Smith theory and the functor T
William G. Dwyer and Clarence W. Wilkerson
University of Notre Dame
Purdue University
x1. Introduction
J.Lannes has introduced and studied a remarkable functor T [L1]
which takes an unstable module (oralgebra) over the Steenrod algebra
to another object of the same type. This functor has played an imp or-
tant role in several proofs of thegeneralized Sullivan Conjecture [L1]
[L2] [DMN] and has led to homotopical rigidity theorems for classify-
ing spaces [DMW1] [DMW2]. In this paper we will use techniques
of Smith theory [DW] to calculate the functor T explicitly in certain
key special situations (see 1.1 and 1.3). On the one hand, our calcula-
tion gives general structural information (1.4) about T itself. On the
other hand, up to a convergence question which we will not discuss here
our calculation produces a direct analogue ofSmith theory (1.2) for
actions of elementary abelian p-groups on certain infinite-dimensional
complexes; this analogue differs from Smith theory only in that "homo-
topy fixed point set" is substituted for "fixed point set".
We will now state the main results, which are completely algebraic in
nature although they have a geometric motivation. Fix a prime p; the
field Fp with p elements will be the coefficient ring for all cohomology.
Let!Ap denotethe mod p Steenrod algebra, and U (resp. K) the category
of!unstable modules (resp. unstable algebras)over Ap (see [L1]). If R is
an!object!of K, an unstable Ap fiR module M is by definition an object
of!U which is also an R module in such a way that the multiplication
map!RM!! M obeys the Cartan formula;we will denote the category
of!Ap fiR modules by U (R). An object of U (R) typically arises from a
map!q!: E ! B of spaces;in this case the induced cohomology map q
makes!H! E an object of U(R) for R =H B .
! LetV be an elementary abelian p-group, ie., a finite-dimensional vec-
tor!space!over Fp , and HV the classifying space cohomology H BV .
Lannes![L1] has constructed a functor TV : U ! U which is left adjoint
to!the!functor given by tensor product (over Fp ) with HV and has shown
that!TV! lifts to a functor K! K which is also left adjoint totensoring
!
Both authors were supported in part by the National Science Foundation.
2 W. Dwyer and C. Wilkerson
with HV . The adjointness property of TV produces for any spaceX a
natural map
X : TV (H X) ! H Hom (BV; X)
which is often an isomorphism [L1][L2] [DS]. Given an object R of K
there is a simple way (see x2)of using a particular K-map f : R ! HV
to single out a quotient TVf(R) of TV(R) or for M 2U (R) a quotient
TVf(M) of TV (M). These quotients correspond via to subspaces of
function spaces; more precisely, if q : E ! B is a map of spaces and
f : H B ! HV is a map inK ,then E induces a quotient map E;f from
TVf(H E ) to the cohomology of the subspace of Hom (BV; E) consisting
of maps r : BV ! E with r q = f.
For any K-map R ! HV, we will let Sf ae Rdenote the multiplicative
subset of R generated by the Bockstein images in R2 of the elements of
R1 which map non-trivially under f. Recall from [W1] [Si] that if M
is an object of U(R) any localization of the form S1f M inherits an
action of Ap , although this action is not necessarily unstable. Denote
the largest unstable Ap -submodule [DW, 2.2] of such a localization by
Un (S1f M).
Theorem 1.1. Let W be an elementary abelian p-group, V a subgroup
of W, and f : HW ! HV the map induced by subgroup inclusion.
Suppose that M is an object of U(HW ) which is finitely-generated as a
module over HW . Then there is a natural isomorphism
TVf(M )= Un S1f (M ):
This theorem has a geometric background. Let V and W be as in 1.1.
Suppose that X is a finite CW-complex with a cellular action of W and
let M be the cohomology of the Borel construction EW X =E WW X
on this action. According to [L2] there is an isomorphism
TVf(M) = H EW (XV )
where XV is the fixed point setof the action of V on X. Similarly, by
Smith theory [DW, 2.3] there is anisomorphism
H EW (XV )= Un S1f (M):
The composite of these two maps isthe isomorphism of 1.1; the present
paper sprang in part from a desire to produce this isomorphism in a
purely algebraic way withoutassuming that M = EW X for a finite