Combinatorics of orbit configuration spaces

TL;DR

This paper studies the topology of orbit configuration spaces arising from group actions, providing a combinatorial framework and explicit cohomological calculations that generalize classical lattice structures.

Contribution

It introduces a combinatorial description of the poset of layers in orbit configuration spaces, extending partition and Dowling lattices, and applies this to compute their cohomology.

Findings

Poset of layers generalizes partition and Dowling lattices.

Explicit cohomological calculations for orbit configuration spaces.

Provides a combinatorial framework for studying these spaces.

Abstract

From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this "orbit configuration space" is the complement of an arrangement of subvarieties inside the cartesian product, and we use this structure to study its topology. We give an abstract combinatorial description of its poset of layers (connected components of intersections from the arrangement) which turns out to be of much independent interest as a generalization of partition and Dowling lattices. The close relationship to these classical posets is then exploited to give explicit cohomological calculations.