Graphing, homotopy groups of spheres, and spaces of links and knots

Abstract

We study homotopy groups of spaces of links, focusing on long links of codimension at least three. In the case of multiple components, they admit split injections from homotopy groups of spheres. We calculate them, up to knotting, in a range depending on the dimensions of the source manifolds and target manifold which roughly generalizes the triple-point-free range for isotopy classes. At the edge of this range, joining components sends both a parametrized long Borromean rings class and a Hopf fibration to a generator of the first nontrivial homotopy group of the space of long knots. For spaces of two-component long links, we give generators of the homotopy group in this dimension in terms of this class from the Borromean rings and homotopy groups of spheres. A key ingredient in most of our results is a graphing map that increases source and target dimensions by one. CONTENTS 1. Introduction 1.1. Previous related results 1.2. Main results and organization 1.3. Acknowledgments 2. Key definitions 2.1. Spaces of embeddings, link maps, and pseudoisotopy embeddings 2.2. Maps between spaces of links 3. Injectivity of graphing 3.1. Injectivity for spaces of embeddings via the λ-invariant 3.2. Injectivity for spaces of link maps in a range via the α-invariant 4. Bijectivity of graphing in a range 4.1. Lemmas on pseudoisotopy embedding spaces 4.2. Bijectivity of graphing and homotopy groups of spaces of long links in a range 4.3. Spherical links and a further calculation for long links 5. Homotopy classes in spaces of long knots and links from joining pure braids 5.1. Configuration space integrals for cohomology of spaces of long 1-dimensional links 5.2. Some low-dimensional graph cocycles 5.3. Homology classes dual to configuration space integrals 5.4. Long Borromean rings, the Hopf map, and classes of long links and knots 5.5. Questions and future directions References

Embedding spaces of split links

“…In recent work, Koytcheff [Koy22] studies the homotopy type of various spaces of links. This work focuses on long links defined in general dimension and codimension, and the results focus on long links of codimension at least 3.…”