TY - JOUR

T1 - Modular flats of oriented matroids and poset quasi-fibrations

AU - Mücksch, Paul

N1 - Funding Information: The author was financially supported by the JSPS during a JSPS fellowship for foreign researchers in Japan. This work was partially supported by the DFG-grant RO1072/19-1. The author was supported by the Open Access Publication Funds of the Ruhr Universität Bochum.

PY - 2024/1/24

Y1 - 2024/1/24

N2 - 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(π,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.

AB - 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(π,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.

KW - math.CO

KW - math.AT

KW - poset quasi-fibration

KW - Oriented matroid

KW - supersolvable lattice

KW - discrete Morse theory

KW - Salvetti complex

UR - http://www.scopus.com/inward/record.url?scp=85184933220&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2211.14083

DO - 10.48550/arXiv.2211.14083

M3 - Article

VL - 11

SP - 306

EP - 328

JO - Transactions of the American Mathematical Society. Series B

JF - Transactions of the American Mathematical Society. Series B

SN - 2330-0000

ER -