Shelling-type orderings of regular CW-complexes and acyclic matchings of the Salvetti complex

TL;DR

This paper introduces shelling-type orderings for regular CW-complexes to construct maximum acyclic matchings, applying the method to the Salvetti complex of arrangements and generalizing to oriented matroids.

Contribution

It develops a new combinatorial stratification and ordering method to explicitly construct maximum acyclic matchings in CW-complexes, including the Salvetti complex.

Findings

Constructed maximum acyclic matchings for the Salvetti complex.

Provided explicit descriptions of critical cells based on linear extensions.

Generalized the method to arbitrary oriented matroids.

Abstract

Motivated by the work of Salvetti and Settepanella we introduce certain total orderings of the faces of any shellable regular CW-complex (called `shelling-type orderings') that can be used to explicitly construct maximum acyclic matchings of the poset of cells of the given complex. Building on an application of this method to the classical zonotope shellings we describe a class of maximum acyclic matchings for the Salvetti complex of a linear complexified arrangement. To do this, we introduce and study a new combinatorial stratification of the Salvetti complex. For the obtained acyclic matchings we give an explicit description of the critical cells that depends only on the chosen linear extension of the poset of regions. It is always possible to choose the linear extension so that the critical cells can be explicitly constructed from the chambers of the arrangement via the bijection to no-broken-circuit sets defined by Jewell and Orlik. Our method can be generalized to arbitraty oriented matroids.