A Serre spectral sequence for the moduli space of tropical curves

TL;DR

This paper develops a spectral sequence framework to compute the rational cohomology of tropical moduli spaces of curves, linking combinatorial graph configurations to algebraic geometric invariants, and provides explicit calculations for certain genus and point counts.

Contribution

It introduces a new spectral sequence for the tropical moduli spaces, enabling explicit cohomology computations and connecting tropical and algebraic geometry.

Findings

Computed weight 0 cohomology for genus 3, up to 9 points.

Partial computations for up to 13 points.

Established a spectral sequence relating tropical and algebraic moduli spaces.

Abstract

We construct, for all $g\geq 2$ and $n\geq 0$, a spectral sequence of rational $S_n$-representations which computes the $S_n$-equivariant reduced rational cohomology of the tropical moduli spaces of curves $\Delta_{g,n}$ in terms of compactly supported cohomology groups of configuration spaces of $n$ points on graphs of genus $g$. Using the canonical $S_n$-equivariant isomorphisms $\widetilde{H}^{i-1}(\Delta_{g,n};\mathbb{Q}) \cong W_0 H^i_c(\mathcal{M}_{g,n};\mathbb{Q})$, we calculate the weight $0$, compactly supported rational cohomology of the moduli spaces $\mathcal{M}_{g,n}$ in the range $g=3$ and $n\leq 9$, with partial computations available for $n\leq 13$.