Chun-Nip Lee

Stable splittings of classifying spaces of compact Lie group

Let G be a compact Lie gorup. In this paper, we study the stable splitting of BG completed at p into a wedge sum of indecomposable spectra. When G is finite, this question has been reduced to understanding the irreducible modular representations of the outer automorphism group Out (Q) for various p-subgroups Q ⊆ G by work of [2], [14] and [18]. The principal tool used by these authors is a generalization of Segal’s Burnside ring conjecture which describes all stable maps between p-completions of the classifying spaces of p-groups. One of the problems in going from finite groups to compact Lie groups is that in the latter case one no longer has a convenient description of all stable maps between classifying spaces. Our solution to this difficulty is to pass from the ring of stable self-maps to the induced self-maps on Fp-homology. It is well-known that in terms of stable splittings, one does not lose any information by this process. There are two advantages to this approach. One is that a result of Henn [8] implies that the ring of induced self-maps on the Fp-homology of BG is finite. Two is that one can now give a more explicit description of all the induced self-maps on Fp-homology for a large class of compact Lie groups which we call p-Roquette. The latter is exactly the class of compact Lie groups for which an appropriate density theorem is valid for a generalized Segal’s Burnside ring conjecture for compact Lie groups as shown by Minami [16] whose result built upon work of Feshbach on the original Segal’s Burnside ring conjecture for compact Lie groups [7]. In particular, every compact Lie group is p-Roquette if p is odd. With these reductions, one can follow a similar procedure for splitting BG∧ p as in the case when G is finite. Recall that a compact Lie group Q is said to be p-toral if it is an extension of a torus by a finite p-group. The main result of this paper is that when G is p-Roquette, one can reduce the stable splitting of BG∧ p to the study

On stable summands of the classifying space of a compact lie group

LET G be a finite group. Recent works of Benson and Feshbach [2] and Martin0 and Priddy [8,9] give a complete account of the question of stable splittings of the classifying space BG into indecomposable summands. This work is in part motivated by the work of Nishida [13] in which the notion of a dominant summand was first introduced. The purpose of this paper is to develop a framework for studying the analogous question for compact Lie groups. As in the case of finite groups where a certain generalization of the affirmative solution to Segal’s Burnside ring conjecture was used, our work begin with a theorem of May, Snaith and Zelewski [lo] which describes the set of stable maps from BQ to BK where Q is finite and K is compact Lie. If there were a similar description for Q compact Lie, one could try to imitate the work of [2] and [8] and reduce the problem to one of representation theory. However, as shown by work of Feshbach [4] and later Bauer [l] on Segal’s Burnside ring conjecture for compact Lie groups, the situation when Q is not finite is much more intricate. The key ingredient in our investigation of the stable summands of BG when G is a compact Lie group is the utilization of certain group-theoretic notions to analyze stable maps between classifying spaces of compact Lie groups. The end result is that we can define group-theoretic invariants for stable summands of BG which are invariant under stable homotopy equivalence. This puts a restriction on what kind of stable summands can occur for a given compact Lie group G. As an application, we use this framework to study two special classes of stable summands. The first is a generalized notion of a dominant summand in BG applicable for compact Lie groups. Let ( )i denote completion at the prime p in the sense of Bousfield and Kan [3]. For technical reasons, we shall add a disjoint basepoint to each classifying space of G, denoting the result by BG, . Our main theorem in this case is _ THEOREM 3.8. Suppose Xi is a dominant summand of BGi+~ where Gi is a compact Lie group with p-Sylow subgroup Ni, i = 1,2. Zf X1 is stably homotopy equivalent to X2, then N1 is isomorphic to N1.

Segal’s Burnside Ring Conjecture for Compact Lie Groups

What is Segal’s Burnside ring conjecture for compact Lie groups? In order to explain it, we will have to go back thirty years when Michael Atiyah proved the following remarkable result in topological K-theory.

On the nilpotence order of β1

For p > 2, β1 ∈ π 2p2−2p−2(S) is the first positive even-dimensional element in the stable homotopy groups of spheres. A classical theorem of Nishida [Nis73] states that all elements of positive dimension in the stable homotopy groups of spheres are nilpotent. In fact, Toda [Tod68] proved β 2−p+1 1 = 0. For p = 3 he showed that β 1 = 0 while β 5 1 6= 0. In [Rav86] the second author computed the first thousand stems of the stable homotopy groups of spheres at the prime 5. One of the consequences of this computation is that β 1 = 0 while β 17 1 6= 0. Our purpose here is to study the problem for larger primes. Our result is the following.