Homology Decompositions for Classifying Spaces of Finite Groups Associated to Modular Representations

For a prime p , a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod‐ p homology. Associated to a modular representation G → Gl( n ; F p ), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so‐called subgroup, centralizer and normalizer decompositions. For an action of G on an F p ‐vector space V , this collection consists of all subgroups of G with nontrivial p ‐Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG .