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We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence.
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This content will become publicly available on January 1, 2026
Graphing, homotopy groups of spheres, and spaces of long links and knots
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.
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- Award ID(s):
- 2405370
- PAR ID:
- 10610258
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 13
- ISSN:
- 2050-5094
- Subject(s) / Keyword(s):
- Spaces of embeddings, long knots, long links, pure braids, graphing, homotopy groups of spheres, configuration spaces, the lambda-invariant, the alpha-invariant, pseudoisotopy embedding spaces, configuration space integrals, immersions
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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