Mathematical Sciences: Topology of Complex Hyperplane Arrangements

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9504833 Suciu The principal investigator, Alexandru I. Suciu, will study topological invariants of the complement of an arrangement of hyperplanes in complex n-space, and their relationship to combinatorial invariants of the intersection lattice of the arrangement. This project, continuing work with Daniel C. Cohen, draws on a variety of techniques from algebraic topology, group theory, algebraic geometry, computational algebra (Groebner basis), and combinatorics (oriented matroids). The starting point of the investigation is a presentation of the fundamental group that closely reflects the topology of the complement. From this presentation, computable algebraic objects are derived: Chen groups, Alexander invariants, characteristic varieties, etc. Some of these topological invariants can be computed directly from the lattice, but some probably require more subtle information, to be extracted from the underlying matroid. Under certain hypotheses, this approach permits the computation of the cohomology of the complement, with coefficients in a local system determined by a linear representation of the fundamental group. Such computations have applications in the theory of singularities (Milnor fibrations of non-isolated singularities, monodromies of complex plane curves), theory of braids (generalized Burau and Gassner representations), differential equations (Knizhnik-Zamolodchikov equations, hypergeometric functions), and low-dimensional topology. In its simplest manifestation, an arrangement is merely a collection of lines in the plane. These lines cut the real plane into pieces, and understanding the topology of the complement is an elementary exercise, which amounts to counting those pieces. In the case of lines in the complex plane (or, for that matter, hyperplanes in complex n-space), the complement is of one piece. But, unlike a disk, for example, this does not mean that it can be drawn back over itself until it shrinks down to a single poi nt: An algebraic invariant that measures this failure is the fundamental group, which, roughly speaking, consists of those loops that can not be shrunk in the complement. A particularly important example is the braid arrangement of "diagonal" hyperplanes in complex n-space. In that case, loops in the complement can be viewed as braids (strings of wire weaving around each other, without backing up), and the fundamental group can be identified with the (pure) braid group. For arbitrary arrangements, the identification of the fundamental group is more complicated, but it can be done in an algorithmic way, using the theory of braids. This theory, in turn, is intricately connected with the theory of knots and links in 3-space, with its wealth of algebraic and combinatorial invariants, and its varied applications to biology, chemistry, and physics. ***