Some Remarks on the Structure of Mackey Functors

All Mackey functors over a finite group G are built up by short exact sequences from Mackey functors arising from modules over the integral group rings of appropriate subquotients W H of G. The equivariant cohomology theories with coefficients in Mackey functors arising from W H-modules admit particularly simple descriptions. Let G be a finite group. The notion of a Mackey functor plays a fundamentally important role in induction theory, both in purely algebraic situations and in equivariant stable homotopy theory. A Mackey functor is a fairly complicated algebraic gadget. We give a quick and elementary structural analysis of Mackey functors, which shows that general Mackey functors are built up inductively out of very simple building blocks. Each building block is constructed in a direct and explicit fashion from an ordinary W H -module V, where W H = N H / H for some subgroup H of G. Our motivation comes from equivariant cohomology theory, and we analyze the cohomology of G-spaces X with coefficients in the cited simple building blocks. In fact, these cohomology groups tum out to be nothing but the homology groups of the obvious cochain complex HomWH(C*(XH) , V). The results here were obtained as a step in our construction of completions of G-spectra at ideals of the Bumside ring [2]. In that application, we were able to prove facts about general Mackey functors by induction starting from the corresponding facts about the Mackey functors constructed from W H-modules, which were relatively easy to prove. We would expect analogous applications in other situations. Actually, Hopkins found an elegant substitute for use of the present theory in the context of [2], but the analysis of Mackey functors should be of independent interest. Much of this analysis is implicit in Lewis's notes [3], which give a more sophisticated study of the structure of Mackey functors. An analysis similar to ours was arrived at independently by Thevenaz and Webb [5]. We are indebted to both Lewis and Webb for useful discussions of this material. We begin by recalling Dress's definition of a Mackey functor [1]. Let GY be the category of finite left G-sets and st'b be the category of Abelian groups. A Mackey functor M consists of a contravariant and covariant functor GY ~ Received by the editors April 5, 1990. 1980 Mathematics Subject Classification (1985 Revision). Prinlary 55N91, 20J99.