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 ℝm into ℝd, where m ≥ 1 and d ≥ m+3. For manifold links we consider the component determined by disjoint copies of a fixed embedding of a closed smooth m-manifold into ℝd−2, where d ≥ 3. In both cases, the rational cohomology groups form finitely generated FI#-modules in each degree with generation arity at most n(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.
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BibTeX
@misc{buryak2026representation,
title = {{Representation stability of string links and manifold links}},
author = {Filipp Buryak},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20470},
eprint = {2609.20470},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20470},
}
RIS
TY - UNPB
TI - Representation stability of string links and manifold links
AU - Filipp Buryak
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20470
UR - https://arxiv.org/abs/2609.20470
AB - 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 ℝ^m into ℝ^d, where m ≥ 1 and d ≥ m+3. For manifold links we consider the component determined by disjoint copies of a fixed embedding of a closed smooth m-manifold into ℝ^d−2, where d ≥ 3. In both cases, the rational cohomology groups form finitely generated FI^#-modules in each degree with generation arity at most n(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.
ER -