Homotopy decompositions of orbit spaces and the Webb conjecture

Jolanta S/lomi/nska¹

¹ Faculty of Mathematics and Information Sciences Technical University of Warsaw Pl. Politechniki 1 00-661 Warszawa, Poland

Fundamenta Mathematicae, Tome 169 (2001) no. 2, pp. 105-137

Résumé

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.

Détail

DOI : 10.4064/fm169-2-2

Keywords: prime number prove compact lie group non trivial p subgroup orbit space cal classifying space category associated g poset cal non trivial elementary abelian p subgroups contractible gives every g cw complex each whose isotropy groups contains non trivial p subgroup decomposition homotopy colimit functor cap mathinner ldotp ldotp ldotp cap defined poset mathop nolimits cal where mathop nolimits barycentric subdivision investigate other equivariant homotopy homology decompositions prove compact lie group non trivial p subgroup map times cal induced g map cal * mod homology isomorphism