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This is to certify that the

dissertation entitled
PIECEwISE LINEAR HOMEOMORPHISMS 0F
PERIOD 2” ON THE SOLID KLEIN BOTTLE

presented by
RAFAEL ALBERTO MARTINEZ PLANELL

has been accepted towards fulfillment
of the requirements for

Ph.D. degreein MATHEMATICS

{EFT/LIA, ,C; 4...,

M ajor professor

Date July 28, 1983

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PIECEWISE LINEAR HOMEOMORPHISMS OF
PERIOD 2n ON THE SOLID KLEIN BOTTLE

BY

Rafael Martinez Planell

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1983

ABS TRACT

PIECEWISE LINEAR HOMEOMORPHISMS OF
PERIOD 2n ON THE SOLID KLEIN BOTTLE

BY

Rafael Martinez Planell

,In this thesis we classify piecewise linear

homeomorphisms of period 2n on the solid Klein bottle.

It is shown that up to equivalence there are five

distinct involutions on the solid Klein bottle, K.

Also, for n > 1, there are only two equivalence

classes of homeomorphisms of period 2n on K.

AC KNOWLEDGMENTS

I wish to thank Professor Kyung Whan Kwun for

suggesting the problem and for his patience and guidance

during the course of my research.

ii

TABLE OF CONTENTS

Page
INTRODUCTION . . . . . . . . . . . . . . . . . 1
CHAPTER
1. NOTATION AND PRELIMINARIES . . . . . . 3
2. STATEMENT AND PROOF OF THE
MAIN RESULTS . . . . . . . . . . . . 8
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . 29

iii

INTRODUCTION

All spaces and maps will be in the PL category.

In this theses we classify piecewise linear
homeomorphisms of period 2n on the solid Klein bottle

X.

Two homeomorphisms f :M.a M and g :N 4 N are
said to be equivalent (written f ~ 9) if there is a
homeomorphism k :M.4 N such that k’lgk = f.

1 i

If k- gk = f for i # 1, then we say that f

and g are weakly equivalent.

Let 0 denote disjoint union and I = [0,1]. Our

results are as follows:

I - The involutions on X are determined up to
equivalence by their fixed point set. The
possible fixed point sets are I 0 pt.,

D2 O I, 5’, an anulus, a MOebius band.

II - For n'z 2, there are exactly two weak
equivalence classes of maps of period 2n

on K.

In proving the above results, we make extensive use

of the following theorems of P.K. Kim and J.L. Tollefson:

Theorem: ([5])
Let F be a compact surface and let h be a
PL involution of FxI such that h(anI) = anI
(I denotes the unit interval). Then there exists a map
9 of F (with g2 = identity) such that h is equivalent
to the involution h’ of F)<I defined by
h'(x,t) = (gtx),>.(t)) for (x,t) e FxI and Mt) = t

or l-t.

Theorem: ([6])

Let h be an involution on a compact 3-manifold M.
Suppose that there exists a prOperly embedded disk D
in M such that 6D lies in a given component B of
OM and 5D does not bound a disk in B. Then there
exists a disk S, properly embedded in M, with the

properties:
(i) as c E
(ii) as does not bound a disk in B

(iii) either hiS) n S = O or h(S) = S and
8 lies in general position with respect

to Fix(h).

CHAPTER 1

NOTATION AND PRELIMINARIES

All spaces and maps will be in the PL category.

A homeomorphism. h :M.» M is said to be periodic

if hm = identity for m an integer greater than 1.
An involution is a homeomorphism of period 2.

A periodic homeomorphism h is said to be free if
h and h1 have no fixed points for all i for which
h1L 5! identity.

Given a periodic homeomorphism h :M.4 M, the

orbit space M/{h> is the quotient Space formed by

identifying x with hlx for all x in ‘M and all i.

The following elementary result will be used without

further notice:

Lemma 1 .
Let h :M.4 M be a free periodic homeomorphism on
a connected manifold M. Let q :M.4 M/Zh) be the natural
projection. Let x E M/<h> and § 6 q'lfx}. Then
M/{h> is a connected manifold, q is a regular covering

map and

4

Nl(M/{h>,x)

 

<h> % ”
q*(N1(M,x))

where (h) denotes the group of homeomorphisms of M

generated by h.

Proof:

See for example section 57 of [10].

