Representation stability of string links and manifold links

Abstract

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.