Algebraic invariants of orbit configuration spaces in genus zero associated to finite groups

TL;DR

This paper computes algebraic invariants of orbit configuration spaces on the sphere minus points, generalizing Arnold's classical results, and establishes a formula linking Poincaré series to the lower central series of the fundamental group.

Contribution

It introduces the computation of cohomology rings and Poincaré series for orbit configuration spaces under finite group actions, extending classical configuration space results.

Findings

Computed cohomology rings and Poincaré series of orbit configuration spaces.

Proved formality of these spaces in rational homotopy theory.

Established an LCS formula relating Poincaré series to fundamental group quotients.

Abstract

We consider orbit configuration spaces associated to finite groups acting freely by orientation preserving homeomorphisms on the $2$-sphere minus a finite number of points. Such action is equivalent to a homography action of a finite subgroup $G\subset \mathrm{PGL}(\mathbb{C}^2)$ on the complex projective line $\mathbb{P}^1$ minus a finite set $Z$ stable under $G$. We compute the cohomology ring and the Poincar\'e series of the orbit configuration space $C_n^G(\mathbb{P}^1 \setminus Z)$. This can be seen as a generalization of the work of Arnold for the classical configuration space $C_n(\mathbb{C})$ ($(G,Z)=(\{1\},\infty$)). It follows from the work that $C_n^G(\mathbb{P}^1\setminus Z)$ is formal in the sense of rational homotopy theory. We also prove the existence of an LCS formula relating the Poincar\'e series of $C_n^G(\mathbb{P}^1\setminus Z)$ to the ranks of quotients of successive terms of the lower central series of the fundamental group of $C_n^G(\mathbb{P}^1 \setminus Z)$. The successive quotients correspond to homogenous elements of graded Lie algebras introduced by the author in an earlier work. Such formula is also known for classical configuration spaces of $\mathbb{C}$, where fundamental groups are Artin braid groups and the ranks correspond to dimensions of homogenous elements of the Kohno-Drinfeld Lie algebras.