR: is a neighborhood

of p(x) in Mi.

Lemma 1.2. Let M. be a compact, orientable n-manifold

 

carrying an involution h with F(h) having homogeneous
codimension one. Then F(h) separates M if and only if

*
Mh is orientable.

Proof: If F separates M, and N(F) is a small,
invariant, regular neighborhood of F, then M - N(F) has
two components M1, M2 such that h(Ml) = M2. p(Mi) =

._.______ * o * . .
p(M — N(F)) = Mh - pN(F) 2.Mh Since pN(F) is a collar of
*
BM;. M; is orientable since Mh 2 Mi c:M.

Suppose F does not separate M. Let B be the
standard n-ball, and r the standard orientation reversing
homeomorphism of B. Let f:B 4 M; be an embedding such

*
that f(B) c:pN(F), a collar of aMh. f lifts in two ways
to embeddings f1, 2
hf2(B) and h is orientation reversing. Since M - F is

f :B 4 M with fins) cN(F). f1(B) =

cOnnected, there is an isotOpy h:B x I 4 (M.- F) x I which

is an isotopy between f1 and fzr. (p x idI)H:B x I 4

0* *
Mh x I is an isot0py between f and fr. Hence, Mh is

non-orientable.

we may now classify the involutions with l-dimensional
fixed point sets on closed surfaces. The analysis is in

two cases, depending on whether or not F(h) separates M.

Case 1. F separates M. As in Lemma 1.1, M - N(F)

O
is a disjoint double cover of M; - pN(F) 2 Mg. Hence,
—— 'k *
X(M) = xm - N(F)) = 2X(Mh) = X(2Mh).

* 1' *
where 2Mh denotes the double of "11' Since M and 2Mh
are orientable, closed surfaces with the same Euler

* *
characteristic, M a 2Mh. Because aMh a F, by Lemma 1.1,

and X(M;) = 1 x(M), M; is completely determined by F.

If P is any bordered manifold, define the standard
reflective involution of 2P as follows. Let Pi' i = :1,
be disjoint c0pies of P, with the points of Pi represented
by the ordered pairs (x,i), x 6 P. Then 2P =

g
x e 6P. Let r:2P 4 2P be given by r((x,t)) = (x,-t).

2
(U Pi) U (BP x [-1,1]), where g(x,i) = (x,i) for all
1

Clearly,
(2?): = prw1 uglapx{1} (a? x [0,1]))
3' Pl UgIBPle} (5P1 X [0,1]).

* * * .—
Let szh 4 Mb U (BMh x [0,1]) homeomorphically. f
* .-
lifts to a homeomorphism f:M -. 2Mh such that h = f 1 o r . f.
That is, h is equivalent to the standard reflective involu-

tion on 2M;.

Suppose M. has genus g, and F(h) consists of k
simple closed curves (hereafter, 100ps). Then XIM) = 2 — 29
implies that X(M;) = 2 - (g + 1). If M? is Obtained from
MI by capping 8M; by k disks, then X(Q*) = 2 - 2(9i%359.
Since 3* is an orientable, closed surface, 9i%:k- is a
nonnegative integer. Hence, it is necessary that k 3:9 + l
and that g + 1 - k is even. Clearly, these are also
sufficient conditions on g and k for the existence of

an involution with separating fixed point set of k loops

on the closed surface of genus 9.

Case 2. F does not separate M. Then k g 9, other-
/\
wise F(h) would separate M. Let M - N(F) be obtained from Mu-N(F)

of capping 5(M.- N(F)) by 2k disks. hIM - N(F) extends

A /\
to h, a free involution on the closed surface M.- N(F)

A * C I * I
of genus g - k. (M.— N(F))fi is homeomorphic to Mh ‘Wlth
5M; capped by k disks and is therefore non-orientable

*
since Mh is non-orientable. Hence, M - N(F) is the unique

/\ * .
orientable double cover of (M - N(F)) and h is the

orientation reversing covering transformation. If g - k is
A *

even, then (M - N(F))Q is the connected sum of a projective

plane and 9/2 tori (figure a). If g - k is odd, it is

the connected sum of a Klein bottle and iii. tori (figure b).

some:
1 N1
w alb

The components of N(F) are the annuli N1....,Nk,

and thj is equivalent to id 1 x r ‘where Nj a S1 x [-1,1]
S

and r(t) = —t for all t 6 [-1,1]. N: is again an

annulus, and
r”\. A* A
u; = (M - N(F))E #ul #Magu;

The 9: may, of course, be attached anywhere according to the
homogeneity of manifolds. There is therefore a homeomorphism
f:M; 4 M;.. where h' is any other involution on M. with
non-separating fixed point set homeomorphic to F(h). f

lifts to a homeomorphism f:M 4 M. such that h = f o h’ o f.
Hence, any involution 'h' on M, ‘with F(h') a non-
separating collection of k loops, is equivalent to the
involution described above. k g g(M) is a necessary and

sufficient condition for the existence of such an involution.

An orientable, closed surface of genus 9 therefore
admits [[2%;;]] nonequivalent involutions with 1-dimensional
fixed point set, 9 'with nonseparating fixed point set, and
[[sggj] ‘with separating. The involution is completely
determined by two pieces of information: the number of

fixed point components, and whether the fixed point set

separates. Equivalently, in view of Lemmas 1.1 and 1.2,
the involution is completely determined by MI, a compact,

*
bordered surface with X(M ) = l - g.

