Stably splitting BG

Abstract

In the early nineteen eighties, Gunnar Carlsson proved the Segal conjecture on the stable cohomotopy of the classifying space BG of a finite group G. This led to an algebraic description of the ring of stable self-maps of BG as a suitable completion of the "double Burnside ring". The problem of understanding the primitive idempotent decompositions of the identity in this ring is equivalent to understanding the stable splittings of BG into indecomposable spectra. This paper is a survey of the developments of the last ten to fifteen years in this subject.