This paper is an exposition of stable splittings for the classifying space of a finite group. These results, obtained over the past half decade, provide p-local splittings BG = vXi for various p-groups G and are distinguished from earlier research by the pervasive use of modular representation theory. Ideally the Xi are indecomposable, thus displaying the homotopy type of BG in simplest terms.

The number of papers in this area has now reached a point that it seems worthwhile to give an overview of some of the basic results, applications, and philosophy. Our goal is twofold. First, we wish to provide a compendium of related splittings which can be used by future workers seeking patterns and general structure theorems. Second, for the neophyte in modular representation theory, we wish to provide enough elementary notions to explain our modest applications of this deep and powerful tool for homotopy theorists. As such, this paper is an outgrowth of lectures presented at the 1985 Durham Symposium on Homotopy Theory. The author wishes to thank the organizers and especially John Jones for his encouragement in writing this paper.