Oriented hairy graphs and moduli spaces of curves

Abstract

We discuss a graph complex formed by directed acyclic graphs with external legs. This complex comes in particular with a map to the ribbon graph complex computing the (compactly supported) cohomology of the moduli space of points ${\mathcal{M}}_{g,n}$, extending an earlier result of Merkulov-Willwacher. It is furthermore quasi-isomorphic to the hairy graph complex computing the weight 0 part of the compactly supported cohomology of ${\mathcal{M}}_{g,n}$ according to Chan-Galatius-Payne. Hence we can naturally connect the works Chan-Galatius-Payne and of Merkulov-Willwacher and the ribbon graph complex and obtain a fairly satisfying picture of how all the pieces and various graph complexes fit together, at least in weight zero.

References in corpus (8)

• Deformation theory of representations of prop(erad)s
• Tropical curves, graph complexes, and top weight cohomology of M_g
• Topology of moduli spaces of tropical curves with marked points
• The rational homotopy of mapping spaces of E${}_n$ operads
• Relative (non-)formality of the little cubes operads and the algebraic Cerf Lemma
• Hairy graphs to ribbon graphs via a fixed source graph complex
• Euler characteristics for spaces of string links and the modular envelope of $L_{\infty }$
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Cited by in corpus (5)

• Tropical curves, graph complexes, and top weight cohomology of M_g
• Topology of moduli spaces of tropical curves with marked points
• Weight 11 compactly supported cohomology of moduli spaces of curves
• Getzler-Kapranov complexes and moduli stacks of curves
• Toward a minimal model for $H_{\ast }(\overline{\mathcal{M}})$