The classification of involutions with 1-dimensiona1
fixed point sets on compact, bordered surfaces follows
quickly. Observe that if the fixed point set F intersects
a boundary component B, it must do so in exactly two
points, since any involution of S1 is free or has two
fixed points. Also, if B n F = ¢, then h(B) n B = ¢L
The alternative is that B is invariant, so that hlB is
revolution through 180°. Then B can be capped with a disk
D, and h extended into D ‘with a single fixed point.
However, an involution on an orientable surface cannot have
both isolated fixed points and loops of fixed points, since
near the first it would be orientation preserving and near

the latter, orientation reversing.

Cap each boundary component of M: by a disk, getting a
closed surface 9, and extend h to ‘h. The resulting
involution has already been classified. Any involution on a
compact bordered surface M, may be constructed from an
involution on the uniquely determined closed surface M.‘by
deleting (i) invariant disks or (ii) disjoint pairs of disks
Dil, Di2 such that h(Dil) = D12. *Again M; is a bordered
surface. In this case, however, Mh does not completely

determine the involution h.

10

A boundary component of M; may be (i) the image of
a boundary 100p of M, (ii) the image of 100p of fixed
points disjoint from 3M, or (iii) a circut of alternating
arcs of boundary point images and fixed point images.
Suppose f:M; 4 Mi, homeomorphically. f lifts to a
homeomorphism f:M.4 M, if and only if fixed point images
are sent to fixed point images and boundary images to boundary
images. The homogeneity of (M;.) guarantees a homeomorphism
of Mi, inducing any desired permutation of the boundary
components. Hence, there exists a liftable f if and only
if there is a one-to-one correspondence between the boundary
components of M; and M;. associating components of exactly

the same type.

Since the number of boundary components of M; con-
taining fixed point images is simply the number of components
of F(h), and since M; is orientable if and only if (Q);
is orientable, h and h' are equivalent only if h“ and
‘9’ are equivalent. The number of components which are
entirely the image of boundary is the number of pairs of
disjoint disks removed from .8. Finally, the different con-
figurations of mixed boundary components correspond to the
different ways the remaining disks can be drawn from the k

components of F(h). It has thus been shown

11

Proposition 1.1. Let M, be the compact, orientable,
bordered surface with d boundary components and
X(M) = 2 - 2g - d. Up to equivalence, the involutions of
M 'with 1-dimensional fixed point sets are in one-to-one
correspondence with the ordered 4-tuples (_,k,p,a), where
i) the blank is filled by s or n according as
whether or not the fixed point set separates.
ii) 05kgg+l and g-k isoddwhenthe
first coordinate is 8,
iii) 0 g:k g.g when the first coordinate is n.
iv) ogpgg, and
v) a is one of the ways d - 2p disks can be with-

drawn from k indistinguishable loops.

Remark. The classification of involutions on non-
orientable, compact, bordered surfaces, with each component
of F(h) is 1—dimensional, is very simdlar to the above
procedure. There are just two significant differences. First,
we may introduce isolated fixed points when capping the
boundary. The number of these fixed points is always even,
and they may be connected, in pairs, by arcs ai such that
the invariant loops ai U h(di) = ‘i are all disjoint and
nonseparating. Invariant regular neighborhoods of these loops
are deleted as are such neighborhoods of fixed point loops.
Second, the resulting free involution h on M‘:’N(FII:IN?U‘11)
is no longer unique, except on the Klein bottle. Instead,

according to [8], there are two inequivalent free involutions.

SECTION II

INVOLUTIONS ON 3-DIMENSIONAL HANDLEBODIES

In this section we prove necessary and sufficient
conditions for an involution h on an orientable handlebody
M, ‘with F(h) homogeneously 2-dimensional, to be equiva-
lent to a product involution for some fibering of M as
A x [-l,l]. These results are generalized for M as a

twisted line bundle. Some technical lemmas are necessary.

Lemma 2.1. Let D be a bordered (n-l)-manifold

 

pr0per1y embedded in the n-manifold X, where BX is
connected and Hn_1(X) = 0. Then D separates X if and

only if an separates BX.

Proof: The "only if" direction is immediate. Now

 

consider the following commutative diagram.

P
.. Hn_1(D) .. Hn_1(D.aD) 41 Hn_2(BD) ..
s 11; :12

4 Hn_1(X) 4 Hn_1(X.BX) p42 Hn_2(ax) 4

5D separates aX implies that i2p1 is trivial, and
therefore that pzi1 is trivial also. But Hn_1(X) = 0
implies that p2 is one-to-one. Hence, i1 is trivial, and

D therefore separates X.

12

13

The Objects of this study are 3-dimensional, orientable
handlebodies, which all have trivial second homology and
connected boundary. Two dimensional fixed point sets there-
fore separate if and only if their boundaries separate.

This fact is used in Theorem 2.2.

The proofs of Theorems 2.1 and 2.3 are by induction.
To apply the induction hypothesis, in each case, we need
the existence of an invariant disk to out along and the
assurance that the result of this cutting is still a handle-

body. The next two lemmas address these prdblems.

Lemma 2.2. Let D be a disk prOperly embedded in an

 

orientable handlebody M, and let N(D) be a regular neighbor-
hood of D. The components of M - N(D) are orientable

handlebodies.

 

£5222: Suppose D does not separate M, M’ = M.- N(D)
clearly has connected boundary. Hence, there is a 3-cell
C in M' withaface E in BM’ such that M’ nN(D) CE.
Then

M = M' U N(D) (M’ - C) U (C LIN(D))

where (MT-:75) n (C UN(D)) = E, a disk. FTC- 2M'

and C L|N(D) esT, a solid torus. Therefore, M is
homeomorphic to the disk sum M} A T [5]. According to Gross
[S], orientable, irreducible 3-manifolds with non-void connected

boundary admit a decomposition into A—prime manifolds, which

14

is unique up to order and homeomorphism. By uniqueness,

M' is an orientable handlebody. If m = M1 U M2,
then ,M 2.M1 A M2 and M1 and M2 are both orientable
handlebodies.

