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
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