Extensions of G-posets and Quillen's complex

Abstract

AbstractWe develop techniques to compute the homology of Quillen's complex of elementary abelian p-subgroups of a finite group in the case where the group has a normal subgroup of order divisible by p. The main result is a long exact sequence relating the homologies of these complexes for the whole group, the normal subgroup, and certain centralizer subgroups. The proof takes place at the level of partially-ordered sets. Notions of suspension and wedge product are considered in this context, which are analogous to the corresponding notions for topological spaces. We conclude with a formula for the generalized Steinberg module of a group with a normal subgroup, and give some examples.