发表机构
Université Paris Cité, Sorbonne Université, CNRS, IMJ-PRG（巴黎西岱大学、索邦大学、法国国家科学研究中心、IMJ-PRG）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明弦链与流形链空间的理性上同调具有表示稳定性，给出生成元数量上界，并得到理性同伦群的显式生成界。
AI 中文摘要
我们证明了欧几里得空间中弦链与流形链空间的理性上同调随链数增长时的表示稳定性。对于弦链，我们考虑将$\nmathbb{R}^{m}$的不相交副本标准嵌入到$\nmathbb{R}^{d}$中的分量，其中$m \geq 1$且$d \geq m+3$。对于流形链，我们考虑由闭光滑$m$-流形的不相交副本固定嵌入到$\nmathbb{R}^{d-2}$所确定的分量，其中$d \geq 3$。在两种情形下，每个度数的理性上同调群构成有限生成的$\mathrm{FI}^{\\#}$-模，且在$n$度上的生成元数量至多为$n\frac{d-2}{d-m-2}$。我们还获得了理性同伦群的显式生成界。证明结合了结构映射的几何构造、嵌入模浸没的毛状图复形模型，以及一个函子框架，该框架允许在遗忘对称群作用后建立生成界。
英文摘要
We prove representation stability for rational cohomology of spaces of string links and manifold links in Euclidean space as the number of links grows. For string links we consider the component of the standard embedding of disjoint copies of $\mathbb{R}^{m}$ into $\mathbb{R}^{d}$, where $m \geq 1$ and $d \geq m+3$. For manifold links we consider the component determined by disjoint copies of a fixed embedding of a closed smooth $m$-manifold into $\mathbb{R}^{d-2}$, where $d \geq 3$. In both cases, the rational cohomology groups form finitely generated $\mathrm{FI}^{\#}$-modules in each degree with generation arity at most $n\frac{d-2}{d-m-2}$ in degree $n$. We also obtain explicit generation bounds for the rational homotopy groups. The proofs combine geometric constructions of structure maps, hairy graph complex models for embeddings modulo immersions and a functorial framework that allows generation bounds to be established after forgetting the symmetric group actions.
