mm: cvcuc GROUP ACTIONS ON 31 xs”

 

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(00000 1‘» T T T T ‘ T T '

 

Thesis for the Degree of Ph. D.
MICHEGAN STATE UNIVERSITY
RICHARD LANHAM FREMON
1969

  
  
   
 
 
   
 
 
 
 
 
 
 
  
   

  
    

L I B R, A R Y
Michigan Statc
University

 
 

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1293 10198 8 19

This is to certify that the

thesis entitled

FINITE CYCLIC GROUP ACTIONS ON Slen

presented by
Richard Lanham Fremon
has been accepted towards fulfillment

of the requirements for

Ph D degree in Mathematics

 

/C ATM“.

Major professor

 

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ABSTRACT

FINITE CYCLIC GROUP ACTIONS ON SIXSn

BY
Richard Lanham Fremon

This thesis is a study of the ways in which a homeo-
morphism of finite period can act on Slxsn.

In chapter I we show by an elementary argument that
the cohomology groups of the fixed point set of such a
homeomorphism are quite restricted. It is shown that they
must be either those of a sphere, two spheres, or of Slxsk.
From this it follows that there are only ten possible fixed
point sets of dimension two or less.
In chapter II we classify those actions on Slxs2

with two dimensional fixed point sets: there are only four.

 

1

FINITE CYCLIC GROUP ACTIONS ON 5 xsn
BY

Richard Lanham Fremon

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of.Mathematics

1969

 

 

- new"

   

ACKNW IEDGMENTS

The author wishes to express his gratitude to Professor
K. W. Kwun for suggesting the problems and for his helpful

guidance during the research.

iii

 

 
 

 

 

 

 

TABLE OF CONTENTS

LIST OF TABLES AND FIGURES . . . . . .
LIST OF SYMBOLS AND ABBREVIATIONS . .
INTRODUCTION.............
CHAPTER
I. FIXED POINT SETS . . . . . . .
BASICCONSTRUOTIONS . . . . . .

CASE I: INVOLUTIONS WITH h‘l‘l =

CASE II 3 hfl = I o o o o o o 0
II 3 ACTIONS ON 81 X52 . . . . . . .
BIBLIOGRAPHY o o o o o o o o 0

iv

 

Page

iv

mun»

IO

16

 

 

FIGURE I.
FIGURE II.
TABLE I.

FIGURE III.

 

LIST OF TABLES AND FIGURES

THEESLEVEL........
THEE’LEVEL........
pOSSIBLE FIXED POINT SETS . .

CONSTRUCTION OF N . . . . . .

 

Page

10

14

 

 

 

clc

[onox7YI B! Y]

 

LIST OF SYMBOLS AND ABBREVIATIONS

cohomology locally connected [1]
Homology locally connected [2]
Singular homology

Singular cohomology

Sheaf cohomology [2] (closed supports)
Boundary of the manifold x

The adjunction of TA to B along f
The set of homotopy classes of maps

f : x 4 y satisfying EA C'B and
fun = Y

The homotopy class of w

[0.1]

The infinite cyclic group of integers

The integers modulo k

vi

 

 

 

 

INTRODUCTION

The study of finite period transformations began in
1938 with the discovery by P. A. Smith that when a prime
period homeomorphism acts on the three-sphere the fixed

0, $1, or a (Cech) homology two-sphere.

point set must be S
Since then various authors have made improvements on the
methods and results. Most of the work has been concentrated
on actions on spheres, projective spaces, and products of
these. In 1964 J. C. Su extended a technique of R. G. Swan
to characterize the cohomology groups of the fixed point
sets of 2p actions on SmxSn. Recently K. W. Kwun proved
that a piecewise linear involution on 81x82 is unique up
to equivalence if it fixes a torus or two two-spheres. In
this thesis we use very basic tools to simplify Su's result
for the case m = 1 and we prove that when there is a two
dimensional component in the fixed point set of an involu-
tion on 51x82 the action is determined by that fixed point

set.