Lemma 2.3. Let M. be an orientable handlebody admitting

 

an involution h 'with homogeneously 2-dimensional fixed
point set F containing a non-disk component. Then there
exists a properly embedded, invariant disk D such that
either (i) D is non-separating and cutting along D lowers
the genus of both M and F, or (ii) D separates M into

two non-trivial components.

Proof: Choose an arbitrary disk satisfying condition

 

(i). If this cannot be done, let E be a disk not separating
M and not lowering the genus of F. By the arguments

which follow, we may assume h(E) n E = ¢L Cutting along
both E and h(E) leaves at least one handlebody .M’, with
g(M’) < g(M), and an involution h’ such that F(h’) con-
tains a non-disk component. Repetition of this process leads
eventually to a 3-ce11 with an involution having a non-disk
fixed point component. But this is impossible since

.23 rank Hi(F:ZZ) g l [4; Thm. 4.3]. Hence, there is a

éigk D satisfying (i). we now show D may be chosen so

that h(D) = D, or else there exists an invariant disk D”

satisfying (ii).

15

A disk D in M is in h-general position modulo F
if D and h(D) are both in general position with respect
to F, and if D - F and h(D) - F are in general position.
When this occurs, there are seven possible types of inter-
section curves in D n h(D): (i) simple closed curves (here-
after, 1oOps) in D - F, (ii) loops in D meeting F in
a single point, (iii) loops contained in F, (iv) arcs with
both endpoints in F, (v) arcs with both endpoints in BM - F,
(vi) arcs with one endpoint in F and the other in 5M - F,

and (vii) arcs contained in F.

Let Z) be the collection of all properly embedded disks
ambient isotopic to D, and in h-general position modulo F.
Define the complexity of D 6 Z) by c(D) = (a,b,c,d), ‘where
a is the number of lOOpS in D n h(D), b is the number of
arcs of type (iv) or (v), c is the number of arcs of type
(vi), and d is the number of arcs of type (vii). Order
the complexities lexicographically and choose D to have
minimal complexity. Claim that c(D) = (0,0,0,l). If not,
we construct D’ 6 23' ‘with c(D’) < c(D), contradicting the
minimality of c(D), or an invariant disk D' £‘Z3 separating

M.

Case 1. a # 0. Let L be an innermost loop on h(D),
bounding the disks E on D and E’ on h(D). E U E’ is
a sphere bounding a 3—cell C, since M is irreducible.

Let N(C) be a small regular neighborhood of C, and E” a

16

regular neighborhood of E in D n N(C). If I is type
(i), N(C) is chosen so that N(C) n hN(C)== ¢: if L

is type (iii), N(C) is assumed invariant. Let Gt be

an isot0py of M, 'with support in N(C), moving E”
through C and off of h(D), and leaving BE” fixed. Let
D’ = G1(D). If t has type (i), then two loops are deleted
from D n‘h(D); if L has type (iii), one 100p is removed.
In each case, D’ n h(D') contains no loops not already in

Dnh(D), so D623 and c(D’) (c(D). If 1. has type

(ii) it is treated like an arc of type (iv) in Case 2.

Case 2. a = O and b # 0. Choose a to be an
innermost arc, of either type, on h(D). That is, the disk
on h(D), bounded by a and an are either in h(D) n F
or h(D) 0 6M, contains no other arcs or 100ps in its in-

terior. The condition a = 0 guarantees its existence.

If a is type (iv), then a U B, B an arc in h(D) n F,
bounds disks E' on h(D) and E on D. Again, E U E'
bounds a 3-cell C ‘with small regular neighborhood* N(C).

Let N be N(C) pinched down along the arc B. N(C) and

N are chosen carefully so that N rth = B. Let G be the

t
isot0py of M, 'with support in N, which moves E through
C and off E', and leaves 6 fixed pointwise. Let

D' = 61(1)). Then 13' n h(D') = D n h(D) - a - h(a) so that

c(D') < c(D). This procedure is also used to eliminate

looPs of type (ii), in Case 1, by allowing B to be degenerate.

17

If a is type (vi), then u U B'. B' an arc in
h(D) n BM, ibounds a disk E’ on h(D). Also, a U B,
B an arc on D n BM, bounds a disk E on D, where
E #‘h(E’). Suppose B U B' bounds a disk E” on 3M,
Then (E U E”) U E’ bounds a 3-cell C 'with face E” in

BM such that N(C) n hN(C) = O} Let G be an isot0py of

t
M, 'with support N(C), which moves E through C, B
across E”, and E off of E’. Let D' = 61(D) and we

have c(D') < c(D).

If B U B’ does not bound a disk on BM, then

E'= E U E' is a disk properly embedded in M such that

 

E n'hE = O. The components of M.- N(Eb - hN(E) (possibly

just one) are handlebodies of lower genus then 'M. Start

 

the entire process over on a component of M - N(E) - hN(E).
On some eventual component, this difficulty will not occur,
and we will go on to find an invariant disk. M is rebuilt
from this component. we may therefore assume that this case

does not occur.

