Homology Representations of Compactified Configurations on Graphs Applied to 𝓜_2,n

Abstract

We obtain new calculations of the top weight rational cohomology of the moduli spaces M_2,n, equivalently the rational homology of the tropical moduli spaces ${\Delta }_{2,n}$, as a representation of S_n. These calculations are achieved fully for all $n\le 11$, and partially—for specific irreducible representations of S_n—for $n\le 22$. We also present conjectures, verified up to n = 22, for the multiplicities of the irreducible representations ${\text{std}}_n$ and ${\text{std}}_n\otimes {\text{sgn}}_n$. We achieve our calculations via a comparison with the homology of compactified configuration spaces of graphs. These homology groups are equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. In this paper, we construct an efficient free resolution for these homology representations, from which we extract calculations on irreducible representations one at a time, simplifying the calculation of these homology representations.

MATHEMATICS SUBJECT CLASSIFICATION (2010)::

• Primary: 05C10
• Secondary: 14H10
• 14Q05
• 14T20
• 55R80
• 55P65

KEYWORDS:

• Tropical curves
• moduli spaces of curves
• compactified configuration spaces on graphs
• outer automorphisms of free groups

Acknowledgments

We thank Eric Ramos for informing us of each others’ work. We also thank Dan Petersen and Louis Hainaut for suggesting to us the connection between our configuration spaces and Hochschild homology, along with many other useful ideas. We thank Victor Turchin, Ronno Das, Philip Tosteson, Orsola Tommasi, and Ben Ward for helpful conversations. Lastly, we thank ICERM and Brown University for generously providing us with the computing resources on which we ran our program.

Notes

1 We’ve learned though private communication that O. Tommasi, D. Petersen and P. Tosteson have independently found the same construction for this calculation. Petersen and Tommasi have also obtained results on the weight-0 compactly supported cohomology of ${\mathcal{M}}_{2,n}$, also using graph calculations. At this moment, we do not know how to directly relate their methods with the ones presented in this paper.

2 https://github.com/ClaudiaHeYun/BCGY

Additional information

Funding

C.B. was supported by NSF DMS-2204299; N.G. was supported by NSF Grant No. DMS-1902762; M.C. was supported by NSF DMS-1701924, CAREER DMS-1844768, and a Sloan Research Fellowship.