On the topology of the moduli of tropical unramified p-covers

TL;DR

This paper investigates the topological structure of the moduli space of unramified cyclic covers of tropical curves, establishing contractibility and simple connectivity, and determining homotopy types for genus 2 cases across all primes.

Contribution

It applies recent techniques to identify contractible subcomplexes and fully characterizes the homotopy type of the moduli space for genus 2 tropical curves.

Findings

The moduli space is contractible in certain cases.

The moduli space is simply connected.

Homotopy types are determined for genus 2 for all primes.

Abstract

We study the topology of the moduli space of unramified $\mathbb{Z}/p$-covers of tropical curves of genus $g \geq 2$, where $p$ is a prime number. We use recent techniques by Chan--Galatius--Payne to identify contractible subcomplexes of the moduli space. We then use this contractibility result to show that this moduli space is simply connected. In the case of genus 2, we determine the homotopy type of this moduli space for all primes $p$. This work is motivated by prospective applications to the top-weight cohomology of the space of prime cyclic \'etale covers of smooth algebraic curves.