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arXiv · 1403.5799

Crossed simplicial groups and structured surfaces

Tobias Dyckerhoff·Mikhail Kapranov
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Abstract

We propose a generalization of the concept of a Ribbon graph suitable to provide combinatorial models for marked surfaces equipped with a G-structure. Our main insight is that the necessary combinatorics is neatly captured in the concept of a crossed simplicial group as introduced, independently, by Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category leads to Ribbon graphs while other crossed simplicial groups naturally yield different notions of structured graphs which model unoriented, N-spin, framed, etc, surfaces. Our main result is that structured graphs provide orbicell decompositions of the respective G-structured moduli spaces. As an application, we show how, building on our theory of 2-Segal spaces, the resulting theory can be used to construct categorified state sum invariants of G-structured surfaces.

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Tobias Dyckerhoff, Mikhail Kapranov. 2014-03-23. Crossed simplicial groups and structured surfaces. https://doi.org/10.1090/conm%2F643%2F12896

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