Let h :M.4 M. The fixed point set of
h = {x 6 M [x = hx} will be denoted by Fix(h) or

by F(h).

It is well known that if h :M 4 M is simplicial
and periodic, and if M” denotes the second barycentric

subdivision, then:

1. For every i, Fix(hl) is a subcomplex

of M”.

2. The natural cell structure of the orbit
Space M”/{h> and the projection

q :MV 4 M”/Zh> are simplicial.
3. q maps each simplex homeomorphically.

4. An h-invariant subcomplex of M”, has an

h-invariant regular neighborhood.

The star of a vertex x of a simplicial complex M,

will be denoted St(x).

A compact, not necessarily connected 2-manifold F

is said to be 2-sided in M if there is an embedding

5

h :F x[-l,l] 4 M with h(x,O) = x for all x 6 F and

h(F x [-1,1]) m BM = thF x {-1.11).

A surface F is properly embedded in M if

FflaM=AF.

Let F be a two-sided surface properly embedded
in the 3-manifold M. The manifold M’ obtained by
cutting M along F is the manifold whose boundary

contains two copies of F, F and F such that there

1 2'

is a natural projection g :(M’, F U F2) 4 (M,F) 'with

1
the property that g |M'--(Fl U F2) is a homeomorphism
onto M-F. If h :F)<[~l,l] 4 M is an embedding then

M’ is homeomorphic to M-h(F)<(-1,l)). So in particular

 

if R is a regular neighborhood of F, M’ e M-R.

A disk D which is properly embedded in M
and such that 8D does not bound a disk in AM, ‘Will

be called a meridional disk.

The boundary of a meridional disk will be called a

meridional simple closed curve (s.c.c.).

We will be using the following spaces:

11 = set of real numbers
C = set of complex numbers
D2 = {z E C ‘Izl g_1}
I = [0,1]
l
S = {z c T ||z| = l}
2

P = projective plane

6

K = solid Klein bottle (as defined below)

2
II

the non-orientable 2-sphere bundle over
S I

By the solid Klein bottle we mean the quotient Space

2
K=-]2-—?f—Ii where (z,t) ~ (E,t+l). Anelement of K

with representative (z,t) is denoted by [(z,t)].

Products of maps are also defined in a standard way,
so for example -2 x(t+ l) : D2 X]! 4 D2 X]? is the function
which sends (z,t) to (-z,t+-1).

When the domain of a product map is not given it

will be assumed to be D2 x R.

It is easy to check that -zx (-t). z x {-t). -Z X (12+ 1).
-z><t and '§)(t all induce maps on X; The induced

maps will be denoted by [-z><(-t)], [z><(-t)], and so on.

The above induced maps are involutions. Their

fixed point sets are given in the following list:

FiX([-zx (-t)l) as I 0 Pt

D201

22

Fix( [2 x {-t)])

22

Fix([—z><(t+-lfl a MOebius band
FiX([-zxt]) NS,

Fix( [3 x tl) as an anulus .

With the above notation, and for n > 1, let

 

cpl= [ZX(t+-;Il-—I)] and $2: [-zx(t+ )]. Then

2n-l

for i = 1,2 and n > 1, ”i is a map of period 2n on

2n-l
X, with Fix(cpi ) z anulus.

If M is a manifold with boundary, the manifold
obtained by taking two disjoint copies of M and
identifying corresponding boundary points is called the

double of M and is denoted 2M.

A 3-manifold M is said to be irreducible if
every embedded 2-sphere in M bounds a 3-ce11.
If a 3-manifold M is irreducible and does not
2 2

contain any two-sided P , then M is said to be P -

irreducible.

CHAPTER II

STATEMENT AND PROOF OF THE MAIN RESULTS

Our main results are:

Theorem 1.
An involution on X is equivalent to exactly

one of:
1 - [-zx(-t)]
2 - [zx(-t)]
3 -- [-zx(t+ 1)]
4 - [-zxt]

5 — [Ext]

Theorem 2.

 

A homeomorphism on K, of period 2n (“.2 2) is

equivalent to exactly one of:

 

l - [2)<(t4- )]

2n-l

 

2-[-zx(t+ {1)}

2n

Both theorems above will be shown simultaneously in

what follows.

Proof of Main Results:

Let h be a homeomorphism of period 2n on X.
n-l
Then h is an involution on K.

