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 any $G \triangleright N$, a compact Lie group and its normal subgroup $A(G \triangleright N)$ is defined to be an appropriate set of the equivalence classes of compact $G$-ENR’s with free $N$-action, in such a way that $\psi :A(G \triangleright N) \simeq \pi _{G/N}^0({S^0};B{(N,G)_ + })$, where $B(N,G)$ is the classifying space of principal $(N,G)$-bundle. Under the "product" situation, i.e. $G = F \times K,\;N = K,\;A(F \times K \triangleright K)$ is also denoted by $A(F,K)$, as it turns out to be the usual $A(F,K)$ when both $F$ and $K$ are 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 map $\alpha _p^ \wedge :A(L,H)_p^ \wedge \to \{ B{L_{ + ,}}B{H_ + }\} _p^ \wedge$ for any compact Lie groups $L,\;K$ when $p$ is odd. (Some restriction is applied to $L$ when $p = 2$.) 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.