Graph complexes and the Moduli space of Riemann surfaces

In this thesis we compare several combinatorial models for the Moduli space of open-closed cobordisms and their compactifications. More precisely, we study Godin’s category of admissible fat graphs, Costello’s chain complex of black and white graphs, and Bödigheimer’s space of radial slit configurations. We use Hatcher’s proof of the contractibility of the arc complex to give a new proof of a result of Godin, which states that the category of admissible fat graphs is a model of the mapping class group of open-closed cobordisms. We use this to give a new proof of Costello’s result, that the complex of black and white graphs is a homological model of this mapping class group. Beyond giving new proofs of these results, the methods used give a new interpretation of Costello’s model in terms of admissible fat graphs, which is a more classical model of the Moduli space. This connection could potentially allow to transfer constructions in fat graphs to the black and white model. Moreover, we compare Bödigheimer’s radial slit configurations and the space of metric admissible fat graphs, producing an explicit homotopy equivalence using a “critical graph” map. This critical graph map descends to a homeomorphism between the Unimodular Harmonic compactification and the space of Sullivan diagrams, which are natural compactifications of the space of radial slit configurations and the space of metric admissible fat graphs, respectively. Finally, we use experimental methods to compute the homology of the chain complex of Sullivan diagrams of the topological type of the disk with up to seven punctures, and we give explicit generators for the non-trivial groups. We use these experimental results to show that the first and top homology groups of the chain complex of Sullivan diagrams of the topological type of the punctured disk are trivial; and to give two infinite families of non-trivial classes of the homology of Sullivan diagrams which represent non-trivial string operations.