THE RELATIVE BURNSIDE MODULE AND THE STABLE MAPS BETWEEN CLASSIFYING-SPACES OF COMPACT LIE-GROUPS

Tom Dieck's Burnside ring of compact Lie groups is generalized to the relative case: For any G right-pointing triangle N, a compact Lie group and its normal subgroup A(G right-pointing triangle 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 right-pointing triangle N) N 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 x K, N = K, A(F x K > 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)(boolean AND) : A(L, H)(p)(boolean AND) --> {BL(+), BH+}(boolean AND)(p) 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.