E’EECEWESE mania m’vetweas
an '92 x s

I
1

III

 

A Dhsor‘mhon
{or fin Dogma of DR. D.
MICHIGM STATE UNWERSITY
Muhammad Arafat Natsheh
I974

   

THES'S

 

 

thesis entitled

PIECEWISE LINEAR INVOLUTIONS

ON P2 5 “‘81 ,

 

 

 

ABSTRACT

PIECEWISE LINEAR INVOLUTIONS

ON P2 x S1

BY
Muhammad Arafat Natsheh

This thesis is to classify the PL involutions on

P2 x 81. The main technique used is the P-equivariant

surgery developed by Tollefson [10] and Tollefson and

Kim [5]. If h is an involution on a 3-manifold M: we

look for an appropriate surface S properly embedded in M
for Which h(S) = S or h(S) n S = ¢, and then cut M along
S U h(S) to get a manifold M' and an induced involution
h': M’ -—> M', where h' is easier to classify than h.

Pasting back what we cut help us to classify h.

In this thesis our manifold M is P2 x_S1 and the

surface we are looking for is an embedded P2 in P2 x 51.

Lemma 1: Let h.:P2-——> P2 be a PL involution.

Then F ¥ ¢, moreover F = a U {a}, where d is a non-

separating simple closed curve in P2.

Lemma 2: Let th'2 x Sl-—-> Pz'x S1 be a PL
involution. Then there exists a projective plane P embedded

in p2 x s1 such that h(P) = p or h(P) n p = ¢.

yd Muhammad Arafat Natsheh

Theorem 3: Up to PL equivalence there are 3 PL
involutions on P2 x I with fixed point sets homeomorphic
to (i) a projective plane, (ii) a disjoint union of a simple
closed curve and a single point, or (iii) a disjoint union
of an annulus and a simple arc.

Theorem 4: Up to PL equivalence there are six PL

involutions on P2 x S1 with fixed point sets homeomorphic

to (1) P2 u 92, (ii) p2 u s1 u *, (iii) 51 x s1 u 51,

(iv) K u 51, (v) s1 u s1 u s0 or (vi) ¢.

 

PIECEWISE LINEAR INVOLUTIONS

ON P2 x S1

BY

Muhammad Arafat Natsheh

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1974

To my mother and father

ii

ACKNOWLEDGMENTS

The author wishes to express his sincere gratitude
to Professor K. W. Kwun for his helpful suggestions and
stimulating guidance during the research. He also thanks
Professorsl<xw. Kwun and J. L. Tollefson for suggesting

the problem.

iii

TABLE OF CONTENTS

INTRODUCTION . . . . . . . . . . . . .
CHAPTER
I. INTRODUCTORY REMARKS
AND P-EQUIVARIANT SURGERY
2 1

II. PL INVOLUTIONS ON P X S

BIBLIOGRAPHY . . . . . . . . . . . .

iv

12

27

INTRODUCTION

In this thesis we will classify the piecewise linear

(PL) involutions on P2 x 81. Tollefson [10] showed that

up to PL equivalence there is only one free PL involution

on P2 x 81, which is the obvious one.

Since S x S1 is the orientable double covering of

P2 x S1 we will make use of the PL involutions on 82 x S1
which was classified by Tao [9], KWun [8], Fremon [3], and
Tollefson [10]. Moreover, we will use the P-equivariant
surgery developed by Tollefson[10] and Tollefson and Kim [5].
The idea is if h : M ——-> M is an involution on a 3-manifold
M, then we look for an appropriate surface S properly
embedded in M for which h(S) = S or h(S) n S = ¢ and
then cut along S U h(S) to get a manifold M’ and an
induced involution h1 : M’ —-> M’ which is easier to handle

than M.

In case h.:P2 x Sl-—-5 P2 x S1 we will be able to find

a P2 embedded in P2 x S1 such that either h(PZ) = P2

in case F ¥ ¢ or h(P2) 0 P2 = ¢ in case F = ¢: and

cutting along P2 U h(Pz) we get M' w P2 x I and

hl.:P2 x I-—> P2 x I. In theorem 2.2 we classify all

involutions h and this leads to the classification of the

1
. . 2 l 2 l .
involutions h : P x S —> P x S where it turns out that

there are up to PL equivalence five PL involutions with

nonempty fixed point set homeomorphic to (i) P2 U P2,

(ii) 92 u s1 u *, (iii) 81 x s1 u sl, (iv) K u 31, or

(v) S1 U S1 U SO. This together with Tollefson's result

of the free case completes the classification of all involutions

on P2 x 31. Thus up to PL equivalence there are six

PL involutions on P2 x 81.

CHAPTER I

INTRODUCTORY REMARKS AND P-EQUIVARIANT SURGERY

We work in the PL category, all manifolds are assumed
to have a piecewise linear structure and all maps are to be
piecewise linear maps unless otherwise stated.

Sn will denote the n—sphere, Pn the real projective
n—space, K the Klein bottle, and I the closed unit interval
[0,1]. We will use "s.c.c." for a "simple closed curve",

x(M) for the Euler characteristic of M, and if 11:bd-——> N
is a map then l(h) will denote the Lefschetz number of h.

If M is an n-dimensional manifold, then a map h :M.-—€>b4

is an involution if h is not the identity and hoh = the

identity map on M: F(h) will denote the fixed point set of h.

