On the Euler characteristic of the commutative graph complex and the top weight cohomology of $\mathcal M_g$

TL;DR

This paper establishes an asymptotic formula for the Euler characteristic of Kontsevich's commutative graph complex, revealing super-exponential growth in related homology and cohomology dimensions of moduli spaces of curves.

Contribution

It provides the first asymptotic formula for the Euler characteristic of the commutative graph complex, linking it to the growth of top weight cohomology of moduli spaces.

Findings

Euler characteristic grows super-exponentially with rank

Commutative graph homology growth is super-exponential

Top weight cohomology dimension of $\\mathcal{M}_g$ grows super-exponentially with genus

Abstract

We prove an asymptotic formula for the Euler characteristic of Kontsevich's commutative graph complex. This formula implies that the total amount of commutative graph homology grows super-exponentially with the rank and, via a theorem of Chan, Galatius, and Payne, that the dimension of the top weight cohomology of the moduli space of curves, $\mathcal M_g$, grows super-exponentially with the genus $g$.