Moduli spaces of metric graphs of genus 1 with marks on vertices

TL;DR

This paper demonstrates that certain moduli spaces of genus 1 metric graphs with vertex marks are homotopy equivalent to tropical curve moduli spaces, using explicit homotopies and the scanning homotopy technique.

Contribution

The paper establishes a homotopy equivalence between moduli spaces of genus 1 metric graphs with vertex marks and tropical curve moduli spaces, introducing explicit homotopies.

Findings

Homotopy equivalence between $MG_{1,n}^v$ and $TM_{1,n}$

Use of explicit homotopies and scanning homotopy

Conjecture on generalization to higher genus

Abstract

In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces $MG_{1,n}^v$, which parametrize the isometry classes of metric graphs of genus 1 with $n$ marks on vertices are homotopy equivalent to the spaces $TM_{1,n}$, which are the moduli spaces of tropical curves of genus 1 with $n$ marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.