Cell Complexes for Arrangements with Group Actions

TL;DR

This paper demonstrates that the Salvetti complex for real hyperplane arrangements is homotopy equivalent to their complexified complements, and explores how this equivalence persists under group actions, with applications to braid arrangements.

Contribution

It extends the understanding of homotopy equivalences of Salvetti complexes under group actions and provides new constructions for orbit complexes in reflection arrangements.

Findings

Salvetti complex homotopy equivalent to arrangement complement

Homotopy equivalence preserved under quotient by reflection groups

New construction of orbit complex for finite reflection arrangements

Abstract

For a real oriented hyperplane arrangement, we show that the corresponding Salvetti complex is homotopy equivalent to the complement of the complexified arrangement. This result was originally proved by M. Salvetti. Our proof follows the framework of a proof given by L. Paris and relies heavily on the notation of oriented matroids. We also show that homotopy equivalence is preserved when we quotient by the action of the corresponding reflection group. In particular, the Salvetti complex of the braid arrangement in $\ell$ dimensions modulo the action of the symmetric group is a cell complex which is homotopy equivalent to the space of unlabelled configurations of $\ell$ distinct points. Lastly, we describe a construction of the orbit complex from the dual complex for all finite reflection arrangements in dimension 2. This description yields an easy derivation of the so-called "braid relations" in the case of braid arrangement.