Case 3. Suppose a = b = O and c ¥ 0. Let a be an
innermost are on h(D) of type (vi). The 100p a U B U Y’,
where B is an arc in h(D) n F = D n F and y' is an arc
in h(D) n BD, bounds a disk E' on h(D) containing no
intersection arcs or loops in its interior. Similarly,

a U B U Y ‘bounds a disk E on D, where v c:D n 5M,

Suppose V U y' bounds a disk E” in BM. Then E U E’ U E'

18

bounds a 3-cell C, with face E’ in BM and regular
neighborhood N(C). Let N be N(C) pinched along B

and chosen so that N n'hN = B. Let Gt be an isotOpy with
support in N which moves y across E', B through C

and off E’, and leaves all points on B fixed. D’ = Gl(D)

and c(D') < c(D), c having been lowered by two.

If y U y’ does not bound a disk on BM, then
E = E U E' and ‘h(E) are disks, properly embedded in M,
meeting, not transversely, in one fixed point arc. This

case may therefore be ignored, by the argument of Case 2.

Case 4. a = b = c = O and d # 0. Let a be an
innermost are on D. The 100p a LJB, with B : aD, bounds
a disk E on D such that E contains no intersection arcs.
Similarly, a U h(B) bounds a disk h(B) on h(D), where
h(E) contains no arcs. If B U h(B) bounds a disk on aM,
then the isot0py of Case 3 provides D' 6 23' with

c(D’) < c(D).

If B U h(B) does not bound a disk on BM, then cutting
along the invariant disk 3 = E U h(E) yields either two
nontrivial handlebodies M1 and M2 ‘with genera less than
g(M) or a single handlebody M3 'with g(Mf) < g(M). If
5' does not separate, then S'n F cuts F as desired since

X(M - N(I'S'H = mm) + 1 and xm - N(B'H X((M - N(B) n F)mod 2

[4: Thm. 4.4].

19

Since the originally chosen D cut F, it cannot be
isotOped to a disk D' ‘with c(D') = (0,0,0,0). Hence, we
are always brought to the situation of the preceding para-

graph, and, therefore, to the conclusion of the lemma.

Theorem 2.1. Let M ibe an orientable handlebody with

 

involution h, where F(h) is a collection of disks and
annuli. There exists a compact, bordered surface A such

that M 2,A x I, and h is equivalent to a x id where

10
a is a nonidentity involution of A.

Proof: Define the order of h by o(h) = (g,a,d),

 

where g is the genus of M, and a and d are the

number of annulus and disk components of F(h), respectively.
The proof is by induction on the lexicographically ordered
orders of h. ‘we "begin" the induction in Case 1 and show

the inductive step in Cases 2 and 3.

Let D be an invariant, properly embedded disk in M.
N(D) is an invariant regular neighborhood of D in M
such that N(D) a D x [-l,1], ‘with D corresponding to
D x {0] and BD x [-l,1] a regular neighborhood of BD in
BM. If D is nonseparating, M’ =m is an orientable
handlebody with an induced involution h'. If D separates,
EFZ‘fiTBT'= M1 U M: where M1 is an orientable handlebody
with an induced involution hi' Let Di denote D x {i} on

Mi, M’, or N(D).

20

Case 1. o(h) = (9,0,0). we must show that every
orientation reversing, free involution of an orientable
solid handlebody M is equivalent to an involution a x idI
on some A x I a M. An involution of this type can be
exhibited for each M. with odd genus g. Let 82 be
endowed with the antipodal involution. Removing 251-in-
variant pairs of disks from 82 gives a bordered surface
is

A ‘with orientation reversing involution a. a x idI

* * i-
the desired involution on .M. .M 2 AG x I, where Ad

2 +1 a x161
is P minus 93-- disks, is the non-orientable handlebody
* 'k
with X(M ) = 1 - g. If M5 is a handlebody for any free
involution h, then h ~'d x idI by the uniqueness of

orientable double coverings.

Let M. be the solid torus. From the proof of Lemma
2.3, there exists a prOperly embedded disk D such that
D n h(D) = O. Then

 

M’=M-N(D)-hn(DT=M1i1M2.

with Mi a 3-cell such that h(Ml) = M2. M1,: = Ml/Ul where

x l'y if and only if h(x) = y and x 6 D U h(D) c:aM1.

h preserves orientation on 3M1, so M; is a non-orientable
handlebody. If g(M) > 1, then M3* is a non-orientable
handlebody by the induction hypothesis, and MI is M’* plus
the handle p(N(D)) = p(hN(D)). Hence, MI is also a non-

orientable handlebody.

21

Case 2. d # 0. Let D be a fixed point component.

Two subcases arise.

Subcase a. D does not separate. By the induction
hypothesis, h’ is equivalent to d’ x idI on .M’ 2.A’ x I,
where d’ is a nonidentity involution of A'. ph. = phIM’,
and ph.(A’) is a compact, bordered surface embedded as the

* *
0-section in the handlebody (M’)h. a (A’hlo x I.

