# Oriented hairy graphs and moduli spaces of curves

Assar Andersson, Thomas Willwacher, Marko Zivkovic

Assar Andersson, Thomas Willwacher, Marko Zivkovic

**1/5/20**Published in : arXiv:2005.00439

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.