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.

50 pages, v3: typos corrected and many clarifications added

References in corpus (3)

• The Euler characteristic of the moduli space of graphs
• Weight two compactly supported cohomology of moduli spaces of curves
• Weight 11 compactly supported cohomology of moduli spaces of curves

Cited by in corpus (1)

• Problems on handlebody groups