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...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001142/type/journal_article |
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| author | Robin Koytcheff |
| author_facet | Robin Koytcheff |
| author_sort | Robin Koytcheff |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-c08c39157e0043c0a17d67941101c762 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-c08c39157e0043c0a17d67941101c7622025-02-12T03:49:18ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.114Graphing, homotopy groups of spheres, and spaces of long links and knotsRobin Koytcheff0https://orcid.org/0000-0002-1056-5318Department of Mathematics, University of Louisiana at Lafayette, Lafayette, 70504 LA, USAWe 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.https://www.cambridge.org/core/product/identifier/S2050509424001142/type/journal_article57K4555Q4057R4055R8058D1081Q3055P35 |
| spellingShingle | Robin Koytcheff Graphing, homotopy groups of spheres, and spaces of long links and knots Forum of Mathematics, Sigma 57K45 55Q40 57R40 55R80 58D10 81Q30 55P35 |
| title | Graphing, homotopy groups of spheres, and spaces of long links and knots |
| title_full | Graphing, homotopy groups of spheres, and spaces of long links and knots |
| title_fullStr | Graphing, homotopy groups of spheres, and spaces of long links and knots |
| title_full_unstemmed | Graphing, homotopy groups of spheres, and spaces of long links and knots |
| title_short | Graphing, homotopy groups of spheres, and spaces of long links and knots |
| title_sort | graphing homotopy groups of spheres and spaces of long links and knots |
| topic | 57K45 55Q40 57R40 55R80 58D10 81Q30 55P35 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001142/type/journal_article |
| work_keys_str_mv | AT robinkoytcheff graphinghomotopygroupsofspheresandspacesoflonglinksandknots |