INTRODUCTION

Let G bea group acting ona space X. The stationary point setof

p

this action, denoted by X —also called the fixed point set of G —is the

set ofall points x e X which are left fixed (stationary) byall elements

of G. Theisotropy subgroup at x, denoted by G , isdefined tobe the

subgroup of G consisting ofelements which leave x fixed. Thestudy of

the properties ofthe stationary point set ofactions onreasonably well-

behaved topological spaces hasbeen anarea of interest totopologists for a

long time. A breakthrough was made by P. A.Smith inthe1940's byhis study

of stationary point setofgroup actions onspaces which homologically

resemble a disk ora sphere. The theory developed by P.A.Smith reveals,

for instance, that ifa finite p-group acts ona complex X with

H*(X;2 ) = 0, then

H*(XG;Z

) = 0 aswell. Also if H*(X;Z ) =

H*(Sn;ZZ

)

for some n 0, then H*(X;Z ) = H*(Sk;ZZ ) forsome n k -1.

It isnatural toask whether such strong restrictions hold for other

types offinite groups acting on spaces having a more general homotopy-type.

E. Floyd's study ofthis question showed that for groups ofnot prime power

order the situation was different. Conner and Floyd proved the existenceof

stationary point free actions of H —the cyclic group oforder

pq (pq) = 1 —ona euclidean space through a clever construction(cf.

Ch. IV) which was later modified by Kister, Conner-Montgomery,and

W. C.Hsiang-W. Y.Hsiang togive further examples ofthis kind. Theisotropy

subgroups ofthis action form the collection {1,2 ,2Z }. However, if we

consider anaction of Z ona complex X with H*(X;Z ) = 0, andwe

require that the collection of isotropy subgroups be { l , Z

n n

} then by

TL

p q

Smith theory i t follows that H*(X p q ; 2 ) = H*(point;Z ). In particular,

X p q * 0 for such actions—called semifree actions in the literature. This

v n