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 Δ2,n${\Delta }_{2,n}$${\Delta }_{2,n}$, as a representation of S_n. These calculations are achieved fully for all n≤11$n\le 11$$n\le 11$, and partially—for specific irreducible representations of S_n—for n≤22$n\le 22$$n\le 22$. We also present conjectures, verified up to n = 22, for the multiplicities of the irreducible representations stdnstdn${\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.

摘要

我们获得了模空间 M _{2, n}的顶权有理上同调的新计算，相当于热带模空间的有理同调Δ2 , n${\Delta }_{2,n}$${\Delta }_{2,n}$，作为S _n的表示。这些计算对于所有的人来说都是完全实现的n≤11 _ _$n\le 11$$n\le 11$，并且部分地——对于S _n的特定不可约表示——对于n≤22 _ _$n\le 22$$n\le 22$。 我们还针对不可约表示的多重性提出了猜想，并在n = 22之前得到了验证标准值标准n${\text{std}}_n$和${\text{标准}}_n\otimes {\text{信号}}_n$。我们通过与图的紧致配置空间的同源性比较来实现我们的计算。这些同调群配备有对称群的交换作用和自由群的外自同构群。在本文中，我们为这些同源表示构建了一种有效的自由解析，从中一次提取一个不可约表示的计算，从而简化了这些同源表示的计算。