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.
Author: | Paul MückschGND |
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URN: | urn:nbn:de:hbz:294-110475 |
DOI: | https://doi.org/10.48550/arXiv.2211.14083 |
Parent Title (English): | De.arxiv.org |
Document Type: | Article |
Language: | English |
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 |
Note: | 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): | Creative Commons - CC BY 4.0 - Attribution 4.0 International |