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


Author: Norihiko Minami
Journal: Trans. Amer. Math. Soc. 347 (1995), 461-498
MSC: Primary 55P42; Secondary 22E99, 55Q91, 55R35, 55R91, 57S15
DOI: https://doi.org/10.1090/S0002-9947-1995-1290726-5
MathSciNet review: 1290726
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1290726-5
Keywords: Burnside ring, classifying space, compact Lie groups, stable homotopy theory, Segal conjecture
Article copyright: © Copyright 1995 American Mathematical Society

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