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The $S_n$-equivariant top weight Euler characteristic of $M_{g,n}$

Melody Chan, Carel Faber, Soren Galatius, Sam Payne

TL;DR

The paper proves Zagier s conjecture by deriving a closed form for the S_n equivariant top weight Euler characteristic z_g of M_{g,n}. It converts top weight cohomology into a graph complex and reorganizes graph contributions via automorphisms into orbigraph data, then eliminates nonstatic contributions through exhalation and inhalation arguments, reducing to a finite sum over reduced static orbigraphs. The final z_g is a finite linear combination of Laurent monomials in the inhomogeneous power sums P_i with denominators P_m^k, with coefficients governed by Bernoulli numbers and Möbius functions, and the result recovers genus specific expressions including explicit genus 0 and 1 formulas. The approach connects tropical and orbifold perspectives through Kontsevich type Euler characteristics and yields genus by genus computability, providing both structural insight and practical computability for these equivariant top weight invariants.

Abstract

We prove a formula, conjectured by Zagier, for the $S_n$-equivariant Euler characteristic of the top weight cohomology of $M_{g,n}$.

The $S_n$-equivariant top weight Euler characteristic of $M_{g,n}$

TL;DR

The paper proves Zagier s conjecture by deriving a closed form for the S_n equivariant top weight Euler characteristic z_g of M_{g,n}. It converts top weight cohomology into a graph complex and reorganizes graph contributions via automorphisms into orbigraph data, then eliminates nonstatic contributions through exhalation and inhalation arguments, reducing to a finite sum over reduced static orbigraphs. The final z_g is a finite linear combination of Laurent monomials in the inhomogeneous power sums P_i with denominators P_m^k, with coefficients governed by Bernoulli numbers and Möbius functions, and the result recovers genus specific expressions including explicit genus 0 and 1 formulas. The approach connects tropical and orbifold perspectives through Kontsevich type Euler characteristics and yields genus by genus computability, providing both structural insight and practical computability for these equivariant top weight invariants.

Abstract

We prove a formula, conjectured by Zagier, for the -equivariant Euler characteristic of the top weight cohomology of .

Paper Structure

This paper contains 13 sections, 30 theorems, 109 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivariant Euler characteristics
  4. Graphs and graph complexes
  5. Orbi-counting and orbi-summation
  6. The contribution of a graph as a sum over automorphisms
  7. From graphs with automorphisms to orbigraphs
  8. From orbigraphs to static orbigraphs
  9. Static orbigraphs
  10. Contribution from reduced static orbigraphs
  11. Conclusion
  12. Genus 0 and 1
  13. Appendix: Orbifold Euler characteristics

Key Result

Theorem 1.1

The $S_n$-equivariant top weight Euler characteristic of $\mathcal{M}_{g,n}$ is where the sum is over integers $k,m>0$ and $r,s\ge 0$, and $s$-tuples of positive integers $a = (a_1,\ldots,a_s)$ and $d = (d_1,\ldots,d_s)$, such that and the product runs over primes $p$ dividing $m$ and all $d_1, \dots, d_s$.

Figures (5)

  • Figure 1: The orbigraph $(X,f) = \mathcal{O}(G,\tau)$, where $\tau$ flips both loops of $G$.
  • Figure 2: Figure accompanying Lemma \ref{['lem:half-open']}, $X$ on the left and $X'$ on the right.
  • Figure 3: Exhaling, from left to right. Inhaling, from right to left.
  • Figure 4: An instance of $(\mathrm{Ex} \downarrow (X,f))$
  • Figure 5: A static orbigraph with maximal tails of lengths $0,0,1,2$.

Theorems & Definitions (79)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 69 more