A surface S in a 3-dimensional manifold M is properly

embedded in M if F H BM = BF: two surfaces S and S

l 2
properly embedded in M are called parallel if there is an

embedding of Sl><[-l,l] in M such that 81 = S1 x -l and

S = S1 X 1. A surface S properly embedded in M is two—

2
sided if there is a neighborhood of S in M of the form

S x [-1,1] with S = S x O and S x [-1,1] n BM = as x [-1.1].

A surface S properly embedded in M is one-sided if S does
not separate any connected neighborhood of S.
Definition 1.1: Let S be a 2-sided surface in a

3-manifold M. The manifold M’ obtained by splitting M at

 

S

4

is the manifold whose boundary contains two copies of S

81 and 82 such that there is a natural projection

p: (M’,S U82) ——> (M, S) with PM - (SIUSZ) is a homeo-

1
morphism onto M-S and M’ is homeomorphic to M - (S x(-1,1)).

If S is one-sided, the manifold N obtained by splitting M

at S is the manifold whose boundary contains S a double

1

cover of S and N—S1 is homeomorphic to M-S.

Definition 1.2: Let h be an involution on a manifold

 

M. The quotient space M/h of M which is obtained by iden-
tifying x in M with h(x) is called the orbit space of h
and the quotient map q:lM-——- M/h is called the orbit map.

Definition 1.3: Let hl, h2 : M —-—> M be two homeomorphisms.

h1 and h2 are equivalent if there is a PL homeomorphism
T : M --> M such that th = Th2, in such a case T is called
PL equivariant with respect to h1 and h2.

Definition 1.4: [Tollefson [10] and Tollefson and Kim [5]]
Let h.:M.-—e>bd be a PL involution on the 3-manifold M,
with fixed point set F. Let S be a surface properly embedded

in M. S is said to be in h-general position modulo F if

 

(i) bOth (S, as) and (h(S), 311(5)) are in general position
with respect to F, (ii) S-F and h(S)-F are in general position,
and (iii) all cuts among S, h(S) and F are locally
piercing cuts.

‘We observe that any properly embedded surface in M can
be put into h-general position modulo F by a series of arbi-
trarily small isotopies, and if S meets F at a nonpiercing

point or curve then S and h(S) can be simultaneously pulled

away from F at this place. This can be done by restriction
of h to a small invariant regular neighborhood of F.

Let h:P2 XSZ"-—->P2 xSl, let 2 be the set of all
projective planes embedded in P2 x S1 which are either invariant

and in general position with respect to F or in h-general

position modulo F. For any P 6 EL define the complexity of

 

P, C(P) = (a,tn, where a = the number of components of
[Pflh(P)] - F and b = the number of components of P H F; we
order the complexities in a lexigraphical order.

Remarks 1.5: Any simple closed curve in P2 either bounds

 

a disk and separates P2 or does not bound and is nonseparating.
A nonseparating s.c.c. in P2 is covered by a s.c.c. in S

(the orientable double cover of P2) which is invariant under
the covering transformation; hence any two nonseparating simple

closed curves in P2 has a nonempty intersection.

Any P2 embedded in P2 x S1 does not separate,for P2

does not bound any manifold.

Any embedded P2 C P2 x S1 is two-sided and P2 x S1 - P2

is homeomorphic to P2 x (0,]J. For if p :S2 x Sl’-—+§ P2 x S1

is the orientable double covering then p-1(P2) = S* c S2 x 81.
where 8* is a two sphere which does not bound a 3-cell:

and S* x [-1,]J is a 2-sided regular neighborhood of S*
which double cover P2 x [-1, l]: moreover, S2 x S1 - S* is
homeomorphic to 82 x (0,1) which double cover P2 x S - P .
hence szsl-PzzP2 x (0,1). Let i:P2 -———B PZXS1

be an embedding then 1*:‘W1(P2)-——> r1(P2)(Sl) is a
monomorphism, j*:‘r1(P2)-——§.WI(P2)(I) is a monomorphism too,

where j is an embedding.

Lemma 1.6: Let h :P'2 -——C>P2 be a PL involution with

Fix (h) = F. Then F ¥ ¢; moreover, F = a U {a} where a
. . . 2
is a nonseparating s.c.c. in P .
Proof: Since x(h) = 1 ¥ 0, then F ¥ ¢, and since F
is a submanifold of P, F is a finite number of disjoint simple
closed curves and points.
By Conner [l], x(F) = x(h), hence x(F) = l: and by Floyd [2]

EdimHi(F:Zz) _<_Z‘dim Hi(P2:Z = 3. Hence F has to contain a

2)
single point a and may have at most one s.c.c.

If a is a s.c.c. in F then a cannot bound a disk
because if so then the disk is invariant and its boundary in
F hence the whole disk is contained in F, which cannot happen.
So if a c F, it has to be a nonseparating s.c.c.

Let J be any nonseparating s.c.c. in P2 such that
a t J and .L,h(J) are in general position. J 0 h(J) ¥ U

and either J C F or J n h(J) is an odd number of points,

for by considering the commutative diagram:

2 2

2 2

1 1
H (P ’ZZ) ® H (P :22)

if [J] generates H1(P2,ZZ), 3 its dual and 22 is the

- *—
generator of H2(P2;zz) then <[J]LJ[hJJ, 22) = 1 E Z2, 'hence
the intersection number of J and h(J) 5 1 (modin, i.e. an

odd integer.

h acts as a permutation of order 2 on the points
J n h(J) whose number is odd, hence there is a fixed point
x E J n h(J).

