The relative Burnside module and the stable maps between classifying spaces of compact Lie groups

Abstract

Tom Dieck’s Burnside ring of compact Lie groups is generalized to the relative case: For anyG▹NG \triangleright N, a compact Lie group and its normal subgroupA(G▹N)A(G \triangleright N)is defined to be an appropriate set of the equivalence classes of compactGG-ENR’s with freeNN-action, in such a way thatψ:A(G▹N)≃πG/N0(S0;B(N,G)+)\psi :A(G \triangleright N) \simeq \pi _{G/N}^0({S^0};B{(N,G)_ + }), whereB(N,G)B(N,G)is the classifying space of principal(N,G)(N,G)-bundle. Under the "product" situation, i.e.G=F×K,N=K,A(F×K▹K)G = F \times K,\;N = K,\;A(F \times K \triangleright K)is also denoted byA(F,K)A(F,K), as it turns out to be the usualA(F,K)A(F,K)when bothFFandKKare finite. Then a couple of applications are given to the study of stable maps between classifying spaces of compact Lie groups: a conceptual proof of Feshbach’s double coset formula, and a density theorem on the mapαp∧:A(L,H)p∧→{BL+,BH+}p∧\alpha _p^ \wedge :A(L,H)_p^ \wedge \to \{ B{L_{ + ,}}B{H_ + }\} _p^ \wedgefor any compact Lie groupsL,KL,\;Kwhenppis odd. (Some restriction is applied toLLwhenp=2p = 2.) This latter result may be regarded as the pushout of Feshbach’s density theorem and the theorem of May-Snaith-Zelewski, over the celebrated Carlsson solution of Segal’s Burnside ring conjecture.