AI 中文总结
本研究通过FA-模的余棒构造确定圆周楔和紧支上同调对应的多项式函子系数，建立其与图同调的关联，给出亏格2 Payne-Willwacher带标记图复形的显式分解，进而推导模空间相关上同调的性质。
AI 中文摘要
Gadish与Hainaut的工作（继Petersen之后）将圆周楔和的紧支上同调建模为多项式函子。我们通过FA-模的余棒构造确定了该函子的系数Φ[n,m]。这一对应从形式上表明这些系数会出现在图同调的计算中，我们利用该结果给出了若干图复形的例子，其同调可嵌入H_c^*(F(S^1∨S^1,n))。\n这其中包括亏格2的Payne-Willwacher带标记图复形，我们针对它给出了一种基于单FA-模的全新显式分解。这使我们能够将gr₁₁H_c^*(ℳ_{2,n})描述为带装饰树复形的上同调，并证明例如gr₁₁H_c^{n+1}(ℳ_{2,n})=0。
英文摘要
Work of Gadish and Hainaut (after Petersen) models the compactly supported cohomology of a wedge of circles as a polynomial functor. We identify the coefficients of this functor, $Φ[n,m]$, via a cobar construction of $\mathbf{FA}$-modules. This identification formally implies that these coefficients will arise in computations of graph homology, and we use this result to give examples of graph complexes whose homology may be embedded in $H_c^\ast(F(S^1\vee S^1,n))$. This includes the Payne-Willwacher marked graph complex in genus 2, for which we give a new, explicit decomposition in terms of $\mathbf{FA}$-modules. This allows us to describe $\mathsf{gr}_{11}H_c^{\ast}(\mathcal{M}_{2,n})$ as the cohomology of a complex of decorated trees and to show, for example, $\mathsf{gr}_{11}H_c^{n+1}(\mathcal{M}_{2,n})=0$.
Commentsv2 with a few minor corrections (eg Example 2.6, typo in Corollary 4.19); comments always welcome