* _ , * * ,)* *
uh _ (M )h. U N(D)h|N(D) where (M h. n N(D)th(D)

ph’(Di)' i = :1. ph|N(D)(Di) and Ph|N(D)(D) are disjoint
disks on B(N(D)*), and N(D)* is a 3-cell. we may assume
that the fibering of M’ is such that ph.(Di) = v x I,
where y is an arc on B(Adt), for there is an isotopy of
Mfit, liftable to M7, carrying ph’(Di) to such a disk.
This follows from the homogeneity of Mfit and the regular
neighborhood theorem [9: Thm. 3.24]. Let G be an isotOpy

t
of BMfit .‘with support off a neighborhood of BF(h’), carrying

I* o ’*
x e ph’(Di) to xo 6 BAG. x I. If y is any are on BAG.
containing xb, then both y x I and Gl(ph’(Di)) are

*

regular neighborhoods of xo in BM£.. Hence, there is an
o I* O
isotOpy Ht of BMh. carrying Gl(ph’(Di)) onto y x I.
(H o G)t extends to an isot0py Kt of M’* *with support

in a collar of the boundary. Deform the original fibering

*
of Mg. 'by K11 to get a new fibering with the desired

proPerty. Since Kt may be constructed to fix a neighborhood

of F(h’), Kt lifts to an isotopy it of M’. Deform the
fibering of M3 by R; so that the projection p is fiber

preserving.

22

N(D) is a 3-cell and F(hIN(D)) is a disk. Therefore,
pN(D) is also a 3-cell. Impose a fibering on pN(D) as
A”* x I so that the fibering agrees with the fiber structure
of Mfit on p(Di) and such that p(D) = 6 x I for an arc

* * * * ,* * .
5 c aA” . Then Mh EFAa x I. where Ad = Aa' UAA” . Lift

_ *
the fibering to M 2.A x I where A = p 1A . Clearly, h

is equivalent to a x idI.

Subcase b. d = l, a = 0 and D separates M.

'k

*
Mh 2 Mi 2 A x I where ph(D) = y x I ‘with y an arc on

*
BA , by the argument of Subcase a. Then M a.A x I where

l

_. *
A = p A , and h «to x idI on A x I.

Case 3. d = 0 and a # 0. Lemma 2.3 provides an in-

variant disk D transverse to F(h).

Subcase a. D does not separate M. In this case,
g(M’) < g(M), and F(h’) contains fewer annuli then F(h),
so o(h’) < o(h). Again, we assume that Mg: is fibered so
that ph.(Di) = Yi x I, with Yi an arc on 5(Aét). Since
D n F(h) #’¢, the isotopy cannot be carried out off a neigh—
borhood of F(h’) as in Case 1. we must be careful that an
h’-invariant neighborhood of F(h') remains set-wise invariant

under the isotopy.

ph.(Di) c ph.(BM'), and ph’(Di) n ph.(F(h’)) is an arc

B1 on a loop component of ph.(BF(h’)). There is an isotOpy

23

of ph.(BF(h’)) taking Bi to an arc of the form x0 x I
o I* I o I*

‘Wlth xo 6 BAa.. This extends to an isot0py Gt of aMh.

which leaves ph.(BF(h )) invariant and takes ph.(Di) to

a set Yi x I, as pictured below.

P

 

 

’* . ,
G extends to an isotOpy Kt of Mh. . leaVing ph.(F(h ))
invariant. Hence, Kt lifts to an isot0py fit of M’.

Deform the fiberings as in Case 2.

p(Di) and p(F(hIN(D)) are disks with a common arc on
B(pN(D)), for i = :1. Impose a fibering on pN(D) as
A”* x I, so that p(Di) = Y{ x I, ‘with Y; an arc on BA”*,
and where vi and Y; are identical before cutting along D.

Also require that p(F(hIN(D)) = n x I. ‘with n c BA”*.
, _ * ,* *
and that Yi n n be a pOint. Then Mh = Mh. U pN(D) = A x I,

'k ’1' ”'k . ’1' I, 'k ’ .
where A = A . U A 'with A . n.A = y. = y.. Lift to a
a a i i
.. *
fibering of M as A x I, where A = p 1 . A is endowed

with an involution a and h ~.a x idI.

24

Subcase b. Lemma 2.3 provides a separating, invariant

 

disk D. Use the argument of Subcase a to fiber each
*
)

*
(M hi x I for Yi é 6A0

*
i as A0 x I so that phi(Di) = y.

i 1 i
*
Fiber pN(D) as A” x I so that p(Di) = y; x I with

*
y; c BA” , i = i1, and the fibering of p(Di) agrees with

* _ *

* 4 * *
where A = A U A” U A . h e'd x id as before.
a1 02 I

* *
1Upmm U(M2)h 2A x1
2

An orientable handlebody can also be fibered as a twisted
line bundle over some non-orientable compact bordered surface.
The proof of Theorem 2.1 may be adjusted to give the following

extended result.

Theorem 2.1’. Let M. be an orientable handlebody with

 

involution h such that F(h) is a collection of disks,
annuli, and Mgbius bands. Then there exists a compact, bordered
surface A such that M is a bundle over A ‘with fiber

I = [-l,l], h is equivalent to a fiber preserving involution
h’, and h’ restricted to the 0-section is a nonidentity

involution d on A.

Proof: Define o(h) = (g,a,d) where a is now the

 

number of annulus and Mgbius band components of F(h). Let

I

q be the bundle projection. we may assume that ph’(Di) =

- ,*
q’ 1(Y). Where y is an arc in BAa’, by the arguments of

25

Theorem 2.1. pN(D) is fibered as before. In Theorem 2.1

the fibers of ph.(Di) had a well defined positive direction,
*

and we joined Mg. and pN(D), preserving the direction

of the fibers.

In Case 1, choose a local positive direction of ph.(Di)
arbitrarily and then carry out the constructions as before.
In Case 2, choose local positive directions on ph’(Di)'

i = :1: so that h’((ao,t)) = (a'(ao),t) for a0 c: yi c: aAht.
If D cuts an annulus component of F(h), repeat the pre-
vious construction. If D cuts a MEbius band, then attach
the handle pN(D) to Mflt by a direction preserving map at
ph.(D1) and a direction reversing map at ph.(D_1). A* =

* '1'
AC; U A” is now a non-orientable, bordered surface. and
l
'k *
Mh is a twisted line bundle over A . Lift this fibering to

l

.. *
M as a twisted line bundle over A = p A . h is fiber

preserving and hIA is a nonidentity involution.