From Smith theory (see Theorem 12.1 of [11]) we
n-l
know that since K is a homology l-sphere then F(h2 )
must be a homology r-Sphere where -1 g_r $.1. Hence

2“"1 O
) must be within the list O, S

2 2

2

F(h ,ptuI,ptuD.

2

I U D , I U I, D U D , 8’, an anulus, a MOebius band.

n-l
Note that 2% z N and so h induces an
A
involution h on N. Going through the list of the

A
possible involutions h on N ([3]). we see that there

A . .
is none with F(h) = S0 O, S0 U 82 or 52 U $2. Hence

11-1 0 o
F(h2 ) cannot be one of SO, pt U D2 or D2 U D2.

Us

We are left then with the following seven cases:
-1
2n 2

F(h ) = O, I U pt, D U I, I U I, 8’, an anulus, a

MOebius band.

Case 1.
2n-l
Suppose Fix(h ) = ¢.

LamaZ.

There is no free involution on K.

2:92;:

Suppose otherwise. Let f be an involution on K
with F(h) = O. Let 0 be the orbit space K/{f> and
p :K’4 0 be the natural projection. Since f is fixed

point free, p is a double covering map.

10

Note that O is compact, non—orientable, irreducible
and has as boundary a 2-dimensional Klein bottle. Further,
0 does not contain any two-sided projective plane P,
since otherwise:

1P double covers P and hence consists

p-
of either two 2-sided copies of P or a
sphere. Since X is Pz-irreducible,

p'lP is a sphere which bounds a 3-cell
C. Then p(C) is a 3-manifold bounded

by a projective plane which is impossible.

Since p is a double cover, P*(W1K) e Z: has index 2

in N10 and so by Lemma 11.4 of [1], E a finite normal

 

N1(O)
subgroup H of ITlO such that H .~O Z or 22 *zzz.
n 0
Now, the exact sequence 1 4 H 4 N10 4-—fi— 4 1 and

the manifold O satisfy the hypotheses of Theorem 11.1
(part 3) of [1]. Thus we must have 0 z X. But this is
impossible since TT1(7() = 2 has a unique subgroup of

index 2 and hence a unique double cover (the orientable

double cover). [3

Hence case 1 does not arise.

Case 2.
2n-l .
Suppose Fix(h ) z I U pt.

We may assume that the isolated fixed point of
n-l
2

F(h ) is a vertex x0.

11

Consider the action of h on the boundary sphere
of St(xo). h |a(St(xO)) acts freely with period 2n

on a sphere. This is only possible if n = 1.

Lemma 3.
An involution h on X’ with Fix(h) z I 0 pt is

equivalent to [-z><(-t)].

Proof:
By [6] either 3 a meridional disk D 9 D n h(D) = ¢
or D = h(D) and D is in general position with respect

to Fix(h).

Suppose D = h(D). Then by general position,
D n Fix(h) consist of a single point x. If x #'xo
then by [5] we get a contradiction. Hence x = x0. Now,
using a small enough regular neighborhood of D, we get

a meridional disk D’ 9 D’ n h(D’) = O.
So we may assume D n h(D) = O.

Cut along D U h(D) to obtain components U1, U2,
each homeomorphic to D2)<I. Since F(h) #’O, we must

have h(Ul) = U and h(Uz) = U Further by [5] we

1 2'
may assume h|Ul~-z—x(l-t) and h‘U2~-zx(l-t),
were both -z-x(1-t) and -zx(l-t) have szI as

domain. Thus we can find two disks D1, D2 with

D C U1, D C U

2 2'
U D

such that h(Di) = D. and

1 1

F(h) c D

l 2'

ll

__. -__._ _._

 

 

12
Let V1 and V2 be the closures of the components
of K-—(Dl U D2). Then we must have h(Vl) = V2.
Let ‘h = [-z><(-t)]. It is easy to check that 'h

is an involution with F(fi) = I U pt.

Let '51 = {[(z,l/2)] |z 6 D2] c X and
D? = [[(z,O)] l2 6 D2] czxp NOte that the '5: are

disjoint disks and Fix(E) C El U 52' Let VI, V2 be

the closures of the components of K-—(Di U 55).

