Related papers: A long exact sequence in cohomology for deleted an…

We study the topology of moduli spaces of closed linkages in \R^d depending on a length vector \ell\in \R^n. In particular, we use equivariant Morse theory to obtain information on the homology groups of these spaces, which works best for…

A hyperplane arrangement in $\mathbb{R}^n$ is a finite collection of affine hyperplanes. Counting regions of hyperplane arrangements is an active research direction in enumerative combinatorics. In this paper, we consider the arrangement…

Let $\mathcal A$ be a hyperplane arrangement in $\mathbb C^n$. We show that the number of decomposition factors as a perverse sheaf of the direct image $Rj_*\mathbb C_U $ of the constant sheaf on the complement $U$ to the arrangement is…

Given a nonnegative integer $m$ and a finite collection ${\mathcal A}$ of linear forms on ${\mathbb Q}^d$, the arrangement of affine hyperplanes in ${\mathbb Q}^d$ defined by the equations $\alpha(x) = k$ for $\alpha \in {\mathcal A}$ and…

We give a long exact sequence for the homology of a graded atomic lattice equipped with a sheaf of modules, in terms of the deleted and restricted lattices. This is then used to compute the homology of the arrangement lattice of a…

We construct a seven-term exact sequence involving low degree cohomology spaces of a Lie algebra $\Lg$, an ideal $\Lh$ of $\Lg$ and the quotient $\Lg / \Lh$ with coefficients in a $\Lg$-module. The existence of such a sequence follows from…

Let $B$ be an arrangement of linear complex hyperplanes in $C^d$. Then a classical result by Orlik \& Solomon asserts that the cohomology algebra of the complement can be constructed from the combinatorial data that are given by the…

We compute the periods associated with a special class of hyperplane arrangements. In particular, we exhibit a combinatorial condition on the intersection lattice of a hyperplane arrangement that ensures that its associated periods are…

Deletion-restriction is a fundamental tool in the theory of hyperplane arrangements. Various important results in this field have been proved using deletion-restriction. In this paper we use deletion-restriction to identify a class of toric…

We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincare duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincare duality in the same…

The resonance arrangement $\mathcal{A}_n$ is the arrangement of hyperplanes in $\mathbb{R}^n$ given by all hyperplanes of the form $\sum_{i \in I} x_i = 0$, where $I$ is a nonempty subset of $\{1,\dots,n\}$. We consider the characteristic…

Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where…

In [5], Hochschild established a 6-term exact sequence for the cohomology of restricted Lie algebras. We generalize this result to restricted Lie superalgebras.

An arrangement is a finite set of hyperplanes in a finite dimensional complex affine space. A complex rank one local system on the arrangement complement is determined by a set of complex weights for the hyperplanes. We study the…

We define and study the Tutte polynomial of a hyperplane arrangement. We introduce a method for computing it by solving an enumerative problem in a finite field. For specific arrangements, the computation of Tutte polynomials is then…

We consider arrangements of n connected codimensional one submanifolds in closed d-dimensional manifold M. Let f be the number of connected components of the complement in M to the union of submanifolds. We prove the sharp lower bound for f…

We describe the ringed-space structure of moduli spaces of jets of linear connections (at a point) as orbit spaces of certain linear representations of the general linear group. Then, we use this fact to prove that the only (scalar)…

We show a connection between a surgery exact sequence in knot Floer homology and the sequence derived in [18]. As a consequence of this relationship we see that the exact sequence in [18] also works with coherent orientations and admits…

The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements.…

For an extension of associative algebras $B\subset A$ over a field and an $A$-bimodule $X$, we obtain a Jacobi-Zariski long nearly exact sequence relating the Hochschild homologies of $A$ and $B$, and the relative Hochschild homology, all…