Stably splitting BG

Abstract

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. The Segal conjectureThe nineteen eighties saw a number of major breakthroughs in homotopy theory. Among the most spectacular are the proofs of the nilpotence conjecture by Devinatz, Hopkins and Smith [10], the Sullivan conjecture by Haynes Miller [24], and the Segal conjecture by Gunnar Carlsson [7]. Two of these three concern the role of finite groups in homotopy theory: the Sullivan conjecture is "unstable", while the Segal conjecture is "stable" with respect to suspension. This report is about the consequences of the Segal conjecture for the stable splittings of the classifying space of a finite group.We begin by setting the scene. Around 1960, Atiyah [3] calculated the Ktheory of the classifying space of a finite group. There is a natural map from the character ring R(G) to K 0 (BG) which sends a complex representation V to the corresponding vector bundle EG× G V → BG. Atiyah proved that this map induces an isomorphismwhere the completionn is with respect to the augmentation ideal I = Ker(dim : R(G) → Z). He also showed that K 1 (BG) = 0. The Segal conjecture is the corresponding statement for stable cohomotopy, in which the representation ring is replaced by the Burnside ring A(G). This is the 1991 Mathematics Subject Classification. Primary 55P.

The Structure of Mackey Functors

“…When X consists of all finite groups and Y consists of the identity group, this ring is known as the double Burnside ring of G, see [3] or [43]. (We have chosen the opposite convention to many authors, who take X = 1 and Y = all finite groups.…”

“…For surveys of the background material to this section see [3] and [43]. We denote by (BG + ) ∧ p the p-completion of the suspension spectrum obtained from the classifying space BG after first adjoining a disjoint base point to give a space BG + .…”

Problems in the Steenrod Algebra

“…Topologists are interested in splitting classifying spaces of finite groups, as we discussed in Subsection 7.4. For general information on this subject, we refer to [24,126,160]. Some recent work on splitting 2-groups can be found in [45].…”

Representations of the double Burnside algebra and cohomology of the extraspecial p-group II

“…Let G be a finite group with Sylow p-subgroup E. Then the multiplicity of X in BG is equal to the dimension of S[G] where [G] is an element of A p (E, E) corresponds to the (E, E)-biset G. See [1], [2], [11] for details.…”