Stable splittings of BP for some P of order thirty-two Original Research Article

In this paper, we use the method of Benson and Feshbach outlined in “Stable splittings of classifying spaces of finite groups” (in Topology, Vol. 31, No. 1 (1992)) to give the complete, 2-complete stable splitting of the classifying space BP for all but two of the groups P of order thirty-two which are not direct products in a nontrivial way. One of these two is Z/32, and B(Z/32) is known to be indecomposable. For the other, P is the semidirect product of (Z/2)4 and image. This is the only nonabelian group of order 32 which is not a direct product and has a subgroup isomorphic to (Z/2)4. We also give some results which refine the method of Benson and Feshbach. We have found that BP is indecomposable for three of these groups, and these are the first examples of indecomposable BP for which P is a nonabelian 2-group. Poincaré series are given for each classifying space using Rusinʹs paper “The cohomology of the groups of order 32” (Mathematics of Computation, Vol. 53, No. 187 (1989)).