Combinatorial Morse theory and minimality of hyperplane arrangements

TL;DR

This paper introduces a combinatorial gradient vector field on the Salvetti complex to analyze hyperplane arrangements, providing new tools for understanding their topology and local homology.

Contribution

It presents an explicit combinatorial gradient vector field on the Salvetti complex and offers a detailed construction for the braid arrangement, advancing combinatorial methods in hyperplane arrangement topology.

Findings

Constructed a combinatorial gradient vector field on the Salvetti complex.

Provided a combinatorial description of singular facets and local homology.

Detailed the case of the braid arrangement with explicit construction.

Abstract

We find an explicit combinatorial gradient vector field on the well known complex S (Salvetti complex) which models the complement to an arrangement of complexified hyperplanes. The argument uses a total ordering on the facets of the stratification of R^n associated to the arrangement, which is induced by a generic system of polar coordinates. We give a combinatorial description of the singular facets, finding also an algebraic complex which computes local homology. We also give a precise construction in the case of the braid arrangement.