Bredon homological stability for configuration spaces of $G$-manifolds

Abstract

McDuff and Segal proved that unordered configuration spaces of open manifolds satisfy homological stability: there is a stabilization map $\sigma :C_n(M)\to C_{n+1}(M)$ which is an isomorphism on $H_d(-;\mathbb{Z})$ for $n\gg d$. For a finite group $G$ and an open $G$-manifold $M$, under some hypotheses we define a family of equivariant stabilization maps ${\sigma }_{G/H}:C_n(M)\to C_{n+|G/H|}(M)$ for $H\le G$. In general, these do not induce stability for Bredon homology, the equivariant analogue of singular homology. Instead, we show that each ${\sigma }_{G/H}$ induces isomorphisms on the ordinary homology of the fixed points of $C_n(M)$, and if the group is Dedekind (e.g. abelian), we obtain the following Bredon homological stability statement: $H_d^G({⨆}_{n\ge 0}{C}_n(M))$ is finitely generated over $\mathbb{Z}[{\sigma }_{G/H}:H\le G]$. This reduces to the classical statement when $G=e$.

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