Homotopy decompositions of orbit spaces and the Webb conjecture
Volume 169 / 2001
Abstract
Let 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.