Topology of Hyperplane Arrangements

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DMS-0105342Alexandru I. Suciu This project is centered around a topological studyof complex hyperplane arrangements, with a view towardsfinding effectively computable invariants of their complements.The goal is to decide whether a given invariant iscombinatorially determined, and, if it is, to expressit explicitly in terms of the intersection lattice ofthe arrangement. An important role is played by thejumping loci for cohomology with coefficients in localsystems, and the related resonance varieties.These varieties have emerged as a central object of study.They provide deep information about the homotopy theoryof the complement of an arrangement, as well asa bridge relating various invariants, in often unexpected ways.Another key role is played by the rational-homotopynotion of formality, which provides the underlyingexplanation 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 theresults, and helps explain what is really peculiar to complexhyperplane arrangements.In its simplest manifestation, an arrangement is afinite collection of lines in the plane. These linescut the plane into components, and understanding the topologyof the complement amounts to counting those components.In the case of lines in the complex plane (or, for thatmatter, hyperplanes in complex n-space), the complementis connected, and its topology (as reflected, for example,in its homotopy groups) is much more interesting.The theory of arrangements is a relatively new branchof mathematics, started in the 1960's with a studyof the classifying space for the pure braid group.The theory has developed at the interface betweentopology, algebra, algebraic geometry, and combinatorics.Hyperplane arrangements, and the closely relatedconfiguration spaces, are used in numerous areas,including robotics, multi-dimensional billiards,graphics, molecular biology, computer vision, anddatabases for representing the space of all possiblestates of a system characterized by many degreesof freedom. There are also deep connections betweenhyperplane arrangements, knot theory, hypergeometricfunctions, conformal field theory, and quantum cohomology.