Therefore there is a nonseparating s.c.c. a C F such that
x 6 a and F = a U {a}.

Lemma 1.7: (P—equivariant surgery) Let

h:P2 x Sl-——> P2 x S1 be a PL involution. Then there exists

2 l 2

a p2 c p x s such that h(Pz) = p or h(PZ) n p2 = 95.

Iggggfi: Let ’Z) be the set of all projective planes embedded
in P2 x S1 which are invariant and in general position with
respect to F, or in h-general position modulo F. If there is
a P2 E Z disjoint from F, then we choose P such that its
complexity is minimal among all such P's in. 23 which are
disjoint from F. If every P EIZ) meets F then choose an
arbitrary P E E with minimal complexity.

We argue that C(P) = (0,0), for if C(P) > (0,0) we can
obtain P’ 6:} of the same type with lower complexity by
performing P-equivariant surgery once on P. Hence our original
choice of P must satisfy h(P) = P or h(P) n P = ¢.

Choose P 618 of minimal complexity and suppose c(P) \ (0.0).
In P n h(P) we have the following types of intersection curves:
(a) an isolated point which is in F, (b) a s.c.c. in P-F.

(c) a s.c.c. with one point in F, (d) a s.c.c. in F, (e) a
simple arc with its end points in F.

First, we rule out case (a) using Tollefson argument

[10, lemma 2]. If x is isolated in F then we move P and

h(P) simultaneously off x. If X is a point of a one-dimen-

sional component of F, then let N be an invariant 3-cell

neighborhood of x such that N n F is an arc. Then th

is simply a rotation about this arc,we adjust P and h(P)
slightly so that P U h(P) is in general position with respect
to EN. There are simple closed curves in P n N and h(P) n N
that bound innermost disks (containing x) R<C P and Q C h(P).
R U Q separates N into three components U, V, W, where

RC 8U, h(U) = V and h(W) = W. Clearly F 0 UC W.

Let D be a disk close to and parallel to R such that
IntDC U and SD = BR. Define P’ = (P-R) U D. The only
difference between P 0 h(P) and P’ n h(P’) is that we have
removed the point x, but because of our choice of P this
case cannot appear.

Second: There is a s.c.c. J in P n h(P) of type (b),
(c), (d) or a simple arc d of type (e): since a ¥ h(d),

a U ha is a s.c.c. in P n h(P).

Case I: There is a s.c.c. J'C P n h(P) which bounds an
innermost disk E in h(P): where a surface E is innermost
in h(P) if E n P = ¢ and AB C P U h(P). There always exists
an innermost disk, for if J bounds the disk E in h(P) which
is not innermost we can find a s.c.c. J” C E where
J” C P n h(P) and bounds an innermost disk E’ in h(P).

Hence J'C P n h(P) bounds an innermost disk E in h(P).
J 0 F may be one of the following: U, J, a single point, or
a couple of points.

2

Since J bounds E in h(P), it bounds a disk in P X S1

and hence it bounds a disk in P, for i* and j* are
monomorphisms in:
1* 2 1 3*
vrl<P)-————>vr1<p XS )<———-——rr1(h(p))

9

Hence J separates P into two components E and E let

1 2’

E be the disk. we have either h(J) = J or h(J) ¥ J in

1
either case let U be a small regular neighborhood of E. In
U find a disk E’ parallel to E and such that:

(i) E’ n h(P) = F n J, (ii) BE’ U J bounds a semi-degenerate
annulus A contained in P, which is pinched along J n F,

(iii) the interior of the 3-cell bounded by E U A U E’ is
disjoint from P U h(P). Take P = E’ U (E-A). If P is

not already in general position with respect to F along the
curve J then move P slightly off J where necessary to
achieve this general position. Then P’ 6'2) and c(P’) < c(P).

Repeating the last process we can assume that there are no
s.c.c. d in P n h(P) such that d bounds a disk in h(P):
and hence we can assume all s.c.c.'s in P 0 h(P) do not
bound in h(P) and intersect in a single point x, otherwise
we are going to find a s.c.c. which bounds an innermost
disk in h(P).

Case II: Let E be an innermost surface in h(P) bounded
by two such nonseparating s.c.c.'s d and B. E is a pinched
annulus pinched at x. Now a U 8 ~ 0 in h(P) and this
implies that h(d) U h(B) ~ 0 in P. So h(d) U h(B) bounds
a pindhed annulus in P.

Subcase (a): h(E) 0 E = x a single point. Let U be
a small regular neighborhood of E. In U find a pinched

I

annulus E (pinched at x) which is parallel to E and such

that: (i) E’ H h(P) = x, (ii) aE’ U a U B bounds two pinched

annuli (each pinched at x) A A both in P, (iii) the

1’ 2

interior of the pinched solid torus (pinched at x) that is

10

bounded by E U E’ U A1 U A2 is disjoint from P U h(P).
Now a U 6 separate P into two components E1 and E2
let 12:2 contain h(a) u h(s). Take 9’ = E’ u [32- (AIUA2)],

in this way we get rid of a, B, h(a), h(B) in P n h(P).

