Homology Approximations for Classifying Spaces of Finite Groups

where D is some small category, F is a functor from D to the category of spaces, and, for each object d of D, F (d) has the homotopy type of BH for some subgroup H of G. An expression like 1.1 is sometimes called a homology approximation to BG or a homology decomposition of BG, and can be used either to make calculations with BG or to prove general theorems about BG by induction. (Of course an induction is likely to work only if the values of F are of the form BH for H a proper subgroup of G!) For example, Jackowski and McClure [15] approximate BG by classifying spaces of centralizers of non-trivial elementary abelian p-subgroups of G. Their result had been anticipated for SU(2) (see [10]) and used to prove a homotopy uniqueness theorem. The p-compact group version of their result [11] is exploited in [7]. Jackowski, McClure and Oliver [16] approximate BG (G compact Lie) by classifying spaces of p-stubborn subgroups of G, and then use the approximation to make beautiful calculations about the space of self-maps of BG. Benson-Wilkerson [4] and Benson [3] use homology approximations to BG, where G is respectively the Mathieu group M12 or Conway’s group CO3, to obtain maps from BG to classifying spaces of 2-compact groups. One goal of this paper is to describe many different homology decomposition formulas (including the ones mentioned above) in terms of a single invariant: an associated poset of subgroups of G. Although we expect to extend the results in a future paper to compact Lie groups and p-compact groups, we concentrate here on finite groups because there are fewer technicalities to get in the way of the basic ideas. We also obtain what seems to be a new homology decomposition for finite groups; this decomposition generalizes a classical theorem of Swan (see 1.21).