The tropicalization of the moduli space of curves II: Topology and applications

TL;DR

This paper investigates the topology of a specific tropical moduli space of stable curves, revealing its high connectivity and linking it to algebraic geometry and mapping class groups, with implications for cohomology and homology structures.

Contribution

It establishes the high connectivity of the tropical moduli space and connects it to the dual complex of the algebraic moduli space boundary, providing explicit homology bases.

Findings

The tropical moduli space is highly connected.

An explicit dual basis in homology is constructed.

Top weight cohomology of M_{1,n} is expressed as a symmetric group representation.

Abstract

We study the topology of the tropical moduli space parametrizing stable tropical curves of genus g with n marked points in which the bounded edges have total length 1, and prove that it is highly connected. Using the identification of this space with the dual complex of the boundary in the moduli space of stable algebraic curves, we give a simple expression for the top weight cohomology of M_{1,n} as a representation of the symmetric group and describe an explicit dual basis in homology consisting of abelian cycles for the pure mapping class group.