Subcase (b): h(E) 0 E = B, h(a) 0 a = x

 

Let U be a small regular neighborhood of E. In U find a

I

pinched annulus E (pinched at x) parallel to E and such

that (i) E’ n h(P) = 5, (ii) aE’ U a bounds a pinched annulus
A.C P, (iii) the interior of the pinched solid torus which is
bounded by E U A U E’ is disjoint from P U h(P). a U B

separate P into two components E and E2 ,let E contain

1 2
h(d). Take P’ = E’ U (E2-A), hence we get rid of a and
h(a) in P 0 h(P).
Subcase (c): E U h(E) = a U B where h(a) = B we get
rid of a and B the same way as in subcase (a).
Subcase (d): Now we can assume that every s.c.c.
a C P n h(P) is nonseparating in h(P) and so in P and
h(a) = a. We have two cases: (i) there exists more than one 0
(ii) there is only one a.
(i) If E is an innermost surface in h(P) which is

bounded by a U B where h(d) = a, h(S) = 6, let h(E) = E1

in P. E U E is either a projective plane which is invariant

1
and we are done (finding invariant projective plane embedded

in P2 x 81) or E U E bounds a pinched solid torus T

l
which is invariant under h. Hence there exists a pinched annulus

E’ c T such that as’ = u. u a and h(E’) = E’. If

h(P) = E1 U E --- U En, h(di) = ai where every ai 18 a

2

11

simple closed curve in BEi, then let P’ = E 1 U E 2 U --- U E’n
P’ 62 and h(P’) = P’.

(ii) P f) h(P)

a a single nonseparating s.c.c. in P
(and so in h(P)). Split P2 x 81 along P we get a space

homeomorphic to P2 x I and the following two cases:

(a) F’ (I (é/ V ‘9"

‘ 0 W1 ‘ 0‘ am

Since P - d z Intlfz, an open disk (a) cannot happen because
h(P) - a is an annulus, (b) only may happen and in this case
P and h(P) does not cross each other. Now a 0 F = ¢,
a or 2 points. a n F = ¢ cannot happen since P is in
h-general position modulo F. In case a C F we move P and
h(P) simultaneously off a, to get P n h(P) = d. So if
a 0 F = [x,y} let U be a small regular neighborhood of
a pinched along x and y. Let A be the semi-degenerate
annulus contained in BU such that BA = BU n P and
Anh(P) =¢. Let A’=Unp andput P’=AU(P-A’).

If P’ is not in general position with respect to F along
a, move P’ slightly off a 0 F to achieve the general position.
P’ E Z) and c(P’) < c(P). Hence in all cases we can achieve

the conclusion of the theorem, i.e. there is a P 6‘23 such

that h(P) = p or h(P) n P = ¢.

CHAPTER II

PL INVOLUTIONS ON P2 X S1

In this chapter we will prove the main theorem of the
classification of PL involutions on P2 x 81. First we
classify the PL involutions on P2 x I, then using this we
prove our final theorem. We will use for P2 the more convenient

one of the two notations: (i) P2 = SZ/~ where x ~ -x V x E 82.

or (ii) P2 = D2,/~ where D2 = {szC (z)=1, ogpgl} and

z ~.-z.
Lemma 2.1: Let h:P2 x I —-> P2 x I be a PL involution. Then

there is an annulus A_C P2 x I, whose boundary components are

non-separating simple closed curves in P2 x o and P2 x l
and such that h(A) = A.

3322:: Let a be a non-separating s.c.c. in P and let
S=axI. (de,axodel) canbe deformed in
(P2 x I, P2 xoU P2 x 1) so that S w a x I is either invarient
and in general position with respect to F (and hence we are
done) or inh-generalpositionmodulo F.

Let Z) be the set of all annuli S c P2 X I. which are
in h-general position modulo F and such that the boundary
components of every S are non-separating simple colsed curves
in P2 x o and P2 x 1. Define the complexity c(S) as before

and choose S EEZ) of minimal complexity.

12

13

Again as in lemma 1.7 we choose E an innermost surface
in h(S). ‘We have the following cases:

Case I: E is a disk in Int(h(S)). Let J = BE. J 0 F
may be one of the following : ¢, J, a simple arc, a point,
two or more components each is a point or a simple arc.

J separates A into two components E and E and

l 2
since J c int(h(S)), J c intss. If E1 and B2 are annuli
then J is homotopic in P2 x I to one of the boundary com-
ponents of S, but each of the boundary components of S is
not null homotopic in P2 x I, hence J is not null homotopic
in P2 x I, a contradiction since J = 8E, E is a disk in
h(S) and so in P2 x I. Hence one of E1, E2 has to be a
disk, let E be the disk. We handle this case the same way

1

as in Lemma 1.7 Case I.

Case II: E is a disk in h(S) which meets one boundary
component ofh(S). Let J=SflE, B=Enah(S). JflF is
the same as in case I. Let U be a small regular neighborhood
of E. In U find a disk E’ parallel to E and such that
(i) E’ U h(S) = J 0 F, (ii) BE’ U J U B bounds a semi-degenerate
annulus A.C S pinched along J 0 F, (iii) the interior of the

I

3-cell bounded by E U A U E is disjoint from S U h(S).

Now J separates S into two components a disk E1 and an

U (E2-A). If S’ is not in gen-

I

annulus E2. Let S’ = E
eral position with respect to Falong J, move 8’ slightly off
J to achieve this general position.

