Elmendorf’s Theorem via Model Categories

In [2], working in the category of compactly generated spaces U , Elmendorf relates the equivariant homotopy theory of G-spaces to a homotopy theory of diagrams using fixed point sets. The diagrams are indexed by a topological category OG with objects the orbit spaces {G/H}H for the closed subgroups H ⊂ G. Although, his general assumption there is that G is a compact Lie group, a formulation of Elmendorf’s Theorem can be found on page 44 in [10] for any topological group in U . A more modern approach has been given by Piacenza in [11] using model categories. For any topological group G in U , he equips the category UO op G of continuous contravariant functors from OG to U with a model category structure, where the weak equivalences are the objectwise weak equivalences. Concerning equivariant homotopy theory, a morphism f in the category of G-spaces UG is defined to be a weak equivalence, if for all closed subgroups H ⊂ G, the map (f)H is a weak equivalence between spaces, where (−)H : UG → U is the H-fixed point functor. That is, for a G-space X one has