Computing homology representations of the tropical moduli spaces $\Delta_{g,n}$ via compactified configurations on graphs

This repository hosts code presented in two papers by Christin Bibby, Melody Chan, Nir Gadish, and Claudia He Yun. The computational data in both papers can be found on this page.

Homology representations of compactified configurations on graphs

arXiv: 2109.03302

Abstract

We obtain new calculations of the top weight rational cohomology of the moduli spaces $\mathcal{M}_{2,n}$, equivalently the rational homology of the tropical moduli spaces $\Delta_{2,n}$, as a representation of $S_n$. These calculations are achieved fully for all $n <= 11$, and partially - for specific irreducible representations of $S_n$ - for $n <= 22$. We also present conjectures, verified up to $n=22$, for the multiplicities of the irreducible representations std_n and its conjugate.

We achieve our calculations via a comparison with the homology of compactified configuration spaces of graphs. These homology groups are equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. In this paper, we construct an efficient free resolution for these homology representations. Using the Peter-Weyl Theorem for symmetric groups, we consider irreducible representations individually, vastly simplifying the calculation of these homology representations from the free resolution.

code

See folder genus2.

A Serre spectral sequence for the moduli space of tropical curves

arXiv: 2307.01960

Abstract

We construct, for all $g\ge 2$ and $n\ge 0$, 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 $\widetilde{H}^{k-1}(\Delta_{g,n};\mathbb{Q}) \cong W_0 H^i_c(\mathcal{M}_{g,n};\mathbb{Q})$, we calculate the weight $0$, compactly supported rational cohomology of the moduli spaces $\mathcal{M}_{g,n}$ in the range $g=3$ and $n\le 9.$

Code

See folder genus3.