Abstract: Given two G-spaces (G a compact Lie group), when are their orbit spaces homotopy equivalent? I will discuss a new and surprisingly elementary method which guarantees such an equivalence. This result gives a quick and elegant proof of Webb's conjecture, namely to the contractibility of the orbit space of the simplicial complex associated to the non-trivial p-subgroups of a finite group G. This conjecture has been proved by Peter Symonds, but his method does not extended to compact Lie groups, unlike the approach I will present.

The ideas can also be used to determine when a map between two G-spaces induces an isomorphism in equivariant cohomology. There results a direct sum decomposition of the cohomology ring of a finite group G, known as Webb's formula. This decomposition is a shadow of a stronger topological result on the classifying space of G. Our methods have the advantage that they extend without difficulty to compact Lie groups.