A Serre spectral sequence for the moduli space of tropical curves

Author:

Christin Bibby, Melody Chan, Nir Gadish, Claudia He Yun

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Algebraic Topology (math.AT), Combinatorics (math.CO)

journal:

date:

2023-07-03 16:00:00

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

PDF: A Serre spectral sequence for the moduli space of tropical curves.pdf