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

TL;DR

This work determines an effective formula for the ${ ext{S}}_n$-equivariant Euler characteristic of the moduli space $ ext{MG}_{g,n}$ of graphs and demonstrates that the rational ${ ext{S}}_n$-invariant cohomology stabilizes for large $n$ (when $n\,oldsymbol{e}oldsymbol{g}$). It develops forested graph complexes to compute the cohomology, introduces orientability constraints, and derives generating-function methods (via plethystic exponentials) to extract the Euler characteristics, with computations aided by FORM. The stabilization result for $H^ullet( ext{MG}_{g,n};oldsymbol{Q})^{ ext{S}_n}$ (and twisted variants) follows from Lyndon–Hochschild–Serre spectral sequences applied to the extension $1 o F_g^n o oldsymbolΓ_{g,n} o ext{Out}(F_g) o 1$, together with an outer-space spine model and Schur–Weyl duality. The paper also reports large-$g$ asymptotics and conjectures relating $ ext{MG}_{g,n}$ characteristics to those of $oldsymbolΓ_{g,n}$, supported by extensive computational data and an explicit FORM implementation.

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 $k$.

Figures (2)

• Figure 1: Examples of forested graphs
• Figure 2: Extended forest with one special component

Theorems & Definitions (42)

• Definition 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Proposition 2.4
• Corollary 2.5
• proof
• Proposition 2.6
• proof
• Definition 2.7