 

 

 

 

CHAPTER I

FIXED POINT SETS
BASIC CONSTRUCTIONS

We shall begin by defining certain basic tools which
will be useful throughout. Let h : Slen 0 (n > 1) be
the transformation satisfying hp = I (p prime) and F =

1

Fh = [x E S xsn I h(x) = x] 7! o’ or Slen. As most of the

work is done in the universal covering space we define

l n

p = ExpxI : 1Rxsn -o S xS where Exp(x) = e21r1x

; and pk :
RXSn p by pk(x,z) = (x+k,z).

We lift h to 11x8n as follows. Pick xo 6F and
y e p‘1(x ). Then let i : (Rxs“.y ) a be the unique
0 o yO . O
lifting of ph through p fixing yo. Thus ph = hy p.

~ ~ 0

When there is no ambiguity "h" and "F" will be used to denote

h and F" . The uniqueness property guarantees that hp
Y0 hy

O
= 'I.

We will also make use of the usual construction, P(X,x)
= [I,1,0;X,X,x], of the universal covering space of a con—
nected, locally path connected, semilocally 1-connected space
X. Details of the» topology on P(X,x) will not be needed in
the sequel. Observe that P(X,x) is a functor and let f‘H‘
denote P(f). Let g : P(X,x) -o x be the projection defined
by e([w]) = (”(1). Then, as RXSn is simply connected, 6 :
P(RxSn,yo) «Rxsr1 is a bijection. As p : leSn 4 Slen is
a covering projection P‘H‘ : P(Rxsn,pk(yo)) a P(Slen,xo) is

2

 

3

also a bijection for any k. Thus we can define qk = p%1 :
P(Slen,xo) 4 P(RxSn,pk(yo)). Then ph = hp yields the rela-
tion fifi = qoh#p# : PCRxSn,yo).p and ~h = ehfie'l = eqoh%p#e_l.

Using this fact we can show that h and pk commute
up to sign : hpk = pikfi’ Pick y éleSn and let [a] =
e'1(y). That is a(0) = yo and a(l) = y. Also let [w] =
e'lpkwo). Then [w-(pkoa)]= e'1<pk<y)) and fipkm =
eqoh#p#[w-(pk°a)] = eq0(hfip#[w]-hfipfi[9k«3]) = eqo(h#[pow]'
h#p#[a]). As pw(0) = pw(l) = x0 h%[pow] depends upon how
h acts on 71(Slen) 2 Z. There are two cases : h# = :1 :

w1(Slen) 9.

CASE I: INVOLUTIONS WITH h# = -I

Assume h%[pow] = -[pow]. ‘We will need 6 E PORxSn,yO)
satisfying 5(1) = p_k(yo). w-(pkoB) is null homotopic: so
0 = P=H[w'(pk°B)] = p+|[w]-p%[pk°B] = p+‘[w]-p+‘[6]. Hence
P#[B] = —p#[w] = h#[pom]. Consider the path B-(p_k°Y)
where [Y] = qohfipNTG] = qohflfpoGJ- pfiTB-(p_kov)] = pfiTBJ-
p#[p_kov] = hfifpowl°hfifpw1]. Therefore hpk(y) =
equ#[B-(p_kov)] = e[B-(p_k°v)] = €[P_k°Y] = p_kv(l) =
p_ke[y] = p_keqoh*p#[a] = p_kh(y). In short hpk = p_kh.

H*CRxSn:ZZ) = H*(Sn;ZZ). So, from Smith Theory [p. 43, 1]
we know that H*(§:Zz) a H*(Sk:zz) for some 0 g k g n;

also [p. 76, 1] F must be a Z -orientab1e cohomology manifold.

2
hpm = P—mh tells us that when h is extended to the two

point compactification of RxSn the resulting map does not

 

 

j":

Jr

 

 

 

 

 

4

fix the added points. Hence S is a compact k-cohomology
manifold. It is also known [p. 72, 1] that F is clcZ2
which implies it is HLC. Hence AH*(§;zz) e H*(§;zz) and
E must be path connected or SO.
~ Let Cxo be the path component of x9 in F. Clearly
p(F) D Cx . Suppose p were not 1-1 on F. Let w be a
path in S with pw(o) = pw(l). Then [pow] is an element

of the image of vlF 4 Ulslxsn. Consider the following dia-

gram.
1% l n
1rlF —__,1r18 XS
I 11% = -I
i# l n
1r1F ._____71rls XS
Its commutativity yields [pow] = -[pow] : pow a O and w(0)

= w(1). Therefore every path component of S projects homeo-
morphically and every component of F is a cohomology sphere
or a point.

