Stably splitting BG

Abstract

In the early nineteen eighties, Gunnar Carlsson proved the Segal conjecture on the stable cohomotopy of the classifying space B G BG of a finite group G G . This led to an algebraic description of the ring of stable self-maps of B G 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 B G BG into indecomposable spectra. This paper is a survey of the developments of the last ten to fifteen years in this subject.