Since both h |D2 and T)- '62 fix only one point
and both h |D1 and ‘Elifil fix a prOperly embedded line

segment, we have a map t :D 4 Di, t :D 4'5’ such

1 2 2
that th = ht. Extend t to a homeomorphism t.:Vi 4 Vi.

On V2, define t by 'hth. Note that the definitions

of t on V1 and on V2 agree on V

(since here th = Ft).

0 V = D U D

1 2 l 2

Let x 6 X. If x 6 V2 then tx = hthx and so
1 then since h(Vl) = V

t"fitx = hx. If x e v

1 2'

'fithx ='EIt |V2)(hx) =‘B(fith)(hx) = tx and so t-Lhtx = hx.

Therefore h ~ {-2 x (-t)] . D
Case 3.

2n'1 2 -
Suppose Fix(h ) z D U I.
2n-l .
Since F(h ) m D2 U I is h-invariant, we must

2 2n—l

have h(D) = D (where D a D c F(h )). Let U be

an h-invariant regular neighborhood of D such that
n—l

U n F(h2 ) = D. Then U n X-—U consists of disjoint

13

meridional disks D’, D” with the prOperty that

h(D’) = D”. Clearly D n (D’ u D”) = O.

Note that h2(D’) = D’ and so h2 leaves a point
n-l
fixed in D’. Then h2 leaves the same point fixed in
n-l
D’. This is impossible unless h2 = h2 = identity.

Hence h is an involution.

Lemma 4.
If h is an involution on X’ with ,Fix(h) z D2)<I

then h ~ [z><(-t)].

Proof:

Let D’ be as above. Then D’ n hD’ = O.

Cut X along D' U hD’ to get two components

U and U each homeomorphic to D2)<I. NOte that

l 2
by [5] we may assume that h |U1 ~ 2)<(1-t) and
h|U2~Ex(1-t) where both zx(1-t) and Exu-t)

are maps on D2><I. Now proceed as in the proof of

Lemma 3. D
Case 4.
2n-l ,
Suppose Fix(h ) z I U I.

Lemma 5.

There is no involution on K leaving fixed two

line segments.

14

Proof:
Suppose otherwise. Let f be an involution on

X with F(f) = I 0 I.

By [5] and [6], E a meridional disk D in

K 3 D n f(D) O. Let N be a regular neighborhood of

DaNnFIf)

O. Then K-N contains two COpies of D,

D and D such that K is obtained back from X-N

l 2’
by identifying D1 with D2 via a homeomorphism O.

 

Note that x-.N-f(N) consists of components Ui'

i = 1,2, each homeomorphic with D2)<I. Further note

that f(U ) = U..
1 1
_ ~ - 2
Let Hui—ti. By [5]. fi zx(l-t) (on D xI).

Now, we may obtain K back from U1 and U2 in

a two step procedure.

First, identify Dl with D via O, to obtain

2

a manifold homeomorphic to D2)<I (see diagram).

15

In this new Space, the map on lel(e Dzyco) induced
by f2 O fl is the same as the map induced by f2f1°
This map (f2f1) can be considered as being from

D2x0 to D2xl in szI.

Finally, we Should obtain K from D2)<I by

identifying x in DZxO with fzflx in D2x1. But

using the fact that fi ~ E)((1-t) ‘we see that hzhl

is orientation preserving, which is impossible.
Thus no involution on X leaves two fixed line

segments. D

From the above lemma, we see that case 4 does

not arise.

Case 5.

Suppose Fix(h ) m 5'.

Lemma 6.
Suppose h is an involution on X ‘with

Fix(h) m S’. Then h ~ [-z)<t].

Proof:
By [5] and [6], 3 a meridional disk D 3 D = h(D)

and Fix(h) is in general position with respect to D.

Cutting K along D we obtain a component
U as D2 x I where K is obtained from D2 x I by
identifying (x,O) with (O(x),l) where O is an
orientation reversing map of D2. Also, by [5],

h ‘U ~ (-z><t) ID2)<I.

16

Let p be the orbit map of h. Let O’ be the
map of D2(z p(D)) induced by O such that O'p = pO.
2

Then the orbit Space of h is viewed as 2—$§1-.

2
Now, 'E—figl is either a solid torus or a solid Klein

bottle. It can't be a solid torus since otherwise p IBK
would be a double cover of the 2-dimensional torus which is
2

D )(I

impossible. Hence X/{h> z z X.

