Tropical moduli spaces as symmetric Delta-complexes

TL;DR

This paper develops new techniques for analyzing the topology of symmetric Delta-complexes and applies them to moduli spaces of tropical curves, revealing their fundamental groups and homology properties.

Contribution

It introduces methods for studying symmetric Delta-complexes and applies them to prove topological properties of tropical moduli spaces, including simple connectivity and homotopy types.

Findings

Delta_g and Delta_{g,n} are simply connected for g > 0

Delta_3 is homotopy equivalent to the 5-sphere

Delta_4 has 3-torsion in H_5

Abstract

We develop techniques for studying fundamental groups and integral singular homology of symmetric Delta-complexes, and apply these techniques to study moduli spaces of stable tropical curves of unit volume, with and without marked points. As one application, we show that Delta_g and Delta_{g,n} are simply connected, for positive g. We also show that Delta_3 is homotopy equivalent to the 5-sphere, and that Delta_4 has 3-torsion in H_5.