Moduli spaces of tropical curves of higher genus with marked points and homotopy colimits

TL;DR

This paper investigates the homotopy types of moduli spaces of higher genus tropical curves with marked points, using homotopy colimits and CW decompositions to analyze specific cases and propose new conjectures.

Contribution

It introduces a novel approach to study the homotopy types of these moduli spaces via homotopy colimits over combinatorial complexes, providing detailed results for genus 2 and 3 cases.

Findings

Complete understanding of $X_{2,0}$, $X_{2,1}$, $X_{2,2}$, $X_{3,0}$

Homotopy colimit representations of the spaces

CW complex decompositions for the moduli spaces

Abstract

The main characters of this paper are the moduli spaces $TM_{g,n}$ of rational tropical curves of genus $g$ with $n$ marked points, with $g\geq 2$. We reduce the study of the homotopy type of these spaces to the analysis of compact spaces $X_{g,n}$, which in turn possess natural representations as a homotopy colimits of diagrams of topological spaces over combinatorially defined generalized simplicial complexes $\Delta_g$, with the latter being interesting on their own right. We use these homotopy colimit representations to describe a CW complex decomposition for each $X_{g,n}$. Furthermore, we use these developments, coupled with some standard tools for working with homotopy colimits, to perform an in-depth analysis of special cases of genus 2 and 3, gaining a complete understanding of the moduli spaces $X_{2,0}$, $X_{2,1}$, $X_{2,2}$, and $X_{3,0}$, as well as a partial understanding of other cases, resulting in several open questions and in further conjectures.