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Abstract

McDuff and Segal proved that unordered configuration spaces of open manifolds satisfy homological stability: there is a stabilization map σ:Cn(M)→Cn+1(M) which is an isomorphism on Hd(−;Z) for n≫d. For a finite group G and an open G-manifold M, under some hypotheses we define a family of equivariant stabilization maps σG/H:Cn(M)→Cn+|G/H|(M) for H≤G. In general, these do not induce stability for Bredon homology, the equivariant analogue of singular homology. Instead, we show that each σG/H induces isomorphisms on the ordinary homology of the fixed points of Cn(M), and if the group is Dedekind (e.g. abelian), we obtain the following Bredon homological stability statement: HGd(⨆n≥0Cn(M)) is finitely generated over Z[σG/H:H≤G]. This reduces to the classical statement when G=e.