Case III: Now we can assume that S U h(S) is a finite

number of disjoint simple arcs each one of them starts at a

14

point in P2 x o and ends at a point in P2 x l. The number

of these simple arcs is odd because the number of components

of S n h(S) equals 1(mod 2), (same reasoning as in lemma 1.6).
If the number of these simple closed curves is greater

than one, let E be an innermost disk in h(S) bounded by

two arcs J1 and J2, J1 U J2 = E n 3.

Let Bo=h(S)flP2xo

2 .
and let do — S n P x o — CO U Yo’ Where Yo U Bo 18 a non-

Bo U 50’ Where B0 = 60 n E,

separating simple closed curve in P2 x 0 (this is possible

since both do and 60 are non-separating simple closed curves

2 x o).

in P
Let U be a small regular neighborhood of E. In U

choose a disk E’ parallel to E such that

(i) E’ U h(S) = (JlLJJZ) n F, (ii) BE U BE’ bounds a semi—

degenerate annulus A pinched along J n F, (iii) the interior

of the 3-cell bounded by E U E’ U A is disjoint from

S U h(S). J1 U J2 separates S into two disks E1 and E2.

= E’ U (E1-A). If S’ is not

let YO<C E1 then take 8’

in general position with respect to F along J1 U J2, move
8’ slightly off F 0 (JlLJJZ) to achieve this general position.

Repeating this process one finally gets h(S) n S = J
a simple arc, since we cannot get rid of all of them for
h(S) n s 5:! ¢.

Now let P : 52 x [0,1] -——> P2 x [0,1] 'be the covering
map, S n h(S) = J a simple arc. P-1(S) = 8* an annulus in
52 x I which is invariant under the covering transformation.
Same for p‘1(h(s)). s" n P-1(h(S)) = ’31 u 32 two copies of

2

J. Cut along 8 we get a manifold M’ z D x [0.1] and the

15

disk h(S)’ c M’ is h(S) cut along J to get J1, J2 two
copies of J in ah(S)’. Cutting again along h(S)’ we get

2 x I and 8D2 x I is

two manifolds each homeomorphic to D
homeomorphic in each one of these to h(S)’ U S’ which are
pasted along two copies of J.

Now P2 x I - (Sth(S)) consists of two components A
and B. If h(A) = B then P C J' and either F = J or
F is a point in Inth. If F = J then h(P2)<o) = P2 x 0
such an involution h[P2 x 0 has fixed points other then that
in J (by lemma 1.6) hence F = J cannot happen. If
F = a point in InthI: then we rule this out too.

Let B be a small invariant 3-cell around the fixed
point such that B 6: int P2 x I. Let X = P2 x I - int B.
h acts freely on X, let M be the orbit space and

q : X -—> M the quotient map, q is a 2 - 1 covering map.

32
32 P1
ql ’ on
x M X g,
2 2
L S 82
M

16

2

’X = S x [0,1] - (BllJBZ) is the covering space of X,

Where each of B1 and 32 are mapped onto B by the covering
82 x I -—+> P2 x I; Let M, be the orientable double covering
of M.

Since K is simply connected it is a universal cover of

M hence there is a covering P1 :3? —-> M such that the

diagram commutes i.e. PP1 = qql.

X — S - (BlLJBZLJB3LJB4) and P1 15 2-1 covering
projection. P1(Sz)(0) = P1(Sz‘xl) = Si C BM. and
~ 2 ~
P1(BBI) - P1(BB2) — S1 C M, hence P1 can be extended to a

covering projection of S3

3

which implies that

M w P - (BiLJBg). The covering transformation on M is a

free involution with both Si and s: are invariant spheres.

This involution can be extended to P3 w M U B? U B3 such

that T(Bl’ = B1 and T(B2) = 32 and T is free on M: hence
the fixed point set of T (in P3) consists of a couple of
points one in Int B1 and the other in Int 82 , such an in-
volution cannot happen, see [KWun 6, 7 and Kim 4].

Hence h(A) = A. Cutting along 8 U h(S) we get two
manifolds A and B each homeomorphic to 02 X [0.1]. Now
BAD (S-J)’ U J1 U J2 U (h(S) -J)’ = L where (S-J)’ comes
from S - J after the cutting, and the same for (h(S)-J)’:
J1 and J2 are two copies of J. 'h: P2 xil-——§ P2 x I
induces h’ : A -—> A defined as follows:
for x 6 A - L let h’(x) = h(x)’
for x’ e (S-J)’ u (h(S) -J)’ let h’(x’) = [h(xH’

for x 6 J let h’(x) = h(x)’ in J

1 2

17

and for x 6 J2 let h’(x) = h(x)’ in J1.

h’ :A -—> A is an involution with h’ (J1) = J2 and
h’((S-J)’) = (h(S)-J)’, hence there exists a disk DC A
Which is invariant under h’ and J1 U J2 C BD. Pasting
back what we cut we get P2 x I and D goes back to an in-
variant annulus S.

Hence always there is an invariant annulus S C P2 x I
whose boundary components are non-separating simple closed
curves in P2 x o and P2 x l, and S is in general position
with respect to F.

