Smith  Theory  Revisited



                          William G. Dwyer

                       Clarence W. Wilkerson



1     Introduction


In the late 1930's, P. A. Smith began the investigationof the cohomological

properties of a group G of prime order p acting by homeomorphisms on a topo-

logical space X. This thread has continued now for almost fifty years. Smith was

successful in calculating the cohomology of the the fixed point sets for involu*
 *tions

on!spheres![Smith 1] and projective spaces [Smith 2].In the 1950's, Smith theory

was!reformulated!by the introduction of the Borel construction XG = EGG X

and!equivariant!cohomology H (XG ) = HG (X). Borel made the key observa-

tion![Borel!1] that the cohomology of the fixed point set was closely related to

the!torsion-free (with respect to HG (pt)) quotient of HG (X) . In the 1960's,
!
this!was formalized as the "localization theorem" ofBorel-Atiyah-Segal-Quillen
!
[Atiyah-Segal],[Quillen].!(The localization theorem is described below.) The lo-

calization!theorem!has previously been used to deduce theactual cohomology of

the!fixed!point set for particular examples in an adhoc  fashion, but ageneral

algorithmic!computation!of the cohomology of the fixed point set has not been

provided!in!the literature.
!
    The research of the authors was partially supported by the NSF, and that of*
 * the

second author by sabbatical funds from Wayne State University.





    The main result of this note is that the localization theorem provides sucha

description of the unlocalized equivariant cohomology of the fixed point set al*
 *so.

One simply has to use the data given by the equivariant cohomology as a module

over both HG(pt) and the mod p Steenrod algebra, Ap.



2     Statement ofResults


The theory is valid for G an elementary abelian pgroup, and X a finite GCW

complex. By default, cohomology is understood to have coefficients in the field

Fp of p elements. For K a subgroup of G,define S = S(K) to be the multi-

plicative subset of H (BG) generated by the Bockstein images in H2(BG) of

the elements x in H1(BG) which restrict non-trivially to H1(BK):
!
!
!
!
!   The general form of the localization theorem appearing in [Hsiang,Chapter

]!specializes!the following statement.

2.1!Theorem! (Localization Theorem) [   ]) The localized restriction map


                       S1 HG (X)  !S1 HG (XK )


is an isomorphism.



    For geometric reasons both HG (X) and HG (XK ) are unstable Apmodules.

By [Wilkerson] the localization S1 HG(X) and S1 HG(X K) inherit Apmodule

structures themselves; these induced structures, however, do not satisfy the in*
 *sta-

bility condition. For any graded Apmodule M let Un(M) denote the subset

of "unstable classes", i.e., Un (M) is the graded Fpvector space defined as

follows:


 (1) if p = 2

                   U n(Mk = fx 2 Mk j Sqi(x) = 0; i > kg


 (2) if p is odd


         Un (M)2k = fx 2 M2k j Pi(x) = 0 (i > k); fiP i(x) = 0 (i  k)g


       Un (M)2k+1 = fx 2 M2k+1 j Pi(x) = 0 (i > k); fiP i(x) = 0 (i > k)g