If a fiber preserving involution of a line bundle over a
compact, bordered surface has a 2-dimensional fixed point set,
then it is either the identity on each fiber or the identity
on the 0-section. The last two theorems gave necessary condi—
tions for the first case: the next two theorems provide

necessary conditions for the latter.

26

Theorem 2.2. Let M be an orientable handlebody with

 

involution h, where F(h) is a connected, orientable,

bordered surface with X(F(h)) = X(M). Then there exists

a compact, bordered surface A, such that M a A x I and

 

h a'idA x r, where r(t) = —t for all t E [-l,l] = I.
Proof: F separates M into homeomorphic pieces M+
and M;. This is clear, by Lemma 2.1, when M is a solid

torus, since F is an annulus and any two loops separate a
torus (BM). Suppose g(M) > 1 and Lemma 2.3 provides a
nonseparating invariant disk D, cutting F(h). X(F’) =
x(M’) where M’=M_:—NTBT and F'=-F—-_—N_(_BT, so that F’
separates .M’ by induction hypothesis. F n N(D) also
separates N(D) into N+ and N_. If F does not separate
M, then N+ and N_ each attach to both M4 and N_. The
handle N(D) is therefore attached equivalently to M’ 'with

a twist, so, contrary to assumption, F is non-orientable.

If Lemma 2.3 provides an invariant disk D, separating
M into M1 and M2. then X(F(hIMi) = X(Mi), i = l or 2.

Otherwise, the restriction that

23 rank Hj(F(h'Mi)’zz) g 23 rank Hj(Mi;ZZ)
i=0 i=0

[4; Thm. 4.3]
is violated for one component. Hence F(thi) separates

Mi' and it follows that F separates M.

27

There is a retraction ri:M 4 Mi defined by ri(x) = x
if x 6 Mi and ri(x) = h(x) if x 5 Mi' ri*:1r1(M) 4 1rl(Mi)
is surjective. Furthermore, i*:w1(BM) 4 11(M), induced by

inclusion, is surjective.

Let BM.= F U F where F1 0 F2 = BM n F = BF. Then

1 2

BMi = F U Fi’ From the commutative diagram

slam ———> "1”‘1’

i*tonto 1j*
"1”” 92:2, 1%)
r.
i

*

it follows that j*:w1(Fi) 4 ”1(Mi)' induced by inclusion,

is onto. Furthermore, 2X(Mi) = x(M) + X(F) and XKBM) =

2x(M), so that X(Fi) X(M) = x(Mi). Hence, by Brown

[2; Thm. 3.4], Mi 2 F x [0,(-1)1] 'with F corresponding to
F x [0}. Therefore, M a F x [-l,1] and h is equivalent

to idF x r.

Theorem 2.2’. Let M. be an orientable handlebody with

 

involution h such that F(h) is connected and X(F(h)) = X(M).
Then there exists a compact, bordered surface A such that
M is a bundle over A with fiber [-l,l], h is equivalent
to a fiber preserving involution g, g is the identity on

the O-section, and g restricted to any fiber is r.

28

Proof: The proof is by induction on the genus of M.
When g(M) = 0, the result is well known. we use the

notation of Theorem 2.1.

Case 1. Lemma 2.3 provides an invariant, nonseparating
disk D. By the induction hypothesis, M’ is a bundle over
A’. Let F(h’) 0 D1 = Yi. we may assume that Di = q’-1(Yi)
where q’ is the bundle projection, by an argument similar
to Theorem 2.1, Case 3, carried out on Mfit and lifted to
M'. 0n N(D). give each Di the fibering inherited as a
subspace of M”, without specifying a positive direction.
This imposes fiberings on pDi, i = :1, on the 3-cell pN(D).
Extend to a fibering of pN(D) as p(F(hIN(D))) x [0,1],
where p(F(hIN(D))) corresponds to p(F(hIN(D))) x {0}. Lift
to a fibering of N(D) as F(th(D)) x [—l,1] so that
th(D) preserves fibers. Reattach N(D) to M' so that

F(th’) u F(th(D)) e- F(h).

Case 2. Lemma 2.3 furnishes an invariant separating
disk D. As above, we assume that ‘Mi is fibered as
F(hi) x [-l,l], where D1 = Yi x [—l,l] for Yi c:BF(hi),

and vi = Di n F(h). Fiber N(D) appropriately and reattach.

That the hypotheses of Theorems 2.1 through 2.2’ are
necessary, as well as sufficient, is immediate. Note that
for handlebodies with g(M) g;2, all possible 2-dimensional

fixed point sets (bordered surfaces satisfying the constraints

29

of Theorems 4.3 and 4.4 of [4]) satisfy the hypotheses of

at least one of these theorems. Hence, all such involutions
are equivalent to fiber preserving involutions. Any handle-
body with g(M) 2 3 admits an involution, with 2-dimension
fixed point set, satisfying the hypotheses of none of the
theorems, and therefore, not equivalent to a fiber preserving

map.

Simple criteria for membership in the class of product
(or fiber preserving) involutions with tame, 2-dimensional
fixed point sets have been given. A full classification of
the members is the natural next step. The classification has
two parts: first, the construction of all members of the

class, and second, criteria for detecting redundancies.