We would like to show that the discrete points of F
pair off to make F the union of cohomology spheres. We

will say that two such points are related, lexz, if hy (y2)
1

= y2 for some choice of yi 6 p-1(xi). As each lifting
hy is uniquely determined by the points it fixes hy =

e i 1~

hy2 and R is symmetric. Suppose leszx3. Then hy1 =
i mdh =fi .fi =fi = " =

p-kh pk = hp_k(y3)'

So h fixes y , y and p_ (y ).
3 y1 1 2 k 3

 

 

 

 

 

 

5

If these points were distinct Ffi and Cx would be coho—
y1

mology spheres of positive dimension. Hence two of the yi
must be identical; the same is true for the xi. Thus the
equivalence classes contain at most two points.

Suppose that one of the equivalence classes was a single-
ton. As F is a cohomology sphere by must fix some point,

L O

Pk(yo) 7 but hY
points come in pairs and we could redefine Cx to include

(P (Y )) = P_ (y ). Therefore the discrete
1 k 1 k 1

both equivalent points.

Suppose now that F includes three distinct cohomology

-1

Spheres Cxl, sz, cxa. Choose yi 6 p (xi) as before and

def' k.. b h . = . . If k.. were even
lne 1] y Yi(Y3) pkij(y3) l]
. = i . = D . . This would ut

Fl (ya) p_lk Yi(y3) Yi(plk .(y3)) P

2 ij 2 ij 2 in
plk (yj) in Ffiy and xj E Cxi. Hence kij must be odd.

2 ij i
By uniqueness of liftings h = p h . So p (y ) =

Y1 k12 Y2 R13 3

hy1(y3) = Pklth2(Y3) = pklzpk23(y3) = p(k12+k23)(y3). This

is impossible as it would require that k13 = k12 + k23.
We have shown that when hfi = -I : Wl(Slen) p Fh is

the disjoint union of at most two cohomology spheres. Further—

more, as the codimension of E can be odd if and only if h

and h reverse orientations [p. 58, 1], when F has more

than one component they must be both odd or both even dimen—

sional cohomology spheres.

 

CASE II: h% = I

Just as in case I we can use Smith Theory to conclude
that H*(F;Zp) s H*(Sr;zp) for some 0 s E g n; but in this
case the dimension of F is not so easily determined.

Let 2 be the two point compactification of ‘Rxsn and
let A extend h to 2. Using an argument like the one
above one can show that hpk = p+ h when h# = I. Hence
~ A ~
h(to) = i0 and F = F U [in]. As 2 is an (n+l)-sphere we

A A

can again use Smith theory to characterize F : H*(F;ZP) 2

* ' A
H (sé;zp). As 9 must be Zp-orientable [p. 76, 1] F and
F lmust be 9 cohomology manifolds. By Poincaré duality
[p. 210, 2] we have
2 A
A ~ A - Z when q = r
Hq(F.F;z ) 2 H9_ (F-F;Z ) 2 P
P q p 0 otherwise.
When this information is combined with the long exact se-
quence for the cohomology of a pair it is seen that
L Z for q = 0.9-1
Hq(F;Z ) a P

P 0 otherwise

A ~
when 9) 2. Otherwise FMSl or $2 and F~1RxSo or

8x51. Thus, in general, S. is an 9 cohomology manifold
and an (é-l)-(co)homology sphere.

1

Now we can characterize F. Consider y 6 p_ F and

. ~ _ _ ~p—l _ . .
deflne k by h(y) — pk(Y). y — h pk(y) - ppk(y) 1mp11es
k F O and y E F. Hence F = p‘lF and F is the orbit
Space of [pk]kez acting on F. If F ~RXS° F must be

. . . . . A
either one S1 or the dlSjOint union of two. For r a 2 we

 

 

 

7

Will use a spectral sequence argument [p. 343, 10].

