TOPOLOGICAL AND COMBINATORIAL PROPERTIES OF NEIGHBORHOOD AND CHESSBOARD COMPLEXES

Author

Matthew Zeckner, University of KentuckyFollow

Date Available

6-9-2011

Year of Publication

2011

Degree Name

Doctor of Philosophy (PhD)

Document Type

Dissertation

College

Arts and Sciences

Department

Mathematics

First Advisor

Dr. Benjamin Braun

Second Advisor

Dr. Carl Lee

Abstract

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 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 SG_n,2. Part two of this dissertation studies simplicial complexes that arise from non-attacking rook placements on a subclass of Ferrers boards that have a_i rows of length i where a_i > 0 and i ≤ n for some positive integer n. In particular, enumerative properties of their facets, homotopy type, and homology are investigated.

Recommended Citation

Zeckner, Matthew, "TOPOLOGICAL AND COMBINATORIAL PROPERTIES OF NEIGHBORHOOD AND CHESSBOARD COMPLEXES" (2011). University of Kentucky Doctoral Dissertations. 163.
https://uknowledge.uky.edu/gradschool_diss/163