Tropical moduli spaces as symmetric ΔΔ‐complexes

We develop techniques for studying fundamental groups and integral singular homology of symmetric Δ Δ -complexes, and apply these techniques to study moduli spaces of stable tropical curves of unit volume, with and without marked points. As one application, we show that Δ g Δ g and Δ g , n Δ g,n are simply connected, for g ⩾ 1 g 1 . We also show that Δ 3 Δ ₃ is homotopy equivalent to the 5-sphere, and that Δ 4 Δ ₄ has 3-torsion in H 5 H₅ .