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