Modular flats of oriented matroids and poset quasi-fibrations

  • 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 \(\it K\)-theory. As a direct consequence, the Salvetti complex of an oriented matroid whose geometric lattice is supersolvable is a \(\it K\)(\(\pi\),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.

Download full text files

Export metadata

Additional Services

Share in Twitter Search Google Scholar
Author:Paul MückschGND
Parent Title (English)
Document Type:Article
Date of Publication (online):2024/03/11
Date of first Publication:2022/11/25
Publishing Institution:Ruhr-Universität Bochum, Universitätsbibliothek
Tag:Open Access Fonds
Oriented matroid; Salvetti complex; discrete Morse theory; poset quasi-fibration; supersolvable lattice
First Page:1
Last Page:27
Article Processing Charge funded by the Deutsche Forschungsgemeinschaft (DFG) and the Open Access Publication Fund of Ruhr-Universität Bochum.
Institutes/Facilities:Lehrstuhl für Algebra / Zahlentheorie
Dewey Decimal Classification:Naturwissenschaften und Mathematik / Mathematik
open_access (DINI-Set):open_access
faculties:Fakultät für Mathematik
Licence (English):License LogoCreative Commons - CC BY 4.0 - Attribution 4.0 International