In view of Section I, the first part is completed by
determining all possible factorizations of the orientable
handlebody M as A x I. we seek all compact, bordered
surfaces A with XJA) = X(M). Such a surface is completely
determined by its Euler characteristic and number of bound-

A A
ary components, b. X(A) + b = X(A), where A is the closed

surface Obtained by capping BA. But xXA) x(M) = l - g(M)
and X(£) = 2 - 29(A), so (*) b + 29(A) = g(M) + 1. Hence,
M 2.A x I where A is constructed by removing b disks
from a closed surface A, satisfying (*). For example,

A
b + 29(A) = 3 has two solutions, b = l and 9 = l, or

30

b = 3 and g = 0. Hence, the double solid torus can be
factored in two ways where A (i) is a torus minus a disk,
or (ii) a sphere minus three disks. The handlebody of genus

9 has [[agl]] such factors, determined with equal case.

The prOblem now arises whether two involutions,

a1 x idI and a2 x idI, might be equivalent even though

a1 and a2 are not equivalent, perhaps even being defined

on different spaces. Homeomorphic fixed point sets can
normally be generated in several different ways. It is common,
in fact, to find homeomorphic fixed point sets with different

separation properties. For example, let M 'be the handle-

body with genus five. Among its involutions are:

(i) d1 x idI on A1 x I where A1 is a torus minus
four disks. a1 is constructed from the separating
involution with two fixed point loops by removing
two invariant disks from one loop, and removing
an invariant pair of disks disjoint from the fixed
point set.

(ii) a2 x idI on A2 x I where A2 is the double torus
minus two disks. a2 is constructed from the
separating involution with three fixed point loops
by removing one disk from each of two lOOpS.

(iii) d3 x idI on A2 x I. GB is constructed from the
nonseparating involution of £2 ‘with two fixed

point 100ps by removing two disks from one 100p.

31

In each case, the fixed point set is two disks and an annulus.
The first two fixed point sets separate, but the third does
not. The following theorem indicates which of these involu-

tions are equivalent.

Theorem 2.3. If h and g are product involutions on M

 

with F(h) a F(g) 2-dimensional, and F(h) and F(g) are

both separating or both nonseparating, then h «:9.

Proof. First we prove the special case when <3(h) =

'k *
(9,0,d). Since h ~.d x id on M.2.A x I, Mb esAa x I

I
where A; is a compact, bordered surface completely
determined by the separability of F(d) and x(F(a)).

Hence, M; is a (possibly non-orientable) handlebody com-
pletely determined by the separability of F(h) and X(F(h)).

*
Therefore, M; E‘Mg under the above hypotheses.

In order to show h «o9 we must find a homeomorphism
f:M; 4 M; which can be lifted to furnish the following

commutative diagram:

M 5 M
ph! lpg

* 4 M*

Mh f g

f can be lifted if and only if fph(F(h)) = pg(F(9)). The

 

"only if" direction is immediate. Now let M: = M - N(F(_)).

32

Then (p_IM:) is a covering projection, and M: is the
o* 'k
unique orientable double cover of M; . Hence, k = fIMfi

lifts to 12m}; 4 Mg’ such that
E(5h(F(h) x (ill) = 59(F(g) x [1]). i = :1.

where N(F(_)) a F(_) x [-1,l], and M = M: U_ N(F(_)).
a

09 o k a db induces a homeomorphism j:N(F(h))* 4 N(F(g))*
which lifts to the homeomorphism j:N(F(h)) 4 N(F(9)).
since hIN(F(h)) «'gIN(F(g)). Then
'f = (E U inn}; U N(F(h)) 4 M’ U N(F(g))
.- - g .-
ah O‘h a9

covers f.

If f does not satisfy the required condition, then
change f as follows. By the homogeneity of BM; and
the regular neighborhood theorem [9: Thm. 3.24], there is
an isotopy of BM; carrying fph(F(h)) to pg(F(g)). This
isotOpy extends to an isotopy G on M;. Replace f by

t
the liftable homeomorphism G1 0 f.

The remainder of the proof is by induction on o(h).
Since h and 9 are product involutions, we can, without
using Lemma 2.3, find invariant nonseparating disks Dh

and D9 cutting F(h) and F(g). respectively. Mg =

 

M - N(F(h)) and Mg have induced involutions h’ and g”.

33

with o(h’) = o(g') < o(h) = 0(9). By the induction
hypothesis, there exist homeomorphisms k:Mfi 4 Mg and

* .—
k:M}_:*. 4 Mg’. such that pg. 0 f = f o pfa. As above, we
assume that kph’(Dh x {1}) = pg’(Dg X (ii). i = :1.

where N(D_) corresponds to D_ x [-l,l].

(pth(Dh))(Dh x i) and (pglN(Dg))(Dg x i) are faces
of the 3-cells th(Dh) and pgN(Dg), respectively,
and k induces an isomorphism k, between. E, extends to
a homeomorphism. £:N(Df)* 4 N(Dg)*, taking fixed point
images to fixed point images. Now let f be

’*

(k U 1?):Mf’

U N(D )* 4 M'* U N(D )*
f 9" 9

taking ph(F(h)) to pg(F(g)). f lifts to sz 4 M,
so that h ~49.