Theorem 2.2: Let h: P2 x I —-> P2 x I be a PL in—

 

volution, then h is equivalent to one of the following
involutions:
(i) h1([pz, t1)
(ii) h2([pz,t]) x I U I
(iii) h3([pz, t]) = [-pz, l-t] with F e S1 U *'

[p2,] -t] with F m P2

1

[-pz, t] with F e S

Egggfi: Since x(h) = 1 ¥ 0, then F ¥ D. By Conner [l]
x(h) = x(F), so x(F) = 1. By Floyd [2]
Zdim Hi(F: 22) gZdim Hi(P2 XI: Z2) = 3. Hence a component of
F may be P2, an annulus, a mobius band, a simple arc, a s.c.c..
or a point.
Since x(F) = 1 and ZdimHi(F: Z2) 5 3, so if we have
a mobius band we have both of P2 x o and P2 x l invariant
and hence F will contain 3 components or more which violates

one or more of the above conditions. Hence this case can-

not happen.

18

2 . .
Case I: P C F, then Since ZhiunHi(P2:Zz)= 3 we have

F = P2 C Int (P2 x I), for F is a properly embedded submanifold

of P2 X I. Now F separates P2 x I into two components

A and B each homeomorphic to P2 x I and h(A) = B. Let

t be any homeomorphism from A onto P2 x [0, -21-] such that

£0”) = P2 x-é- and let h1 : P2 X I ——> P2 X I be defined by
2

h1([pz,t]) = [pz,l-—t]. Define T:.P x1I-——> P2 x I as
follows: for u E A, let T(u) = t(u), and for u E B, let
T(u) = hlth(u). Then hT = Th1 and hence h is equivalent
to hl'
Case II: There exists a simple arc component J’C F, then
h(P2><o) = P2 x o and h(P2)<l) = P2 x 1. Hence Edi a non-
separating curve in P2 x i, i = 0,]. such that do U d1 C F.
and since ZdimHi(F:z2) _<_ 3 there is an annulus A C F
with BA = do U d1. A U J = F because we can not have any—
thing else in F. Cut along A to get D2 x I and consider

the following diagram:

 

 

D2 x I h > D2 x I
q 1’ JL g
h
P2 x I > P2 x I
1

Define h(z,t) to be equal to q- hq(z,t) for (z,t) t BD2 x I

and for (z,t) 6 BD2 X I let h be the covering transformation.

Suppose h’ :P2 X I -—+> P2 xil be any other involution with

2

F’ = A’ U J’ then define h’ : D x I -—-> D2 x I as above.

19

Since h.~ h’ on D2 x I there exists t:D2 X I--> D2 x I

such that fit = th’. Consider the following diagram:

 

 

 

 

 

 

 

 

D2 X I h ' > D2 X I
"'I
D2 X I h > D x I
q q
q’ q’
2 I h 2 \)
P X I > P X I
tI
I
P2 X I h > P x I
. , 2 2
Define t :‘P x I-——§ P x I as follows:
2

for [z,t] e P x I - A let t’([pz,t]) = q't’q‘1([pz,t])
for u 6 A let q-1(u) = [ul, “2} we have h(ul) = 112 and

since h't(u 1) = t(uz) then q’(t(u1)) = q'(t(u2)),

)
1
define t’(u) = q’(t(u1)) = q’(t(u2))

the last diagram commutes and hence h ~ h’ ~ h2 where

h2<rpz.t1) = [-pz.t1.

Case III: There is an isolated point a E F. As in lemma 2.1
F ¥ a and we cannot have an annulus as another component of
F because then there would be more components in F which
violates Zdim Hi(F’ZZ’ g 3. Hence the only possibility is

to have d a s.c.c. in F and hence F = d U {a}.

20

Let x €,d and P-1(x) = {x1,xé} C S1 x I then there

exists a unique involution B such that the diagram commutes

 

(82x1, x1) > (3x1,x1)
2 ’ h 2
(P XI, x) > (P XI,x)

 

If d ~ 0 in P2 x I then p-1(d) = d1 U d2 C F(h) and

there is no such involution with d1, d2 as components of the

fixed point set [5, lemma 6.3] and hence d'/ o in P2 X I.

By Lemma 2.1 there is an invariant annulus AkC P2 x I Whose
boundary components are non-separating s.c.c.'s in P x o
and P2 x 1. A 0 F ¥ ¢ otherwise hlP2><I-(ALJP2)<{o,l})

is an involution of the open 3-ce11 with fixed point set

d U {a}, such an involution does not occur. We have either

d C A, or d n A = x and a E A. (A n d has an odd number

of components and fix(h]A) is either a circle or a couple

of points.)

 

Subcase I: d C A. Let P-1(a) = {a1,a2}. There is a
unique involution E that makes the following diagram

commute:

 

2 h A 2
l P 1’13
2 h 2
(P XI, a) A; (P xI, a)

 

-1 * , - * * *
P (A) = A an annulus for which h(A ) = A and T(A ) = A

where T is the covering transformation.

21

Let P-1(d) = d*, ii(d*) = d* and T(d*) = d*

2 * - - 2
SxI-A—KlUK2,K1~K2~DXI,let a1€K1,a2€K2.
h(Kl’ = K1 and since h(al) = a1 we have h1(a2) = a2 and
F(h) = {a1,a2}. Now h[K1 is an involution with d* invariant

and d*<: BK and has fixed point set [a1], hence there is

1

an invariant disk D with BD = d*

and a1 6 D.
S

Let s = D U T(D) then T(S) = s and 3(3) = for

ET = Th. p(S) = P2 invariant in P2 x S1 and d U {a} C P2.

