Topology of moduli spaces of tropical curves with marked points

Abstract

We study a space of genus $g$ stable, $n$-marked tropical curves with total edge length $1$. Its rational homology is identified both with top-weight cohomology of the complex moduli space $M_{g,n}$ and with the homology of a marked version of Kontsevich's graph complex, up to a shift in degrees. We prove a contractibility criterion that applies to various large subspaces. From this we derive a description of the homotopy type of the tropical moduli space for $g = 1$, the top weight cohomology of $M_{1,n}$ as an $S_n$-representation, and additional calculations for small $(g,n)$. We also deduce a vanishing theorem for homology of marked graph complexes from vanishing of cohomology of $M_{g,n}$ in appropriate degrees, and comment on stability phenomena, or lack thereof.