Let f = {-2)<t]. The above argument gives
K/<f> e K-

Let q :X 4 K/Kf) be the orbit projection. Note that
we may assume that p(F(h)) is the "core" of X. Similarly
for q(F(f)). Then there is a homeomorphism

t :X/<h> 4 K/{f> mapping p(F(h)) onto q(F(f)).

Let S be the 2—dimensional Klein bottle. Clearly
t°p=9r-F(h) 4 X/<f>-q(F(f)) and q =X-F(f) 4 X/<f>-
q(F(f)) are double covering maps. Since I
w1(K/Kf>-q(F(f))) e W1(S) has a unique subgroup of
index 2 isomorphic to N1(S), E a homeomorphism 'E

making the following diagram commute:

x-an ——£——> K-FM)

q

K/<h> - p(F(h)) ——t———> X/<f> - q(F(f))

17

Since t maps p(F(h)) homeomorphically onto
q(F(f)), ‘we may extend 'E to a homeomorphism on all

of X 3 q? = tp.

The following diagrams are commutative:

 

 

r—i—ex
N/P
X
Hence P = EE‘IEE and Since on X-—F(h), p is a

double covering map, it follows that 'Eblfg is the unique
non-trivial covering translation. That is, 'Ehle = h on
K-F(h). Also, Since '3 maps F(h) onto F(f), 'E-le = h
on all of y. D
Lemma 7.

There is no homeomorphism h of period 4 on X

with F(hz) e s’.
Proof:
Suppose otherwise.

Since h2 is an involution, the same argument of

Lemma 6 gives that X/{h2> a K. Clearly h induces an

18

involution 'h on K/{h2>. It is easy to see that
F(E) c P(F(h2)) e 5’. Further, by [3], F(h) can't be

O-dimensional. From case 1, F(h) #’O. Hence F(h) m 5’.

Applying Lemma 6 again to X'm K/Zh2> and '3' we
get that X/{h> z K/Zh) z X; Let P1 :x'4 x7kh2> and
P2 :K/{h2> 4 x/Zh> be the orbit projections.

Note that “F ~ [-z><t] and so 3 a meridional disk
D in x/<h> such that p;1(n) is a disk D’. The

set P11(D’) consists of either a single disk or two

disks meeting at a common interior point.

Consider Pl |5X. This is a double cover of ex by
OK. Let a :I 4 OK be a loop which traverses once around

I _ _
OD . Let bO — q(O) — 0(1). Then Nl(aK,bO) z

1

<a,b [bab- = a—l> where a is represented by a and

b is represented by an orientation reversing loop B

-1

).

which meets a transversely once at b0. Let b0 6 P 0

Since 71(BX) has a unique subgroup of index 2

isomorphic to 71(OX). we must have (P1 IBK)*(N1(BK,bO)) =
(a2,b |bab'l = a'1>.

Let y :I 4 3% be a loop which travels once around
the component of Pil(aD’) containing 50' with

y(0) = y(1) = 50. Then (pl |aK)*([y]) = a2 and so

-1

I
1 (D )

P1 |y(I) double covers q(I) = OD'. Therefore P
is an invariant meridional disk D”.

Now cut K along D” to obtain a component

U s D2)(I. It follows that h |U ~ (iz,t). This is

l9

. . . . D
lmpOSSIble Since h |u Should induce a map on .__¢__'

where O is an orientation reversing map

(O :D2)<O 4 D2)<1), but h does not commute with O.

Therefore,there is no h as assumed.

Lemma 8.
There is no homeomorphism h of period 2n(n‘2 2)

2n-l

on X’ with Fix(h ) m 5’.

Proof:

The proof is by induction on n.

Lemma 7 provides the initial step of induction.

A
Suppose there is no periodic homeomorphism h :X’4 K
n-l
of period 2n with F(h2 ) m S’ (“.2 2).
, . . n+1
We 11 Show that no homeomorphism of period 2
n
on K exists, with F(h2 ) a 5’.
2n
Suppose otherwise. Then h is an involution on
2n 2n
x with F(h ) e s’ and as in Lemma 6, K/<h > as x.
n
Now, h induces a map 'h on X/(h2 > of period 2n. Let
n
p::X’4 K/{hz > be the orbit projection. It is straight-
n-l n
forward to Show that F(h2 ) c P(F(h2 )) ~ 8’. Further,
n-l n
Since h. is an involution on K/th > e K, previous
n-l
arguments apply to Show that F(h2 ) is neither
n-l

O-dimensional nor empty. Hence F(h2 ) a 8’.

