Homotopy equivalence between diagrams of spaces
Any discrete group G can be considered as a small category having a single object and having G itself as the set of endomorphisms (in fact automorphisms) of that single object. An action of G on a space X can then be thought of as a functor from G to Top which takes X as its value on the single object of G. \textit{G. E. Bredon} [Equivariant cohomology theories (Lect. Notes Math. 34) (1967; Zbl 0162.272)] proved that if \(f: X\to Y\) is a G-map between G-CW complexes, then it is a G-homology equivalence if and only if for each subgroup H of G the induced map \(f^ H: X^ H\to Y^ H\) on fixed point sets, is a homotopy equivalence. By a careful analysis of Bredon's result, the authors extend it to apply to diagrams of spaces indexed by an arbitrary small category, D. This requires them to generalize the concepts of orbits and fixed point set to the case of a general diagram. These concepts are then applied to give the generalization of Bredon's result together with an application to diagrams of simplicial complexes.
scientific article; zbMATH DE number 3990393