Topology of moduli spaces of tropical curves with marked points

TL;DR

This paper investigates the topology of moduli spaces of tropical curves with marked points, simplifying their structure via deformation retraction and computing Betti numbers for genus 1 cases.

Contribution

It introduces a deformation retraction technique for moduli spaces and analyzes the genus 1 case as a quotient of a torus, providing explicit topological invariants.

Findings

Simplified the topology of moduli spaces using deformation retraction.

Represented genus 1 moduli space as a quotient of a torus with a Z2-action.

Computed Betti numbers of the genus 1 moduli space with Z2 coefficients.

Abstract

In this paper we study topology of moduli spaces of tropical curves of genus $g$ with $n$ marked points. We view the moduli spaces as being imbedded in a larger space, which we call the {\it moduli space of metric graphs with $n$ marked points.} We describe the shrinking bridges strong deformation retraction, which leads to a substantial simplification of all these moduli spaces. In the rest of the paper, that reduction is used to analyze the case of genus 1. The corresponding moduli space is presented as a quotient space of a torus with respect to the conjugation ${\mathbb Z}_2$-action; and furthermore, as a homotopy colimit over a simple diagram. The latter allows us to compute all Betti numbers of that moduli space with coefficients in ${\mathbb Z}_2$.