When Z operates properly on a pathwise connected space,
X, there is a first quadrant spectral sequence, E, with
2:]. = Hi(Z:Hj(X;ZP)) a soc/2:21,).
Because Z is free Hi(Z;-) = 0 for all i > 1 [p. 458, 6]:
so Eij = 0 unless i = 0,1 and j = 0,;. It is known
[p. 457, 6] that when the action of Z on the coefficient
group is trivial, for example when j = O or p = 2,
Hi(Z:Hj(F;Zp)) a Hj(F;Zp). When this action is not trivial
Hi(Z;Hj(F;Zp)) a Hi(SI;U) where u is a local system of
coefficients with stalks fix = Hj(F;Zp) and the action of

1 " . .
w1(S ,l) on Hj(F;Zp) lsdetermlned by[pk]k€z. Hence (see

 

 

 

Hilton and Wylie-—-p. 351 for the construction) Bi; = E3;
= 0 s
‘ s a s
Es+l = H (Es = ker(Eij * Ei—s,j+s-l) = B?.
i+s,j-s—l ij
lpi=0
s . 3
Bot Elf
B _ \
El-s,s-1 _ 0 tr
o‘\\\\\\
as
s s
E
to .
__r O Arlo :7 3 = 0

FIGURE I

THE E8 LEVEL

 

 

 

 

 

 

 

 

o 2
Therefore Ei' — Eij'
i=0
i+j=£‘ I \
‘N \
\\ a \
\
TKEOI‘ \ Elf
i+j=l \\ i+j=f
\ a
VKEOI
\
\
\
‘\
\
E" \ E"
00 x lo 9' j=o
FIGURE II
TI-E E“ LEVEL
As Egg : H(F;ZP) Ea is the graded module associated with

. . . a _
some filtration of H*(F,Zp). That Is Ei,k-i —

2 .
Hk(F’Zp)i/Hk(F’Zp)i-l' Hence H§(F,Zp) 2 E1?‘ This group
must be non-zero as F is Zp-orientable; so the action
induced by {pk} on Hj(F;Zp) must be trivial. In this

case H9(F;Zp) 2 2 HE(F;ZP) 2 Zp; clearly HO(F;ZP) 2 Z

Elf Po
~ 0 a
When r ) l the complexes E*’f_* and E1+*'_* have only

one non-zero group: H1(F;Zp) 2 E2 2 Z and Hf(F;Zp) 2

10 p
Elf 2 Zp. When E = 1 we have the short exact sequence
2 2 .
O 4 E10 4 H1(F,Zp) 4 E014 0. Hence, In every case H*(F,Zp)
2 H*(Slxsf:Zp). We have proven the following.

 

 

 

 

 

9

THEOREM: Suppose h : S1

xSn P so that hP = I and ¢ # F
# Slen then the cohomology of F is either that of Sk,
kl' k2 l k-l

S US , or S xS where O 2 k, k1, and k

2 2 n and
kl 5 k2 (mod 2).

 

 

 

 

 

 

 

 

 

 

 

  

.

 

CHAPTER II

ACTIONS ON Sle2

1x82 the distinc-

When we restrict our attention to 5
tion between cohomology manifold and manifold disappears:

each component of F is locally Euclidean. Hence we are

left with the following possible fixed point sets ."1
1
h preserves orientation h reverses orientation ex
h# = I $1051 3le1
s1 Klein bottle
h=H _ -I Sll‘Jsl SZUSZ
S1 SOLIS2
SOEJSO
SO
TABLE I

POSSIBLE FIXED POINT SETS

To construct examples of the above actions we need the
following maps. Define C : I {D by C(x) = l - x. This in-

duces conjugation on S1 c C when 0 and l are identified.

Use the standard embedding of 52 c E3 to define R1, R2,

T 3 52 a by R1(X1,X2,X3) = (—x1.-x2,x3). R2(x1'x2'x3) =
(x1.-x2.-x3). and T(x1.x2.x3) = (x1.x2.-x3).

