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Topological And Combinatorial Properties Of Neighborhood And Chessboard Complexes, Matthew Zeckner

University of Kentucky Doctoral Dissertations

This dissertation examines the topological properties of simplicial complexes that arise from two distinct combinatorial objects. In 2003, A. Björner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SG_n,k is homotopy equivalent to a k-sphere. Further, for n = 2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of SG_n,k contains as a deformation retract the boundary complex of a simplicial polytope. Part one of this dissertation …