TOPOLOGICAL AND COMBINATORIAL PROPERTIES OF NEIGHBORHOOD AND CHESSBOARD COMPLEXES

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 SGn,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 SGn,k contains as a deformation retract the boundary complex of a simplicial polytope. Part one of this dissertation provides a positive answer to this question in the case k = 2. In this case it is also shown that, after partially subdividing the neighborhood complex, the resulting complex deformation retracts onto a subcomplex arising as a polyhedral boundary sphere that is invariant under the action induced by the automorphism group of SGn,2. Part two of this dissertation studies simplicial complexes that arise from non-attacking rook placements on a subclass of Ferrers boards that have ai rows of length i where ai \u3e 0 and i ≤ n for some positive integer n. In particular, enumerative properties of their facets, homotopy type, and homology are investigated

Simplicial Complexes of Triangular Ferrers Boards

We study the simplicial complex that arises from non-attacking rook placements on a subclass of Ferrers boards that have ai rows of length i where ai> 0 and i ≤ n for some positive integer n. In particular, we will investigate enumerative properties of their facets, their homotopy type, and homology.