Homology representations of compactified configurations on graphs applied to ${\mathcal{M}}_{2,n}$

Abstract

We obtain new calculations of the top weight rational cohomology of the moduli spaces ${\mathcal{M}}_{2,n}$, 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 $n\le 10$, 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 ${\mathrm{std}}_n$ and ${\mathrm{std}}_n\otimes {\mathrm{sgn}}_n$. 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.

18 pages, minor edits

Cited by in corpus (1)

• The $S_n$-equivariant top weight Euler characteristic of $M_{g,n}$