I will discuss the topology of a space of stable tropical curves of genus $g$ with volume 1. The reduced rational homology of this space is canonically identified with the top weight cohomology of $M_g$ and also with the homology of Kontsevich's graph complex. As one application, we show that $H^{4g-6}(M_g)$ is nonzero for infinitely many $g$. This disproves a recent conjecture of Church, Farb, and Putman as well as an older, more general conjecture of Kontsevich. We also give an independent proof of a recent theorem of Willwacher, that homology of the graph complex vanishes in negative degrees, using the identifications above and known vanishing results for $M_g$. Joint work with M. Chan and S. Galatius.