On models of orbit configuration spaces of surfaces

Abstract

We consider orbit configuration spaces $C_n^G(S)$, where $S$ is a surface obtained out of a closed orientable surface $\bar{S}$ by removing a finite number of points (eventually none) and $G$ is a finite group acting freely continuously on $S$. We prove that the fibration $\pi_{n,k} : C_{n}^G(S) \to C_k^G(S)$ obtained by projecting on the first $k$ coordinates is a rational fibration. As a consequence, the space $C_{n}^G(S)$ has a Sullivan model $A_{n,k}=\Lambda V_{C_k^G(S)}\otimes \Lambda V_{C_{n-k}^G(S_{G,k})}$ fitting in a cdga sequence: $\Lambda V_{C_k^G(S)}\to A_{n,k} \to \Lambda V_{C_{n-k}^G(S_{G,k})},$ where $\Lambda V_X$ denotes the minimal model of $X$, and $C_{n-k}^G(S_{G,k})$ is the fiber of $\pi_{n,k}$. We show that this model is minimal except for some cases when $S\simeq S^2$ and compute in all the cases the higher $\psi$-homotopy groups (related to the generators of the minimal model) of $C_n^G(S)$. We deduce from the computation that $C_n^G(S)$ having finite Betti numbers is a rational $K(\pi,1)$, i.e its minimal model and $1$-minimal model are the same (or equivalently the $\psi$-homotopy space vanishes in degree grater then $2$), if and only if $S$ is not homeomorphic to $S^2$. In particular, for $S$ not homeomorphic to $S^2$, the minimal model (isomorphic to the $1$-minimal model) is entirely determined by the Malcev Lie algebra of $\pi_1 C_n^G(S)$. When $A_{n,k}$ is minimal, we get an exact sequence of Malcev Lie algebras $0\to L_{C_{n-k}^G(S_{G,k})}\to L_{C_{n}^G(S)}\to L_{C_k^G(S)}\to 0$, where $L_X$ is the Malcev Lie algebra of $\pi_1X$. For $S \varsubsetneq \bar{S}=S^2$ and $G$ acting by orientation preserving homeomorphism, we prove that the cohomology ring of $C_n^G(S)$ is Koszul, and that for some of these spaces the minimal model can be obtained out of a Cartan-Chevally-Eilenberg construction applied to graded Lie algebra computed in an earlier work.