Homology Representations of Compactified Configurations on Graphs Applied to 𝓜 _2,n (Q6657558)

The authors obtain new computations of the top-weight rational cohomology of the moduli space \(\mathcal M_{2,n}\) of \(n\)-marked genus \(2\) curves. This is done via the canonical \(S_n\)-equavariant isomorphism between the reduced homology of the boundary complex \(\Delta_{g,n}\) of the Deligne-Mumford-Knudsen compactification \(\overline{\mathcal M}_{g,n}\) and the top-weight rational cohomology of \(\mathcal M_{g,n}\): \N\[\N\tilde{H}_{k-1}(\Delta_{g,n};\mathbb Q)\cong \operatorname{Gr}^W_{6g-6+2n}H^{6g-6+2n-k}(\mathcal M_{g,n};\mathbb Q),\N\]\Nby \textit{P. Deligne} [Publ. Math., Inst. Hautes Étud. Sci. 40, 5--57 (1971; Zbl 0219.14007); Publ. Math., Inst. Hautes Étud. Sci. 44, 5--77 (1974; Zbl 0237.14003)].\N\NThe reduced homology of \(\Delta_{2,n}\) is completely determined by the authors for \(n\leq 11\) and partially for \(n\leq 22.\) Their calculations are achieved by relating \(\Delta_{2,n},\) as a tropical moduli space, with the one-point compactification of the configuration space of \(n\) distinct points on the Theta graph.