But then 'h is a map of period 2n on X’ with
n—l
F(hz ) e S’, a contradiction to the induction

hypotheses.

20
This completes the proof of Lemma 8. l]

The above three lemmas cover all possibilities

in case 5.

Case 6.
2n-l "
Suppose Fix(h ) z a Moebius band.

Lemma 9.
If h is an involution on K’ with Fix(h) z a MOebiuS

band then h ~ [-Z)<(t+-1)].

Proof:

Let h1 = h, h2 = [-zx (t+ 1)], Mi = F(hi’ and

pi :K'4 x/<hi> be the orbit projections. By [S] and [6],

3 meridional disks D. 9 D. = h.(D.). For i = 1,2,
l l l I

cut K along Di to obtain a component Ui z D2)<I.

By [5], hi|Ui ~ zxt.

Note that pi(Mi) is a MOebiuS band in a(X/{hi>).

Since we can obtain x/<hi> as an identification Space

2

from ui/<hi> (~ D )(I), we must have that X/Khi> e K.

Let Di be hi-invariant meridional disks in Ui'
Note that Pi(Di) is a meridional disk in X/{hi>. Let

— I —
oi - a(Pi(Di)) and Bi — a(Pi(Mi)). Then Bi separates

8(K/<hi>) into two MOebius bands. Also, Bi meets oi

in exactly 2 points in such a way that 3(X/{hi>)-ai-Bi

consists of two open rectangles.

21

Now take any homeomorphism from a1 U Bl onto
c2 U 32 and extend it to a homeomorphism

t :B(K/{hl>) 4 B(K/<h2>) in such a way that P (M1)

1
goes onto P2(M2). We can further extend t on plDi.
Finally, noting that K/{h1>-a(Y/<hl>)-p1Di is an
Open 3-ce11, we can extend t to a homeomorphism

t =X/<h1> * X/<h2>.

Since Bi separates a(x/<hi>) then aMi separates
5K and so Mi separates X’ into two components. Each
of the components of x-Mi is mapped homeomorphically

onto X/{hi> by Pi. Hence, since t(P1M1) = P we

2M2 '
have a homeomorphism I? such that the following diagram

 

 

commutes:
K .E *9 K
.1 l .2
K/<hl> t > K/(h2>
Now as in Lemma 6 we can conclude that 'Ehlhéf = hl' [3
Lemma 10.

 

There is no homeomorphism h of K of period 2n
n-l ‘

2

(n > 1) with Fix(h ) w MOebiuS band.

Proof:

Suppose otherwise.

22

n-l
Let c = 5(F(h2 )). Let N be an h-invariant

regular neighborhood of c in 8(X). From Lemma 9,

we see that N is an anulus. So we have that h [N is
n-l
2

a map of period 2n on an anulus with (h |N) orientation

reversing. Since n > 1, this is impossible. I]

Lemmas 9 and 10 complete case 6.

Case 7.
2n-l
Suppose Fix(h ) z anulus.

n-l

As before, we have D 9 D = h (D) and such that

K cut along D is a component U m D2)<I ‘with

2n-l _. 2 2n-l
h ‘U ~ (zxt) |D xI. Then K/(h > is either a

solid torus or a solid Klein bottle.

n-l
Let p :X'4 X/{hz > be the projection onto the

orbit Space.

Lemma 11.
If h is an involution on K leaving an anulus

fixed, then h ~ [Ext].

Proof:
Let h1 = h, P1 = P and h2 = [2)<t]. Let
P2 :X’4 x/Zh2> be the natural projection. Let D1 = D
and Similarly define D2 for h2. Both hi are involutions

with F(hi) an anulus.

It is known (see [7] or [8]) that up to equivalence

there is a unique involution on the 2-dimensional Klein

23

bottle leaving two circles fixed. So in particular,

hl IaK ~ h2 lax. Using this, it is easy to see that
aX-—a(F(hi)) is an Open anulus Ai. Also, Pi(F(hi)) is
an anulus Fi in a(K/<hi>). From the remarks preceding
the statement of this Lemma, we know that a(X/Zhi>)

is either a torus or a Klein bottle. Further,

a(K/<hi>) = Fi U Pi(Ai) where PiAi is either an open
anulus or an open MOebiuS band. Hence the only
possibility is for PiAi to be an open anulus and so

2

a(K/Zhi>) is a torus. Thus K/{hi> z D )(S’.

