Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Robin Koytcheff

TL;DR

The paper addresses the problem of understanding the homotopy groups of spaces of long links in codimension at least 3 by connecting them to spheres via the graphing map and Goodwillie–Weiss calculus. It develops injectivity and bijectivity results for graphing in metastable ranges, and provides explicit geometric generators for the nontrivial homotopy groups in those ranges, arising from long Borromean rings, the Hopf map, and homotopy groups of spheres. A central technique is iterated graphing that increases both source and target dimensions, paired with closure/joining constructions, to relate multi-component link spaces to knot spaces and spheres. The work yields a detailed program for realizing and classifying homotopy classes in spaces of long links and knots, with explicit cocycle and homology duals provided by configuration-space integrals. These results illuminate the structure of embedding spaces in high codimension and connect to broader questions about graph complexes and the rational and torsion components of homotopy groups. The findings have implications for understanding how complex link configurations generate higher-dimensional homotopy classes and contribute to the interplay between knot theory, link maps, and manifold embedding theory.

Abstract

We study homotopy groups of spaces of long links in Euclidean space of codimension at least three. With multiple components, they admit split injections from homotopy groups of spheres. We show that, up to knotting, these account for all the homotopy groups in a range which depends on the dimensions of the source manifolds and target manifold and which roughly generalizes the triple-point-free range for isotopy classes. Just beyond 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 equidimensional long links of most source dimensions, we describe generators for the homotopy group in this degree in terms of these Borromean rings and homotopy groups of spheres. A key ingredient in most of our results is a graphing map which increases source and target dimensions by one.

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

TL;DR

The paper addresses the problem of understanding the homotopy groups of spaces of long links in codimension at least 3 by connecting them to spheres via the graphing map and Goodwillie–Weiss calculus. It develops injectivity and bijectivity results for graphing in metastable ranges, and provides explicit geometric generators for the nontrivial homotopy groups in those ranges, arising from long Borromean rings, the Hopf map, and homotopy groups of spheres. A central technique is iterated graphing that increases both source and target dimensions, paired with closure/joining constructions, to relate multi-component link spaces to knot spaces and spheres. The work yields a detailed program for realizing and classifying homotopy classes in spaces of long links and knots, with explicit cocycle and homology duals provided by configuration-space integrals. These results illuminate the structure of embedding spaces in high codimension and connect to broader questions about graph complexes and the rational and torsion components of homotopy groups. The findings have implications for understanding how complex link configurations generate higher-dimensional homotopy classes and contribute to the interplay between knot theory, link maps, and manifold embedding theory.

Abstract

We study homotopy groups of spaces of long links in Euclidean space of codimension at least three. With multiple components, they admit split injections from homotopy groups of spheres. We show that, up to knotting, these account for all the homotopy groups in a range which depends on the dimensions of the source manifolds and target manifold and which roughly generalizes the triple-point-free range for isotopy classes. Just beyond 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 equidimensional long links of most source dimensions, we describe generators for the homotopy group in this degree in terms of these Borromean rings and homotopy groups of spheres. A key ingredient in most of our results is a graphing map which increases source and target dimensions by one.
Paper Structure (22 sections, 29 theorems, 83 equations, 7 figures)

This paper contains 22 sections, 29 theorems, 83 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Previous related results
  3. Main results and organization
  4. Acknowledgments
  5. Key definitions
  6. Spaces of embeddings, link maps, and pseudoisotopy embeddings
  7. Stacking, graphing, closure, and joining maps
  8. Restriction and inclusion maps
  9. Injectivity of graphing for spaces of 2-component long links
  10. Bijectivity of graphing in a range
  11. Lemmas on pseudoisotopy embedding spaces
  12. Bijectivity of graphing and homotopy groups of spaces of long links in a range
  13. Spherical links and further calculations for long links
  14. Homotopy classes in spaces of long knots and links from joining pure braids
  15. Configuration space integrals for cohomology of spaces of long 1-dimensional links
  16. ...and 7 more sections

Key Result

Theorem A

If $\ 0 \leq p \leq q \leq n-1$ and $i\geq 0$, then $\pi_i \mathrm{Emb}_c(\mathbb{R}^p \sqcup \mathbb{R}^q, \, \mathbb{R}^n)$ contains a direct summand of $\pi_{i+p}S^{n-q-1}$. An inclusion of this summand is given by composing maps induced by a based homotopy equivalence $S^{n-q-1} \rightarrow \mat

Figures (7)

  • Figure 1: Picture of the long Borromean rings (i.e., a pure braid commutator) $[b_{21},b_{31}]$ and the results of joining its components in the classical setting where $p=1$ and $n=3$.
  • Figure 2: (a) The singular 3-strand braid $f=(f_1,f_2,f_3)$ used to build the family $F$. (b) The singular 3-strand braid $f'=(f_1',f_2',f_3')$ used to build the family $F'$.
  • Figure 3: The singular 2-component long link $\ell=(\ell_1, \ell_2)$obtained by joining strand 2 to strand 3 in the singular braid $f=(f_1,f_2,f_3)$. An isotopy takes this singular long link to a singular 2-strand pure braid.
  • Figure 4: The singular 2-component long link $\ell'=(\ell_1', \ell_2')$ obtained by joining strand 2 to strand 3 in the singular braid $f'=(f'_1,f'_2,f'_3)$.
  • Figure 5: The singular 2-component braid $h=(h_1,h_2)$ used to construct the family $H$.
  • ...and 2 more figures

Theorems & Definitions (77)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 67 more