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Modular flats of oriented matroids and poset quasi-fibrations.
- Source :
- Transactions of the American Mathematical Society, Series B; 1/30/2024, Vol. 11, p306-328, 23p
- Publication Year :
- 2024
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 23300000
- Volume :
- 11
- Database :
- Complementary Index
- Journal :
- Transactions of the American Mathematical Society, Series B
- Publication Type :
- Academic Journal
- Accession number :
- 175133938
- Full Text :
- https://doi.org/10.1090/btran/168