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

We study the topology of the moduli space of tropical unramified Z/p-covers of tropical curves of genus g ≥ 2 where p is a prime number. We use recent techniques by Chan--Galatius--Payne to identify a contractible subcomplex of the moduli space. We then use this contractibility result to show that this moduli space is simply connected for all g and p. In the case of genus $2$ we fully 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.