SECTION III

RELATIONSHIP TO A LARGER PROBLEM

Information about involutions on 3-dimensional
orientable handlebodies contributes to a limited knowledge
of involutions on arbitrary, compact, orientable 3-
manifolds. Suppose that X is a closed, orientable 3-
manifold carrying a PL involution h where F(h) is
2-dimensional. X; is a boardered, compact 3-manifold.
The triangulation of X; lifts to a triangulation of X
such that the l—skeleton is h-invariant. Let N be an
invariant regular neighborhood of the l-skeleton. By
[10; p.219], N and X7:—N' are invariant homeomorphic
handlebodies. Then, any involution on X is constructed
and N

from two handlebodies, N with involutions h1

l
by attaching N1 and N

2!

and h along their boundaries

2' 2
by a homeomorphism f such that hllBN1 = f-1(h2IBN2)f.
Downing [3] has extended the result of [10], proving
that every compact 3-manifold X, ‘with non-void boundary,
contains a handlebody which is homeomorphic to the closure
of its complement. If Bi' i = l,...,n, are the boundary

components of X, let X’ = X Ug (UiMi), where Mi is the

34

35

handlebody with BMi 2.8.

1, and ngi maps Bi homeomor—

phically to BMi. X' is a closed 3-manifold, and, according-

ly has a decomposition into handlebodies Hi and H’ =

2
X' - Hf. Via [3: Lemma 1], each M1 is isotOped in X’

 

onto Ni' where N1 = Ni 0 H5, i = l or 2, are specially
j
positioned in H1 and H2. It is shown that X alx’ - U Ni
1
-— ’ = ' ...
and that H1 - H1 - 9 Nil and H2 H2 U Ni are
i 2

homeomorphic handlebodies with H2 = (X' — U Ni) - H1 .

The isotopy of X’ moving each M, to Ni yields a
homeomorphism f:X’ 4 X’ such that f(Ni) = M1 for all i.

f(Hl) and f(Hz) are homeomorphic handlebodies such that

 

f(H2) = X - f(Hl). we do not know that f(Hl) and f(Hz)
are invariant under h. Hi and H5 may however, be assumed
to be invariant. Hence, if h can be extended to an involu-
tion h’ of X’, and if the above mentioned isotopy can

be constructed to commute with h’, then f(Hl) and f(HZ)

are invariant.

To extend h to h’ 'we need only know that any involu-

tion (free or with l-dimensional fixed point set) on BMi

is equivalent, by 9;, to the restriction of an involution

l

on Mi' Attaching M1 to X ‘by ngi = 91 allows h to

be extended into "1' Let A be the compact, bordered surface

A .
with A x I 2 Mi and A a 82. If hIBMi is free or has a

separating fixed point set, then hIBMi is equivalent to

36

(a x idI)IB(A x I), where o is the free involution on

A or an involution with only arcs of fixed points. Suppose
h!BMi is nonseparating with k fixed point loops. Let

B be such that B x I E‘Mi and B a S1 x 81. Let B be
the involution on B constructed by removing k invariant
disks from F(g) and the appropriate number of invariant

pairs of disks disjoint from F(g). Then (5 x idI) [as x I) ~

hIBMi.

Presently, A x [-l,l] is embedded in X’, by i, such
that h’li(A x [-l,l]) preserves the induced product structure.
Using an invariant collar C on Bi in X', i can be
extended to an embedding, also called i, of A x [-l,3]
into X’, such that i(A x [-1,3]) n B1 = i(A x l), and so
that h’li(A x [-l,3]) is fiber preserving. By [6, Thm. l3],
hlc ~ (hIBi) x id[1’3] or c 213i x [1.3]. Since A x {1}
is an invariant subspace of Bi' i(A x [1,3]) is accordingly
an invariant subspace of C 'with h|i(A x [1,3]) fiber
preserving. If T x [-l,3] of [3: Lemma 1] is replaced by
A x [—l,3], then the isotoPy taking Mi to Ni' with
support in i(A x [-l,3]), may be assumed to commute with
h’. If B1 is not invariant under h, then any isotopy

G 'with support in a small neighborhood of Mi' ‘will suffice

t0
so long as h’Gt is applied near Mj = hMi° we have shown:

37

Proposition 3.1. Let X be a compact, orientable,

 

3—manifold with PL involution h, where F(h) is homogene-
ously 2-dimensional. Then there exist homeomorphic h-
invariant handlebodies H1 and H2 in X such that

we have shown that involutions, with 2-dimensional fixed
point sets, on arbitrary, compact, orientable, 3-manifolds,
can be built up from involutions on handlebodies. Unfortunately,
this is not done is a unique way. The relationship between
the pieces and the whole seems, in fact, almost hopelessly

complicated.

BIBLIOGRAPHY

BIBLIOGRAPHY

[1] Bing, R.H., An alternate proof that 3-manifolds can
be triangulated, Ann. of Math. (2) 69 (1959), 37-65.

[2] Brown, E.M,, Uhknotting in M? x I, Trans. Amer. Math.
Soc. 123 (1966). 480-505.

 

[3] Downing, J.S., Decomposing compact 3-manifolds into
homeomorphic handlebodies, Proc. Amer. Math. Soc.
24 (1970). 241-244.

[4] Floyd, E.E., Periodic maps via Smith theory, Seminar on
Transformation Groups, Princeton university Press,
Princeton, 1960, pp.35-47.

 

[5] Gross, J.L., A unique decomposition theorem for 3-
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[6] Kim, P.K., and Tollefson, J.L., PL involutions of
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[7] Kwun, K.w;, An involution of a closed 3—manifold with
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[8] Orlik, P., On the extensions of the infinite cYclic
group by a 2-manifold group, Illinois J. Math. 12
(1968). 479- 82.

[9] Rourke, C.P., and Sanderson, B.J., Introduction to
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[10] Seifert, H., and Threlfull, w., Lehrbuch der Topologie,
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38

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