Localizations of Unstable Alpha-Modules and Equivariant Mod rho Cohomology

摘要

Let p be a fixed prime and G be either a compact Lie - group (not necessarilyconnected) or a discrete group of finite virtual cohomological dimension (f.v.c.d. for short), e.g. the general linear group over the ring of S-integers in a number field or the mapping class group of an orientable surface. In a paper from 1971, Quillen considered the category A(G) whose objects are the elementary abelian p--subgroups of G and whose morphisms are given as compositions of inclusions and conjugations. He showed that the restriction homomorphisms from the mod p cohomology H(star)BG of the classifying space BG to H(star)BE for E a member of A(G) induce a map q(G) : H(star)BG -> lim(sub A(G) sup op) H(star)BE such that the kernel of q(G) consists of nilpotent elements and such that for each element in the limit a sufficiently large p(sup n)-th power is in the image of q(G). In the paper the authors will consider H(star)BG as an unstable module over the mod p Steenrod algebra A (unstable module for short) and show how the structure can be used to refine Quillen's description and describe a finite sequence of approximations to H(star)BG starting with Quillen's map q(G) and ending with a genuine isomorphism.