The 𝕊n-equivariant Euler characteristic of the moduli space of graphs

Borinsky, Michael; Vermaseren, Jos

doi:10.2140/agt.2025.25.4229

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ISSN (electronic): 1472-2739

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

Michael Borinsky and Jos Vermaseren

Algebraic & Geometric Topology 25 (2025) 4229–4255

DOI: 10.2140/agt.2025.25.4229

Abstract

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

Keywords

moduli space, graphs, Euler characteristic, cohomological stability, graph complex, $\mathrm{Out}(F_{n})$

Mathematical Subject Classification

Primary: 14D22, 18G85, 58D29

Secondary: 05E18, 14T99, 20F28, 20F65, 20J06

Supplementary material

Larger_versions_of_Tables_2--7

FORM_program_for_Theorem_2.18

References

Publication

Received: 12 February 2024

Revised: 15 October 2024

Accepted: 2 November 2024

Published: 29 October 2025

Authors

Michael Borinsky
Perimeter Institute for Theoretical Physics Waterloo Canada

Jos Vermaseren
Nikhef Amsterdam Netherlands

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Volume 25, issue 7 (2025)