Graphing, homotopy groups of spheres, and spaces of long links and knots
arXiv:2205.00635 · doi:10.1017/fms.2024.114
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.
42 pages, 7 figures. Accepted for publication in Forum Math. Sigma. Main changes from v2: moved old Theorem B to Appendix; improved clarity in statements and proofs of Theorems A and B; added Proposition 4.6; added content about restriction maps; added Figure 1 with joining long Borromean rings' components; and made statement about Haefliger trefoil in even codimension into Corollary 5.8