Examples of F ~ 51 or Slstl with h# = I can be in-

2 2

duced on Sle by Ile : 1x8 :3 when 0x32 is identified

10

 

 

 

 

 

 

 

 

 

 

11

with 1x32

5le1

by either I or R2. Similarly we get F u

and K from IxT : 1x82 P. Examples of the other
two-dimensional fixed point sets are found by considering

CxI : 1x52 cu CxR induces an action with fixed point set

1
SOUSO. It is suspected, but not known, that involutions

l

with fixed point set S and h = -I are non-existant.

A simple application of the Lefshitz fixed point theorem to

2

the compliment of any S0 c Sle shows that no homeomor-

O

phism leaves just S fixed.

It is known [8] that in the piecewise linear category

actions with F In Slxs1 or F as 82052 are unique up to

equivalence: any involution with one of these fixed point

sets must be a conjugate of the standard action above (i.e.

1 l 2

h = fhsf- for some f : 5 x8 s». We will now show that

this is also the case for the other two-dimensional fixed

point sets K and $2082.

Consider first the case of F ~ K. Suppose that 81x52

has been triangulated so that h is simplicial. Let x be

the orbit space of h with the induced triangulation and

let q : Sle2 4 X be the identification map: q(x) = qh(x).

The triangulation of Sle2 also induces a local polyhedral
structure on IRxSZ, the covering space. As F is twO dimen-

2-F must be a

sional h must be an involution and qlSle
two to one covering projection. Recall that in this case
9 ~ 32. When h is piecewise linear F is locally polyhe-
dral except at in. J. C. Cantrell has shown [4] that the

. . A
complimentary domains, A and B, of F c 2 u S3 are open

 

 

 

 

 

 

‘lI',

 

12

three cells. Thus either component, A, will be a universal

covering space for SleZ-F and X-qF. The translations of

qp‘A : A 4 X-qF form a subgroup of those of p : IRsz 4

51x52: hence, as the latter is infinite cyclic, the former
and 1r1(X) s.- 1r1(X-qF) 2 Z. This suggests that X fibers

over 81. All that needs to be shown is that each tame two

Sphere in X bounds a three cell (i.e. X is irreducible).
As X is homeomorphic to the compliment of an Open collar
of the boundary we need only worry about spheres interior

to X. These can be lifted to- A which is irreducible.

Hence [13] X fibers over S1 'with fiber D2. As h re—

verses orientation X must be the non-orientable disc bundle
over 81.

Now consider two involutions h and h' ‘with F u F'

~ K and the corresponding identifications q : Sle2 41X

and q' : 81x82 4~X. Using the lifting theorem we get a

homeomorphism f : SleZ-F 4 SleZ-F' satisfying q = q'f.

Q'h'-1fh = q'h'fh = q'fh = qh = q: so, by uniqueness of
lfh and h' = fhf-l. f extends trivially

to Sle2 as q'-1q is singlevalued on F. Thus h and

liftings f = h"

h' are equivalent.

Uniqueness of actions which fix SOUS2 is still simpler
2, and their
fixed point sets. F1 and F2. Cutting along the $2 (:1?1

to prove. Consider two such actions, h1 and h

yields two connected, compact manifolds Xi with boundary
SZUSZ. The Xi are-connected because the two spheres can

not separate; otherwise they would bound cells. As hi

 

 

 

 

     

e», I:
!

   

13

reverses orientation on Sle2 it must switch the sides of

the $2 CIFi and hence the induced map on Xi switches the

boundary components. Thus when we cone over each 82 c 8X1

to attach two balls Bi and Bi to Xi forming Y1 =

l 2 . , . . .
Xan Ban B1 the extention, hi . Yi.;n of this induced map

1 2 j l 2 3
switches the B1 : hiBi = B.. As Yi n S [proof to prop. 4,

i

8] and Rh, = so, (and if G. R. Livesay's proof [9] is cor-
i

. . . . 3 _

rect) hi is equivalent to T . S ;> by T(x1,x2,x3,x4) —

l

("xll 'xzt-x30x4): hi = fi Tfio

If £181 = szg for j = l or 2 ‘we'would be done; in
this case fglfllxi ‘would induce the equivalence between hl
and h2. When lei czszg the proof is almost as easy.
Suppose this is true for j = l and let B1 = lei, 32 =
sz;. and B3 be a derived neighborhobd [16] of B2 in S3.

