Two classifications of simple Mackey functors with applications to group cohomology and the decomposition of classifying spaces

Abstract

We describe a method of computing the group cohomology (with trivial coefficients) of finite groups, and also give a new proof of a theorem of Benson-Feshbach and Martino-Priddy on the stable splitting of BG. In both cases the approach uses structural properties of Mackey functors in a crucial way: we consider the simple functors, composition series of Mackey functors, and projective covers of the simple Mackey functors. A feature of the method for computing group cohomology is that one obtains simultaneously the p-part of the cohomology of all finite groups with a given Sylow p-subgroup.