Given integers g ≥ 0, n ≥ 1, and a vector w ∈ (Q ∩ (0, 1]) n such that 2g − 2 + w i > 0, we study the topology of the moduli space ∆ g,w of w-stable tropical curves of genus g with volume 1. The space ∆ g,w is the dual complex of the divisor of singular curves in Hassett's moduli space of w-stable genus g curves M g,w . When g ≥ 1, we show that ∆ g,w is simply connected for all values of w. We also give a formula for the Euler characteristic of ∆ g,w in terms of the combinatorics of w.

We study how changing the weight datum A = (a 1 , ..., a n ) ∈ (Q ∩ (0, 1]) n affects the topology of tropical moduli spaces M trop g,A , M trop g,A and ∆ g,A and the homology of the latter one. We show that for fixed g and n, there are particular filtrations of these topological spaces which we can use to compute the reduced rational homology of ∆ g,A and the top weight cohomology of the moduli space M g,A of smooth (g, A)-stable algebraic curves.

We study the topology of moduli spaces of weighted stable tropical curves Δg ,w with fixed genus and unit volume. The space Δg ,w arises as the dual complex of the divisor of singular curves in Hassett’s moduli space M g ,w of weighted stable curves. When the genus is positive, we show that Δg ,w is simply connected for any choice of weight vector w. We also give a formula for the Euler characteristic of Δg ,w in terms of the combinatorics of the weight vector.

We consider tropical versions of Hassett’s moduli spaces of weighted stable curves M_{g,\mathcal{A}}^{\operatorname{trop}} , {\overline{M}_{g,\mathcal{A}}^{\operatorname{trop}}} and \Delta_{g,\mathcal{A}} associated to a weight datum \mathcal{A}=(a_1,\ldots,a_n)\in(\mathbb{Q}\cap(0,1])^n , their associated graph complexes {G^{(g,\mathcal{A})}} , and study the topology of these spaces as \mathcal{A} changes. We show that for fixed g and n , there are particular filtrations of these topological spaces and their graph complexes which may be used to compute the reduced rational homology of \Delta_{g,\mathcal{A}} and the top weight cohomology of the moduli space \mathcal{M}_{g,\mathcal{A}} of smooth (g,\mathcal{A}) -stable algebraic curves.

Let M g,m|n denote Hassett's moduli space of weighted pointed stable curves of genus g for the heavy/light weight data 1 (m) , 1/n (n) , and let M g,m|n ⊂ M g,m|n be the locus parameterizing smooth, not necessarily distinctly marked curves. We give a change-ofvariables formula which computes the generating function for (S m × S n )-equivariant Hodge-Deligne polynomials of these spaces in terms of the generating functions for S n -equivariant Hodge-Deligne polynomials of M g,n and M g,n .