Salvetti complexes for conditional oriented matroids

TL;DR

This paper provides a new, choice-free proof that the complement of the complexified real hyperplane arrangement is homotopy equivalent to its Salvetti complex, extending to conditional oriented matroids.

Contribution

It introduces a simplified, visualizable proof and generalizes the homotopy equivalence to the broader setting of conditional oriented matroids.

Findings

Homotopy equivalence between arrangement complement and Salvetti complex

Choice-free and visualizable proof method

Extension to conditional oriented matroids

Abstract

We give a new proof of the fact that the complement of the complexification of a real hyperplane arrangement is homotopy equivalent to the Salvetti complex of the associated oriented matroid. Our proof involves no choices, is relatively easy to visualize, and generalizes to the setting of conditional oriented matroids.