I will compare two combinatorial models of the Moduli space of two dimensional cobordisms. More precisely, I will construct direct connections between the space of metric admissible fat graphs due to Godin and the chain complex of black and white graphs due to Costello. Furthermore, I will construct a PROP structure on admissible fat graphs, which models the PROP of Moduli spaces of two dimensional cobordisms. I will use the connections above to give black and white graphs a PROP structure with the same property.
If there is an extension of the talk I would suggest the following.
Talk part II
Title: Other combinatorial models of the Moduli space of Riemann surfaces
Abtract: I will mention how B\"{o}digheimer's model of radial slit configurations fit into the picture of the first talk; and how this shows that the space of Sullivan diagrams, is homotopy equivalent to B\"{o}digheimer's Harmonic compactification of Moduli space. If time permits I will mention a reinterpretation of Sullivan diagrams and admissible graphs in terms of arc complexes and some new computational results.
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