Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group

Abstract

We compute the weight 2 cohomology of the Feynman transforms of the cyclic (co)operads $\mathsf{BV}$ and $\mathsf{HyCom}$, and the top$-2$ weight cohomology of the Feynman transforms of $D\mathsf{BV}$ and $\mathsf{Grav}$. Using a result of Giansiracusa, we compute, in particular, the top$-2$ weight cohomology of the handlebody group. We compare the result to the top$-2$ weight cohomology of the moduli space of curves $\mathcal{M}_{g,n}$, recently computed by Payne and the last-named author. We also provide another proof of a recent result of Hainaut-Petersen identifying the top weight cohomology of the handlebody group with the Kontsevich graph cohomology.