Combinatorial Models for Manifolds and Vector Bundles

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9803615Anderson This project centers on two combinatorial models for manifolds andvector bundles: Gelfand and MacPherson's theory of combinatorial differentialmanifolds and matroid bundles and Mnev and Mani's theory of locally polytopalmanifolds. The primary focus is on the bundle theory and classifying spaceswithin each model. In both of these models, a bundle-theoretic perspectiveleads to deep and unexpected new relations between topology and combinatorics:the primary objective of this research is to expand on these relations. Theproject's goals include both combinatorial applications in topology (forinstance, combinatorial formulas for topological invariants) and topologicalresults in combinatorics (particularly on the topology of posets of orientedmatroids and polytopes). The classifying spaces for Gelfand and MacPherson'stheory derive from certain partially ordered sets of oriented matroids, andhave close ties to the real Grassmannian. Among the objectives of thisresearch are a closer knowledge of the topology of the combinatorialGrassmannian and applications of this knowledge both to real vector bundles and to various combinatorial problems. Mnev and Mani's theory also appearsto give rise to a classifying space of interest in both topology andcombinatorics, relating PL manifolds and convex polytopes. The author willalso study a new type of ``non-Euclidean'' pathology in oriented matroids andits implications for combinatorial differential manifolds. Much of the power of these combinatorial models for topology derivesfrom the elegance with which topological concepts translate into combinatorial analogs. One can, for instance, derive combinatorial formulas for topologicalinvariants by simply translating each element of a topological formula intoits combinatorial equivalent. The translation from combinatorics to topologyhas proven equally useful: these models have provided a new framework in whichto view various combinatorial topics. Both models have revealed avector-bundle perspective on various combinatorial problems that originallyappeared to have no connection to real vector bundles, leading to unexpected topological proofs. This project will build on recent results in this areawhose depth lies in uniting the combinatorial side of mathematics, rooted incounting things, to the topological side, rooted in plasticity and spatialcontinuity.***