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 (2)

• The rational homotopy of mapping spaces of E${}_n$ operads
• Relative (non-)formality of the little cubes operads and the algebraic Cerf Lemma

Cited by in corpus (1)

• Toward a minimal model for $H_{\ast }(\overline{\mathcal{M}})$