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 $K$-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.Comment: 27 paper, 7 figure

An equivariant discrete model for complexified arrangement complements

We define a partial ordering on the set $\mathcal {Q}=\mathcal {Q}(\mathsf {M})$ of pairs of topes of an oriented matroid $\mathsf {M}$, and show the geometric realization $\vert\mathcal {Q}\vert$ of the order complex of $\mathcal {Q}$ has the same homotopy type as the Salvetti complex of $\mathsf {M}$. For any element $e$ of the ground set, the complex $\vert\mathcal {Q}_e\vert$ associated to the rank-one oriented matroid on $\{e\}$ has the homotopy type of the circle. There is a natural free simplicial action of $\mathbb{Z}_4$ on $\vert\mathcal {Q}\vert$, with orbit space isomorphic to the order complex of the poset $\mathcal {Q}(\mathsf {M},e)$ associated to the pointed (or affine) oriented matroid $(\mathsf {M},e)$. If $\mathsf {M}$ is the oriented matroid of an arrangement $\mathscr {A}$ of linear hyperplanes in $\mathbb{R}^n$, the $\mathbb{Z}_4$ action corresponds to the diagonal action of $\mathbb{C}^*$ on the complement $M$ of the complexification of $\mathscr {A}$: $\vert\mathcal {Q}\vert$ is equivariantly homotopy-equivalent to $M$ under the identification of $\mathbb{Z}_4$ with the multiplicative subgroup $\{\pm 1, \pm i\}\subset \mathbb{C}^*$, and $\vert\mathcal {Q}(\mathsf {M},e)\vert$ is homotopy- equivalent to the complement of the decone of $\mathscr {A}$ relative to the hyperplane corresponding to $e$. All constructions and arguments are carried out at the level of the underlying posets.We also show that the class of fundamental groups of such complexes is strictly larger than the class of fundamental groups of complements of complex hyperplane arrangements. Specifically, the group of the non- Pappus arrangement is not isomorphic to any realizable arrangement group. The argument uses new structural results concerning the degree-one resonance varieties of small matroids

Arrangements of Submanifolds and the Tangent Bundle Complement

Drawing parallels with the theory of hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection \A of locally flat codimension $1$ submanifolds that intersect like hyperplanes. To such an arrangement we associate two posets: the \emph{poset of faces} (or strata) \FA and the \emph{poset of intersections} L(\A). We also associate two topological spaces to \A. First, the complement of the union of submanifolds in $X$ which we call the \emph{set of chambers} and denote by \Ch. Second, the complement of union of tangent bundles of these submanifolds inside $TX$ which we call the \emph{tangent bundle complement} and denote by M(\A). Our aim is to investigate the relationship between combinatorics of the posets and topology of the complements. We generalize the Salvetti complex construction in this setting and also charcterize its connected covers using incidence relations in the face poset. We also demonstrate some calculations of the fundamental group and the cohomology ring. We apply these general results to study arrangements of spheres, projective spaces, tori and pseudohyperplanes. Finally we generalize Zaslavsky\u27s classical result in order to count the number of chambers