Modular flats of oriented matroids and poset quasi-fibrations

TL;DR

This paper explores the combinatorics and topology of modular flats in oriented matroids, establishing new fibration properties and asphericity results for their Salvetti complexes, especially in the supersolvable case.

Contribution

It introduces the notion of poset quasi-fibrations for modular flats, generalizes classical hyperplane arrangement results, and constructs many non-realizable supersolvable oriented matroids.

Findings

Salvetti complex of supersolvable oriented matroids is a K(π,1)-space

Fundamental group is an iterated semidirect product of free groups

Provides a construction method for supersolvable oriented matroids

Abstract

We study the combinatorics of modular flats of oriented matroids and the topological consequences for their Salvetti complexes. We show that the natural map to the localized Salvetti complex at a modular flat of corank one is what we call a poset quasi-fibration -- a notion derived from Quillen's fundamental Theorem B from algebraic $K$-theory. As a direct consequence, the Salvetti complex of an oriented matroid whose geometric lattice is supersolvable is a $K(\pi,1)$-space -- a generalization of the classical result for supersolvable hyperplane arrangements due to Falk, Randell and Terao. Furthermore, the fundamental group of the Salvetti complex of a supersolvable oriented matroid is an iterated semidirect product of finitely generated free groups -- analogous to the realizable case. Our main tools are discrete Morse theory, the shellability of certain subcomplexes of the covector complex of an oriented matroid, a nice combinatorial decomposition of poset fibers of the localization map, and an isomorphism of covector posets associated to modular elements. We provide a simple construction of supersolvable oriented matroids. This gives many non-realizable supersolvable oriented matroids and by our main result aspherical CW-complexes.