On models of orbit configuration spaces of surfaces

TL;DR

This paper studies the rational homotopy type of orbit configuration spaces on surfaces, showing they are often rational K(π,1) spaces, computing their minimal models, and analyzing their cohomology and Malcev Lie algebras.

Contribution

It provides a detailed rational homotopy model for orbit configuration spaces on surfaces, including minimal models, fiber sequences, and conditions for being rational K(π,1), extending previous understanding.

Findings

C_n^G(S) is a rational K(π,1) iff S is not S^2.

Computed minimal models and higher ψ-homotopy groups for these spaces.

Proved cohomology ring is Koszul for certain cases.

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.