The topology of Baues complexes and flip graphs

Abstract

This thesis contains several results on the connectedness of Baues complexes and flip graphs, which are topological spaces modeling certain sets coming from geometric combinatorics. These sets include the triangulations of a polytope, the tilings of a zonotope, and the extensions of an oriented matroid. Some long-standing conjectures are resolved, including the connectedness of triangulations of a product of two simplices and the sphericity of extension spaces of realizable oriented matroids. This thesis covers the main construction which is common to these proofs, but defers the details specific to each problem to other papers.