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

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 (eventually none). We compute the cohomology ring and the Poincaré series of these spaces. This generalizes the work of V. Arnold for “classical” configuration spaces of points of the plane. The results imply that the spaces we consider are formal in the sense of rational homotopy theory. We also prove the existence of an LCS formula relating the Poincaré series of such spaces to the ranks of quotients of successive terms of the lower central series of their fundamental group. Such formula is known for the spaces studied by V. Arnold, where fundamental groups are Artin pure braid groups.