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

TL;DR

This paper computes specific weight 2 cohomology groups of graph complexes related to cyclic operads and the handlebody group, revealing connections to moduli space cohomology and Kontsevich graph cohomology.

Contribution

It provides new calculations of weight 2 cohomology for Feynman transforms of cyclic operads and relates these to the handlebody group and moduli space cohomology, offering alternative proofs.

Findings

Computed weight 2 cohomology of Feynman transforms of BV and HyCom operads.

Determined top-2 weight cohomology of the handlebody group.

Connected cohomology results to moduli space and Kontsevich graph cohomology.

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.