Topology of the tropical moduli spaces \(\Delta_{2,n}\)
Isomorphism classes of tropical curves are naturally parametrized by a tropical moduli space \(M_{g,n}^{\mathrm{trop}}\), in analogy with classical complex curves. As a generalized cone complex, this space is contractible, however its link \(\Delta_{g,n}\) is topologically nontrivial, and it actually coincides with the boundary complex of the Deligne-Mumford compactification of \(M_{g,n}\). In this paper, the author proves that \(\Delta_{2,n}\) is \(n\)-connected and computes the two top \(\mathbb{Q}\)-cohomology groups for \(n\) up to \(8\). These result fit into a series of recent papers by the author and others on the topology of \(M_{g,n}\), such as [\textit{M. Chan} et al., J. Am. Math. Soc. 34, No. 2, 565--594 (2021; Zbl 1485.14045)].
scientific article; zbMATH DE number 7495602
scientific article