Homotopy decompositions of orbit spaces and the Webb conjecture

Tom 169 / 2001

Jolanta S/lomi/nska Fundamenta Mathematicae 169 (2001), 105-137 MSC: 55P91, 55U40, 18G10, 18G40. DOI: 10.4064/fm169-2-2


Let $p$ be a prime number. We prove that if $G$ is a compact Lie group with a non-trivial $p$-subgroup, then the orbit space $(B{\cal A}_p(G))/G$ of the classifying space of the category associated to the $G$-poset ${\cal A}_p(G)$ of all non-trivial elementary abelian $p$-subgroups of $G$ is contractible. This gives, for every $G$-CW-complex $X$ each of whose isotropy groups contains a non-trivial $p$-subgroup, a decomposition of $X/G$ as a homotopy colimit of the functor $X^{E_n}/(NE_0\cap \mathinner {\ldotp \ldotp \ldotp }\cap NE_n)$ defined over the poset $(\mathop {\rm sd}\nolimits {\cal A}_p(G))/G$, where $\mathop {\rm sd}\nolimits $ is the barycentric subdivision. We also investigate some other equivariant homotopy and homology decompositions of $X$ and prove that if $G$ is a compact Lie group with a non-trivial $p$-subgroup, then the map $EG\times _G B{\cal A}_p(G)\to BG$ induced by the $G$-map $B{\cal A}_p(G)\to *$ is a mod $p$ homology isomorphism.


  • Jolanta S/lomi/nskaFaculty of Mathematics and Information Sciences
    Technical University of Warsaw
    Pl. Politechniki 1
    00-661 Warszawa, Poland

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