The topology of bounded degree graph complexes and finite free resolutions

Bounded degree graph complexes are simplicial complexes of graphs whose node degrees are bounded from above. The most well-known example is the matching complex, which has arisen in a variety of contexts in the literature. The bounded degree graph complexes have a close connection with the finite free resolutions of quadratic Veronese rings and modules. In the present thesis we study the topology of bounded degree graph complexes. As applications we get results about the minimal free resolutions of quatratic Veronese modules that are independent of the field characteristic. We first show how the combinatorial Laplacian can be used to give an elegant proof of Bouc's theorem on the Betti numbers of matching complexes. This leads to a simple proof of Reiner-Roberts' theorem on the Betti numbers of general bounded degree graph complexes. We then prove various results about the homotopy types of bounded degree graph complexes, and deduce corresponding characteristic-free results about those finite free resolutions. In particular, we characterize the set of multidegrees which support at least one higher syzygy in such a resolution. The answer turns out to be independent of the field characteristic. Finally we show that one of our theorems about bounded degree graph complexes can be generalized to a duality theorem for the squarefree divisor complexes of arbitrary normal semigroup rings. This generalization turns out to have many interesting applications.