Topology of Hyperplane Arrangements

Northeastern University, Boston MA

Investigators

Abstract

DMS-0105342 Alexandru I. Suciu This project is centered around a topological study of complex hyperplane arrangements, with a view towards finding effectively computable invariants of their complements. The goal is to decide whether a given invariant is combinatorially determined, and, if it is, to express it explicitly in terms of the intersection lattice of the arrangement. An important role is played by the jumping loci for cohomology with coefficients in local systems, and the related resonance varieties. These varieties have emerged as a central object of study. They provide deep information about the homotopy theory of the complement of an arrangement, as well as a bridge relating various invariants, in often unexpected ways. Another key role is played by the rational-homotopy notion of formality, which provides the underlying explanation for many of the encountered phenomena. Whenever possible, the study is done in a more general setting, which includes certain types of subspace arrangements, both real and complex, as well as certain links in the 3-sphere. Such a point of view enlarges the range of applicability of the results, and helps explain what is really peculiar to complex hyperplane arrangements. In its simplest manifestation, an arrangement is a finite collection of lines in the plane. These lines cut the plane into components, and understanding the topology of the complement amounts to counting those components. In the case of lines in the complex plane (or, for that matter, hyperplanes in complex n-space), the complement is connected, and its topology (as reflected, for example, in its homotopy groups) is much more interesting. The theory of arrangements is a relatively new branch of mathematics, started in the 1960's with a study of the classifying space for the pure braid group. The theory has developed at the interface between topology, algebra, algebraic geometry, and combinatorics. Hyperplane arrangements, and the closely related configuration spaces, are used in numerous areas, including robotics, multi-dimensional billiards, graphics, molecular biology, computer vision, and databases for representing the space of all possible states of a system characterized by many degrees of freedom. There are also deep connections between hyperplane arrangements, knot theory, hypergeometric functions, conformal field theory, and quantum cohomology.