Subcase 1!: Let d n A = x. Choose y 6 d -A, and left

 

h to h, as before the diagram commutes:

 

 

2 _
(S xI,y) h >(SZXIiy)
1 1
P] [.
(P2XI, y) h a (P2xI,y)

... * ..
h(A) = A* and T(A*) = A . Let P 1(d) = d* and let

2 * - - 2 ..
S x I - A - K1 U K2, so K1 m K: m D x I and h(Kl) — K1

* . - * .-
T(K1) — K2 hence d — FIXIL Let d 0 K1 = d1 and
-1 * -— . . . -
P (a) — {a1,a2} C A . h[K1 is an involution on Kl'~ D2 x I

with fixed point set = d1 a simple arc. Since h(A*) = A*
3 a s.c.c. B such that (dlflA*) U {a1,a2} C B and T(B) = B.

B consists of two simple arcs B and B2 where

1

T(Bl) = 62 = h(Bl) where the end pOints of B1, B2 are
'k -

d1 n A . Let E be any disk in K with BB = d1 U 61.

h(E) is a disk with Bh(E) = dl U 62 and let E,h(E) be

in general position. E D h(E) = d1 U a finite number of

simple closed curves each bounds a disk in h(E) we can pull

22

E and h(E) apart along these s.c.c.'s by performing
P-equivariant surgery once on each s.c.c. to get E such

that h(E) n E = d1. Let D = E U h(E) and S = D U T(D)

2 2

then T(S) = S h(S) and P(S) = P C P x I an invariant

projective plane such that d U [a] C P2. Hence in any case

there exists P2<: P2 x I such that d U {a} C P2 and

2 . . . .
h(PZ) = P . 'We can define an equivariant homeomorphism

2

tzlP XII-—> P2 x I as we did in case II to get th = h t,

3
where h3([pz,s]) = [-pz,l-S].

Corollary 2.3: If h : 92 x s1 —-> p2 x s1 is a PL

involution with fixed point set F ¥ ¢, then there exists a

projective plane P<Z P2 x S1 such that h(P) = P.

Proof: By lemma 1.6 there exists a PC P2 x S1 such

that h(P) = P or h(P) n P = U. If h(P) n P = ¢, cut along

P U h(P) to get two manifolds A and B each homeomorphic
to P2 x I. h(A) = A, otherwise if h(A) = B we have F = 0.

By theorem 2.2 there is an invariant projective plane P*<: intzx.

Paste back A and B we get P2 x S1 and P* C P2 x S1 is

the invariant projective plane.

Remark 2.4: If g: P2-—§ P2 is a homeomorphism then

we can get P2 x S1 from P2 x I by the identification

[pz,o] ~ [g(pz),l]. In theorem 2.5 we will use the following

types of identifications (i) P2 x S1 = P2 x I/[Pz,o] ~ [p2,1],

(ii) P2 x s1 = P2 x I/[pz,o] ~ {-92.1}.

(iii) P2 x s1 = P2 x I/[pz,o] ~ [p2,1].

Theorem 2.5: Let h:P2 x S1 -——> P2 x S1 be a PL

 

involution with fixed point set F. Then h is equivalent

23

to one of the following six involutions:

(i) h1([pzl,22]) [pzl,-22] with F z ¢,

2 2

(ii) h2([pzl,22]) = [p21,22] with F a P U P ,
(iii) h3([pz,t]) = [pz,l-t] with F m P2 U S1 U *,

. - . 1

(iv) h4([pzl,zz]) = [-pzl,22] With F m S U S1 U So,
(v) h5([pzl,zz]) = [p21,22] with F m S1 X S1 U 81,

(vi) h6([pzl,t]) = [-pzl,t] with F e K u 51.
Where in (iii) P2 x I/[pz,o] ~ [-pz,l] and in
(Vi) P2 x S1 = P2 x I/[pz,o] ~ [p2,1].

Proof: By Tollefson [10], there is only one free involution

on P2 x 81, the obvious one, and case (i) is settled. By

Cor. 2.3 there exists a projective plane P C P2 x S1 such

that h(P) = P. Split P2 x S1 along P to get a manifold

homeomorphic to P2 x I and an induced involution

2XI—5P2XI.

h! :P
By Lemma 2.2 h’ is equivalent to one of the three
involutions: (i) hi([pz,t]) = [pz,l-t],

(ii) h§([92.t) = [-pz,1-t]. (iii) h§([pZ.t]) = [-pz,t]-

2 l 2 1
In P x 5 let M+— [[pzl,zz]€P xS [Re2220}, and let
M_ = C1(P2)<Sl-M+), then M+ w M; w P2 x I, also let
P1 = P2 X l C P2 x S1 and P2 = P2 x{-1}C P2 x 81. Now we

have the following cases:

Case (I): h’ leaves P2 X %- invariant and interchange
2

P x o and P2 x 1. In this case after pasting back we get

two invariant projective planes P’ and P” and

P2 X S1 - (P’LJP”) = N U N2 where N N N m P2 X I and

l 1 2

h(Nl) = fiz.

