Tropical moduli spaces as symmetric Delta-complexes

TL;DR

The paper develops a framework of symmetric $\Delta$-complexes and symmetric CW-complexes to study the integral homology and fundamental groups of moduli spaces of stable tropical curves, focusing on the links $\Delta_g$ and $\Delta_{g,n}$. By constructing large contractible subcomplexes and employing a skeleta-based spectral sequence, the authors establish simple connectivity for most $(g,n)$, identify $\Delta_3$ with $S^5$, and exhibit torsion phenomena in $\Delta_4$ (notably $3$-torsion in $H_5$ and $2$-torsion in $H_6,H_7$). They further relate the rational homology to graph homology and the topology of toroidal dual complexes, and provide explicit computational methods (including computer-assisted calculations) for the sphere-quotient contributions, as well as a relative framework for studying homology relative to contractible subcomplexes. These results illuminate the intricate topology of tropical moduli spaces and their connections to algebraic geometry and geometric group theory.

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.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Definition 2.1
• Definition 2.2
• Example 2.3
• Example 2.4