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

Abstract

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

arXiv comment20 pages, tables of Euler characteristics and program code are included in the ancillary files; v4: accepted version to be published in Algebraic & Geometric Topology

References in corpus (11)

• Representation theory and homological stability
• The tropicalization of the moduli space of curves
• On a theorem of Kontsevich
• Tropical curves, graph complexes, and top weight cohomology of M_g
• Assembling homology classes in automorphism groups of free groups
• The Euler characteristic of $\mathrm{Out}(F_n)$
• Feynman amplitudes on moduli spaces of graphs
• The $S_n$-equivariant top weight Euler characteristic of $M_{g,n}$
• Commutative hairy graphs and representations of $Out(F_r)$
• Integral Euler characteristic of Out F_{11}
• The Euler characteristic of the moduli space of graphs