Hairy graphs to ribbon graphs via a fixed source graph complex

Abstract

We show that the hairy graph complex $(HG{C}_{n,n},d)$ appears as an associated graded complex of the oriented graph complex $(OG{C}_{n+1},d)$, subject to the filtration on the number of targets, or equivalently sources, called the fixed source graph complex. The fixed source graph complex $(OG{C}_1,d_0)$ maps into the ribbon graph complex $RGC$, which models the moduli space of Riemann surfaces with marked points. The full differential $d$ on the oriented graph complex $OG{C}_{n+1}$ corresponds to the deformed differential $d+h$ on the hairy graph complex $HG{C}_{n,n}$, where $h$ adds a hair. This deformed complex $(HG{C}_{n,n},d+h)$ is already known to be quasi-isomorphic to standard Kontsevich's graph complex $G{C}_n^2$. This gives a new connection between the standard and the oriented version of Kontsevich's graph complex.