Higher limits via subgroup complexes

Jesper Grodal

Status: MIT preprint

Abstract:

We study the higher derived functors of the inverse limit of a functor
F: D --> Z_{(p)}-mod, where D is one of the standard categories which
arise when studying the homotopy theory of the classifying space of a
finite group G, e.g., the orbit category or the Quillen category of
G.  These higher limits are of importance e.g., for the study of maps
between classifying spaces as well as for group cohomology. 

We show that these higher limits can be identified with the
G-equivariant Bredon cohomology of the subgroup complex of p-subgroups
in G (i.e., the nerve of the poset of p-subgroups in G) with values
in a G-local coefficient system. We examine when smaller complexes can
be used e.g., taking only p-radical subgroups, p-centric subgroups,
elementary abelian p-subgroups or various subcollections thereof. 

Since the subgroup complexes are finite complexes, and often rather small,
this provides concrete, computable formulas for these higher limits,
generalizing earlier work of especially Jackowski-McClure-Oliver. It also
gives a conceptual explanation of high dimensional vanishing results
previously established in more indirect ways. 

As an application we look at the special case where all the higher limits
vanish, as for example is the case for group cohomology. If F is a
functor on the orbit category our formulas for the higher limits in this
case yield five different expressions of F(G) in terms of values of F
on proper subgroups. Two of these are `classical' namely Webb's exact
sequence of Mackey functors and a formula for calculating stable elements,
previously obtained using Alperin's fusion theorem.
Examining this case also leads to improvements of sharpness results of
homology decompositions due to Dwyer and others.

Central to many of the proofs are properties of the Steinberg chain
complex of a finite group G, as well as other concepts from the emerging
Lie theory for arbitrary finite groups of Alperin, Webb, and others.