Define g : $3.p as follows. On B let 9 be a homeo-

3
morphism fixing BBB and carrying 82 onto Bl: let

ngB3 = TgTITB3; and let g|s3-33—T33 = I. Then gT = Tg
I _ -1 g -1 -1 -1 "1 1 _
and hl — f1 gfzhzf2 9 f1. Furthermore f2 9 £181 —
-1 -1 _ -l _ 1 -1 -l . -
f2 g B1 — f2 B2 — B2 and f2 9 f1 induces the required

equivalence between h1 and h2 on 81x82.

For the general case we will construct a third pair
of balls, N and 'TN, associated with an action equivalent to

g- . l j . _ '
both hl and hz. If lel n £232 5:4 g! for j - lor 2. let N

be a ball in the intersection; otherwise conStruct N as
follows.' Let Nl be a derived neighborhood of T(BlUBZ) and

N2 a three ball neighborhood of BlUBZ in S3—N1. To Obviate

NnTN #’¢ let N be a derived neighborhood of TN2 and N

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

14

a three ball neighborhood of BIUB2 - this time in S -N .

 

N2

/: e
((0. r,
u.

FIGURE III

 

 

 

 

CONSTRUCTION OF N

O 0
Now remove N and TN from S

by T to form a space Z ~ 81x82. T induces an involution

3 and attach aN to TaN

on Z which, by the above argument, is equivalent to h1 and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

15

h2' Thus h1 and h2 are equivalent and we have proven

the following.

THEOREM: Let h be a piecewise linear involution of 81x82.

If any component of the fixed point set of h is two-dimen-

sional then it uniquely determines h up to equivalence.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[7]
[8]
[9]

[10]
[11]
[12]

[13]

[14]

[1]

[2]

[3]

[4]

[5]

[6]

[7]
[8]
[9]

[10]
[11]
[12]

[13]

[14]

 

BIBLIOGRAPHY

Armand Borel, Seminar pp Transformation Groups,
Princeton University Press, 1960.

G. E. Bredon, Sheaf Theory, McGraw-Hill, 1967. '

, "Cohomological Aspects of Transformation
Groups," Proceedings 9; the Conference pp Transfor—

mation Groups, Springer-Verlag New York Inc., 1968.

“‘2'.

 

F
“A c

J. C. Cantrell, "Almost locally polyhedral 2-spheres
in S3," Duke Mathematics Journal 30 (1963), 249-252.

Roger Godement, Topologie Algebraic £3 Théorie des
Faisceaux, Hermann, 1964.

P. Hilton and S. Wylie, Homology Theory, Cambridge
University Press, 1960.
D. Husemoller, Fibre Bundles, McGraw—Hill, 1966.

Kyung Whan Kwun, "PL Involutions of 81x82," (to appear)

 

G. R. Livesay, "Involutions with two fixed points on
the three-sphere," Annals of Mathematics 18 (1963),

S. MacLane, Homology, Academic Press, 1963.

P. A. Smith, "Transformations of finite period. II,"
Annals of Mathematics 40 (1939), 690-711.

E. H. Spanier, Algebraic Topology, McGraw-Hill, 1966.

John Stallings, "On fibering 3-manifolds," Topology g;
3-manifolds and Related Topics (Edited by M. K. Fort,
Jr.), Prentice Hall, 1962, 95—100.

J. C. Su, "Periodic transformations on the product of

two spheres," Transactions of the American Mathematical
Society 112 (1964). 369—380.

16

 

 

 

 

 

 

 

 

 

 

 

 

 

r.

 

17
[15] J. L. Tollefson, Thesis, Michigan State university, 1968.
[16] E. C. Zeeman, "Seminar on Combinatorial Topology,"

(mimeo), Institut des Hautes Etudes Scientifiques,
1963.

—. m. N'—

 

 

   

 

 

 

 

 

 

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