Mathematical Sciences: RUI: The Homotopy Theory of Arrangements

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The investigator will continue to study the dependence of topological invariants of the complement of a union of hyperplanes in complex n-space on the combinatorial structure of the associated arrangement. Invariants to be studied include the homology of the universal abelian cover, the higher homotopy groups, the homology of the Milnor fiber, and the cohomology ring. These will be described in terms of the underlying matroid, the oriented matroid and its generalization to non-real arrangements, and the bounded complex. The results will be applied to unsolved problems concerning the matroid stratification of the Grassmannian, the homology classification of hyperplane complements in terms of combinatorial invariants, the combinatorial structure of free arrangements, and the factorization properties of the Orlik-Solomon algebra. The initial focus of this research will be on complexified arrangements in dimension three. The main topological tool is the 2-complex model T of the universal cover constructed by the investigator. The structure of T is determined in a simple way by the complex of bounded faces of the real part of the arrangement. On the one hand, this facilitates a geometric analysis of the second homotopy group. At the same time, this universal cover complex yields an algebraic description of the homology of abelian covers in terms of modules over polynomial rings. The study of free arrangements will center on the various notions of formality introduced by the investigator and his co- workers. The complement of an arrangement of hyperplanes in n-space is just the collection of chunks into which n-space falls when cleaved by a number of (n-1)-spaces. This seemingly innocuous structure has called forth a surprising amount of heavy machinery from various parts of algebra and topology and led to some gratifyingly attractive analysis and general combinatorial principles. Although the work is theoretical and leads to rigorous proofs, at intermediate stages it is useful to experiment and test examples. Some machine computation is indispensable at this point, particularly using such symbolic manipulators as Cayley, for which a Sun SPARCstation is available to the investigator.