The homology of a compactified configuration space of a graph is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We construct an efficient free resolution for these homology representations. Using the Peter-Weyl Theorem for symmetric groups, we consider irreducible representations individually, vastly simplifying the calculation of these homology representations from the free resolution.As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli spaces M 2,n , equivalently the rational homology of the tropical moduli spaces ∆ 2,n , as a representation of S n acting by permuting point labels for all n ≤ 10. We further give new multiplicity calculations for specific irreducible representations of S n appearing in cohomology for n ≤ 17. Our approach produces information about these homology groups in a range well beyond what was feasible with previous techniques.

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.

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