Note that P1(D1) and P2(D2) are meridional disks
in X/{h1> and X/Zh2> respectively. Let Si be the
segment Pi(Di) n Pi and s; = a(Pi(Di)) .51 so that

S. U Si = a(Pi(Di)). Let t be a homeomorphism of S

l 1

onto 82. Extend t homeomorphically in steps as

follows: first on F1 onto F2, then on Si onto

8 next on PlDl onto P2D2, followed by an

I
20
extension on the Open rectangle a(X/<hi>)-Fl(= plAl)

onto the Open rectangle a(K/{h2>)-F2(= P2A2)' finally

across the remaining Open cell in x/zhl>.

Since Pi |(X-—F(hi)) double covers X/'<h1>--Fi
and since 1T1(X/<hi>-Fi) :3 Z , we can lift t to

a homeomorphism '3 such that:

24

? ‘A

Pi is

t
Y/<h1> - F1 ———-) K/<h2> - F2

 

commutes.

Since PgltPl maps F(hl) homeomorphically onto

F(hz), we can extend 'E on F(hl’ so that Pi? = tPl.
The simplicial nature of the maps insures the continuity
of 'E.

1-_
hzt — hl. D

We can now conclude that 'E-

The following result is well known (see [9]). Since
a proof of it does not seem to appear in print, we will
include it for completeness. For a possible alternate

approach to the proof see [10] together with [11]. We

will need to make use of it later on.

Lemma 12.
Let k be a homeomorphism of period m on
D2 xS ’ . If (k) acts freely on D2 xS’ then k is

weakly equivalent to one of:

 

kl(zl.22) = (zl.w22)
or

k2(zl.22) = (zl.w22)

2Ni 2
I
where w=em . Further, szsz’ and
<k1>

szS’

zK.
<k2>

25

Proof:
D2 8’
Let 0 be the orbit Space I—Z£%—- and let P be

the projection onto the orbit Space. Clearly, P is
an m—fold covering map. Also, P |a(D2)<S’) is an
m-fold cover and so depending on whether k preserves
or reverses orientation, we get a0 m S’)(S’ or

ao w 5% respectively.

Also note 0 is compact, irreducible and does not

admit a 2-Sided P2.

By Lemma 11.4 of [4], we have a Short exact sequence

2 *222 and

N is finite normal. By Theorem 11.1 of [4], we see that

1-0N47T1(C5)4Q-01, where sz or Z

N = l and also if k is orientation reversing
(preserving), O z K (respectively 0 a D2)<S’). This

implies k is unique up to weak equivalence. D

We are now ready for the last result of this paper.

Lemma 13.

 

If h is a self homeomorphism of K of period
n-l
(n‘z 2) and if Fix(h2 ) is an anulus, then either

1
or h ~ [-zx(t+2n_1)].

2n

 

 

h ~ [Z X (t+ 2111-1)]

Proof:

Let h :X’4 X be a homeomorphism of period 2n.
n-l
Suppose Fix(h2 ) is an anulus.

If x E F(h) then we may assume that x is a vertex

in the interior of K and so a(St(x)) is an h-invariant

26

n—l
2-Sphere. Since F(h2 ) n a(St(x)) m S’ and n‘2 2,
n-l
we get the contradiction that h is an orientation

reversing map on a(St(x)). Hence F(h) = O.

Arguments similar to the above permit us to conclude

that F(hl) = O for l g_i < 2n-l.

n-l
Since h2 is an involution, Lemma 11 applies to
2n-l 2 2n-l
give that X/{h > a D )(S’. Let K/{h > = I} Since

F(hl) = O for l g_i < 2n-l' h induces a free action

n-l

h. of period 2 on :3

Now we break the proof into cases depending on

whether '3 is orientation preserving or reversing.

Case a:

Suppose h is orientation preserving.

n

 

_ 1
Let h2 — [z><(t+-2n_l)]. Then h2

n-l 2n-l

has period 2

and F(h: ) w an anulus. Hence x/Khz > z a solid
torus 35. Also, h}, the map induced by h2 on
2n-l

K/{hz > is free, orientation preserving and of period

211—1. By Lemma 12, ”-2/61-2) as D2 xS’ as f/(h).

