Alexander Invariants of Complex Hyperplane Arrangements
نویسنده
چکیده
Let A be an arrangement of n complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism, α : Fs → Pn. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of A. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of A. We also provide a combinatorial criterion for when these lower bounds are attained.
منابع مشابه
Geometric and Homological Finiteness in Free Abelian Covers
We describe some of the connections between the Bieri–Neumann– Strebel–Renz invariants, the Dwyer–Fried invariants, and the cohomology support loci of a space X. Under suitable hypotheses, the geometric and homological finiteness properties of regular, free abelian covers of X can be expressed in terms of the resonance varieties, extracted from the cohomology ring of X. In general, though, tran...
متن کاملTutte polynomials of hyperplane arrangements and the finite field method
The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement, which answers a wide variety of questions about its underlying object. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a h...
متن کاملTranslated Tori in the Characteristic Varieties of Complex Hyperplane Arrangements
Abstract. We give examples of complex hyperplane arrangements A for which the top characteristic variety, V1(A), contains positive-dimensional irreducible components that do not pass through the origin of the algebraic torus (C∗)|A|. These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhibit a pair of arrangements for which the resonance varieties of the Or...
متن کاملOn Milnor Fibrations of Arrangements
We use covering space theory and homology with local coefficients to study the Milnor fiber of a homogeneous polynomial. These techniques are applied in the context of hyperplane arrangements, yielding an explicit algorithm for computing the Betti numbers of the Milnor fiber of an arbitrary real central arrangement in C3, as well as the dimensions of the eigenspaces of the algebraic monodromy. ...
متن کاملFundamental groups, Alexander invariants, and cohomology jumping loci
We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. The jump loci of a space X come in two basic flavors: the characteristic varieties, or, the support loci for homology with coefficien...
متن کامل