Sharp Homology Decompositions 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 ∈ D, F (d) has the homotopy type of BH for some subgroup H of G. The operator “hocolim” is the homotopy colimit in the sense of Bousfield and Kan [4], which amounts to a homotopy invariant method of gluing together the values of the functor F according to the pattern of the maps in D. There is a systematic study of some of the more common homology decompositions in [6] (see 1.3 below). Here we go further along the same lines. Bousfield and Kan associate to 1.1 a first quadrant homology spectral sequence [4, XII.5.7]