24

Subcase (a): P’ U P” = F. Let t be any homeomorphism
from N1 onto M+ . Define T: P2 x 81 —-> P2 x 81 as

follows: T(x) = x for all x 6 N and T(x) = hzth(x) for

1

Subcase (b): F = P’ U d U {a}, where d U {a} C P”. Let

x 6 N2. Then hT = Th

f be any homeomorphism from N1 onto P2 x [o,%-] which

takes P’ onto P2 x-gé- and d onto [[z,o]]z€P2,[z\=l}

and a to [0,0]. Define

T: P2 x S1 -—> P2 X 81 = P2 x I/[PZ,0] ~ [-pz,1] as follows:

T(x) = t(x) for x 6 N1 and T(x) = h3th(x) for x 6 N2.

Then hT = Th3 and h ~ h3.

Subcase (c): F = d U [a] U B U {b} where d U {a} C P’

and B U {b} C P”. Let t be any homeomorphism from N1 onto

M+, Which takes P’ onto P1 and P” onto P2 and F onto
Fix(h4). Define T as in the last subcase and conclude
hT = Th4 and hence h ~ h4.

Case (II): h’([pz,t]) = [-pz,t]. Let

 

Fix(h’) = F’ a S1 X I U I and let

E = {[92]€P2|o<p<l} ‘8 51 x I. Let g:P2 -—> P2 be a
homeomorphism such that g([—pz]) = -g([pz]). glE is either
orientation preserving, or orientation reversing. Let

q : P2 X I -—-> P2 x SI be the quotient map where

q([pz,o]) = q([g[pz],l]). In case g is orientation

preserving on E we have q(F’) N S1 x S1 U S1 and in
case 9 is orientation reversing we get q(F’) m K U 81.
Subcase (a): Let ‘h: P2 X S1 -—e> P2 x S1 be a PL

 

involution with F = T U d m S1 x S1 U 81.

 

25

1
Let q : (P2 X 81, d) -—-é> (D2 X S , d) be the projection map

2
onto the orbit space. Let h :P2 x S1 ——5 P X S1 be

5
. . 1 l 1
defined h5([pzl,22[) = [-pzl,22[: F1X(h5) = S x S U S ,

l 1 l

2
let q1 : (P2 xS , $1) —-> (D XS , S ) be the projection

map onto the orbit space.
1

Now let t :S -—€>c1 be any homeomorphism. Extend
t to a homeomorphism t : (D2 x SI, 81) -—> (D2 x 51, d) and
define t: P2 x S1 -—€> P2 x S1 as follows:

Choose a i 81 x S1 U 81, and let q-ltq1(a) = [u,v}.

Let t be the unique lefting of

1 1U51)),a)-—>(szsl-(q(T)Ua).tq1(a))

which takes a to u. For y E S1 X S1 U S1 let

tq1:((P2xS -(Sle
t(y) = q-ltq1(y). Then t is well defined and qt = tql.

and hence ht = ths, by the commutativity of the diagram:

 

 

 

 

 

 

 

P2 x s1 _ h > P2 x s1 _
\ t ‘ \t
\ \\\
\2‘) 1 hs $2 1
P X S > P X S
ql ql [
q q
W W
132 x 31 1d ,‘2 I)2 x s1
w . >
1
132 x81 id > 132 x s

 

26

Spbcase (1)): Let h : P2 X S1 --> P2 x S1 be a PL

involution with fixed point set K1 U d, K1 is a Klein
bottle. Then from what we discussed before the orbit space
is N the non-orientable disk bundle over 81. Let

q: (P2><Sl,cn -——> (N,d) be the projection onto the orbit

space. Let h, :P2 x Sl«—--->P2 x S1 be defined as in the

6
theorem, Fix(h6) = K U S"l and let q1 : P2 x S1 -—> (N,Sl)

be the projection map onto the orbit space. Now let
t.:S1 -—€>c1 be a homeomorphism and extend t to a homeo-

. . ~ 2 2
morphism t : (N,Sl) ——->> (N,d) . Define t : P x S1 -—-?> P x S1
exactly as we did in subcase (a) and using a similar argument

we get h N h6.

BIBLIOGRAPHY

10.

BIBLIOGRAPHY

P. E. Conner, Concerning the Action of a Finite Group,
Proc. Nat. Acad. Sci., 42 (1956), pp. 349-351.

 

E. E. Floyd, 0n Periodic Maps and the Euler Characteristic
of Associated Spaces, Trans. Amer. Math. Soc., 72 (1952),

 

pp. 138-147.

. . . . n
R. L. Fremon, Finite Cyclic Group Actions on S1 x S ,

Thesis, Michigan State University, 1969.

P. K. Kim, PL Involutions on Lens Spaces and Other
3-manifolds, To appear in Proc. Amer. Math. Soc.

 

P. K. Kim and J. L. Tollefson, PL Involutions on
3-manifolds, To appear.

 

K. W. KWun, Scarcity of Orientation-reversing, PL
Involutions of Lens Spaces, Mich. Math. J., 17 (1970),
pp. 355—358.

, Sense-preserving PL Involutions of Some
Lens Spaces, Mich. Math. J., 20 (1973), pp. 73-77.

 

. . . . 2
, Piecewise Linear Involutions of S1 x S ,

Y. Tao, 0n Fixed Point Free Involutions of S1 X 82.
Ojaka J. Math., 7 (1962), pp. 145-152.

J. L. Tollefson, Involutions on S1 x S2 and other
3-manifolds, Trans. Amer. Math. Soc., 183 (1973),
pp. 139-152.

27

MICHIGAN STATE UN

I mum [[u[l[u[[[l[[i[[[[

312 L