# Topology of moduli spaces of tropical curves with marked points

## TL;DR

This paper explores the topology and homology of tropical moduli spaces of marked curves, linking them to complex moduli space cohomology and graph complexes, and provides new results on their homotopy types and cohomological properties.

## Contribution

It establishes a connection between tropical moduli space homology, complex moduli space cohomology, and graph complexes, offering new insights into their topological and algebraic structures.

## Key findings

- Identifies rational homology of tropical moduli spaces with top-weight cohomology of $M_{g,n}$
- Provides a contractibility criterion for large subspaces of tropical moduli spaces
- Calculates the homotopy type for genus 1 tropical moduli space and describes top weight cohomology as an $S_n$-representation

## 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.