A Serre spectral sequence for the moduli space of tropical curves

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 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 ${\tilde{H}}^{i-1}({\Delta }_{g,n};\mathbb{Q})\cong W_0{H}_c^i({\mathcal{M}}_{g,n};\mathbb{Q})$, we calculate the weight $0$, compactly supported rational cohomology of the moduli spaces ${\mathcal{M}}_{g,n}$ in the range $g=3$ and $n\le 9$, with partial computations available for $n\le 13$.