Homotopy decompositions of orbit spaces and the Webb conjecture (Q2759150)

The author studies homotopy and homology decompositions which are associated to the equivariant structure of a \(G\)-CW-complex \(X\) where \(G\) is a Lie group. He generalizes and streamlines techniques of such decompositions. Let \(p\) be a prime number. He proves that if \(G\) is a compact Lie group with a nontrivial \(p\)-subgroup, then the orbit space \((BA_p(G))/G\) of the classifying space of the category associated to the \(G\)-poset \(A_p(G)\) of all nontrivial 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 \(X^{E_n}/(NE_0\cap \cdots \cap NE_n)\) defined over the poset \((sd A_p (G))/G\), where \(sd\) is the barycentric subdivision. The author also investigates some other equivariant homotopy and homology decompositions of \(X\) and proves that if \(G\) is a compact Lie group with a non-trivial \(p\)-subgroup, then the map \(EG\times_G BA_p(G)\to BG\) induced by the \(G\)-map \(BA_p(G)\to *\) is a mod \(p\) homology isomorphism.