The ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs

TL;DR

This paper derives a formula for the ${ m S}_n$-equivariant Euler characteristic of the moduli space of graphs and shows that its rational ${ m S}_n$-invariant cohomology stabilizes as the number of marked points increases.

Contribution

It provides a new explicit formula for the ${ m S}_n$-equivariant Euler characteristic and proves cohomology stabilization for the moduli space of graphs.

Findings

Derived a formula for the ${ m S}_n$-equivariant Euler characteristic.

Proved stabilization of the rational ${ m S}_n$-invariant cohomology for large n.

Established isomorphisms in cohomology groups for n ≥ g ≥ 2.

Abstract

We prove a formula for the ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs $\mathcal{MG}_{g,n}$. Moreover, we prove that the rational ${\mathbb S}_n$-invariant cohomology of $\mathcal{MG}_{g,n}$ stabilizes for large $n$. That means, if $n \geq g \geq 2$, then there are isomorphisms $H^k(\mathcal{MG}_{g,n};\mathbb{Q})^{{\mathbb S}_n} \rightarrow H^k(\mathcal{MG}_{g,n+1};\mathbb{Q})^{{\mathbb S}_{n+1}}$ for all $k$.