Topology of the tropical moduli spaces $M_{2,n}$

Abstract

We study the topology of the link $M_{g,n}^{\mathrm{trop}}[1]$ of the tropical moduli spaces of curves when g=2. Tropical moduli spaces can be identified with boundary complexes for ${\mathcal{M}}_{g,n}$, as shown by Abramovich-Caporaso-Payne, so their reduced rational homology encodes top-weight rational cohomology of the complex moduli spaces ${\mathcal{M}}_{g,n}$. We prove that $M_{2,n}^{\mathrm{trop}}[1]$ is an $n$-connected topological space whose reduced integral homology is supported in the top two degrees only. We compute the reduced Euler characteristic of $M_{g,n}^{\mathrm{trop}}[1]$ for all n, and we compute the rational homology of $M_{2,n}^{\mathrm{trop}}[1]$ when $n\le 8$, determining completely the top-weight $\mathbb{Q}$-cohomology of ${\mathcal{M}}_{2,n}$ in that range.

29 pages, 4 figures