  We develop techniques for studying fundamental groups and integral singular
homology of symmetric Delta-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 Delta_g and Delta_{g,n} are simply
connected, for positive g. We also show that Delta_3 is homotopy equivalent to
the 5-sphere, and that Delta_4 has 3-torsion in H_5.
