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 -