Let h1 = h,.fl

be the natural projections. Note that Ii/Zhi> w K/{hi>.

=1“ and Pi:}(4.7'i,qi:.7'i4.7'i/<hi>

It follows from Lemma 12 that for i = 1,2, 3 a

meridional disk Di in Ii such that, for L = 2n -1,

_. .2 “—1 . . .
Di' hiDi' hiDi""'hi(Di) are all mutually dlSjOlnt.

27

Then Di projects down via qi onto a disk qiDi

in X/<hi>.
2n—l
Also, Pi(F(hi )) is an anulus Ai 1n ail.
2n-l __
Since F(hi ) is hi-invariant, Ai is hi-invariant.

Since a(X/<hi>) z S’)<S’, ini is an anulus in

a (X/<hi>).

Now, using the Di's, it is easy to construct a
homeomorphism. t :x/Khl> 4 K/Kh2> which maps qlAl

onto q2A2.

Since the qi are covering projections we may lift
t to a homeomorphism 'E':Ji 4 :5 which maps Al onto
A2 and such that qé? = tql. Finally, exactly as with

the homeomorphism t of Lemma 11, we may lift 'E to a

homeomorphism E which makes the following diagram commute:

rtll

WI

ql

(r——- q £————:¥

X/<hl> ———t———) K/<h2>

Note that since ql is a covering projection and

Since

 

t t
K/<h1> 7‘ K/<h2> <————— K/<h1>
commutes, we get that for some 6, 'EblhiE = hi. Further,
. _ —- . . 5 _ —fi _
Slnce Pihi — hipi' it follows that. plhl — hlp1 —
t hztp1 — t h2p2t — t pzhzt - plt h2t. Thus 1f
n-1 n-1 - _
x z Fix(h2 ) then hi+2 (x) = E-1h2E(x). Also,
2n-l 2n-l
Since pl |F(h1 ) is a homeomorphism, for x E F(h1 )
n-l _ _
we get: hi+2 (x) = hi(x) = E'1h2E(x). Therefore
hl ~ h2 (weakly) as desired.
Case b:

Suppose h is orientation reversing.

Then a proof Similar to that of case a yields

 

h ~ [-Zx(t+ )] (weakly). [‘3

l
2n—l
Lemmas 11 and 13 complete case 7 and with this we

finish the proof of our classification theorems.

BIBLIOGRAPHY

[l]

[2]

[3]

'[4]

[5]

[6]

[7]

[8]

[9]

BIBLIOGRAPHY

J. Hempel, 3-manifolds, Ann. of Math. Studies 86,
Princeton University Press, Princeton,
New Jersey, 1976.

P.K. Kim, Involutions on Klein spaces M(p,q).
Trans. Amer. Math. Soc. 268 (1981), 377-409.

, PL involutions on the nonorientable

 

2-sphere bundle over $1, Proc. Amer. Math.
SOC. 55 (1976): 449-452.

, Cyclic actions on lens Spaces, Trans.

 

Amer. Math. Soc. 237 (1978), 121-144.

P.K. Kim and J.L. Tollefson, PL involutions of
fibered 3-manifolds, Trans. Amer. Math. Soc.
232 (1977), 221-237.

, Splitting the PL

 

involutions on nonprime 3-manifolds, Michigan
Math. J. 27 (1980), 259-274.

K.W. Kwun and J.L. Tollefson, Extending a PL
involution of the interior of a compact manifold,
Amer. J. Math. 99 (1977), 995-1001.

M.A. Natsheh and F. Abu Diak, PL involutions of
the Klein bottle, Iraqi J. Sci. 21 (1980),
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G.X. Ritter and B.E. Clark, Periodic homeomorphisms
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Lecture Note Series, 26 (1976), 154-168.

[10] H. Seifert and W. Threlfall, Lehrbuch der Topologig,

Teubner, Leipzig, 1934: Chelsea, New York, 1947.

[ll] P.A. Smith, Fixed points of periodic transformations,

Appendix B in Solomon Lefschetz'S Algebraic
Topoloqy, New York, 1942, (Amer. Math. Soc.,
Colloquium Publications, vol. 27).

 

 

29