Homotopy versus isotopy: 2-spheres in 5-manifolds
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909422678179840 |
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| author | Kosanović, Danica Schneiderman, Rob Teichner, Peter |
| author_facet | Kosanović, Danica Schneiderman, Rob Teichner, Peter |
| contents | In this note we give a complete obstruction for two homotopic embeddings of a 2-sphere into a 5-manifold to be isotopic. The results are new even though the methods are classical, the main tool being the elimination of double points via a level preserving Whitney move in codimension~$3$. Moreover, we discuss how this recovers a particular case of a result of Dax on metastable homotopy groups of embedding spaces. It follows that ``homotopy implies isotopy'' for 2-spheres in simply-connected 5-manifolds and for 2-spheres admitting algebraic dual 3-spheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16905 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Homotopy versus isotopy: 2-spheres in 5-manifolds Kosanović, Danica Schneiderman, Rob Teichner, Peter Geometric Topology Algebraic Topology In this note we give a complete obstruction for two homotopic embeddings of a 2-sphere into a 5-manifold to be isotopic. The results are new even though the methods are classical, the main tool being the elimination of double points via a level preserving Whitney move in codimension~$3$. Moreover, we discuss how this recovers a particular case of a result of Dax on metastable homotopy groups of embedding spaces. It follows that ``homotopy implies isotopy'' for 2-spheres in simply-connected 5-manifolds and for 2-spheres admitting algebraic dual 3-spheres. |
| title | Homotopy versus isotopy: 2-spheres in 5-manifolds |
| topic | Geometric Topology Algebraic Topology |
| url | https://arxiv.org/abs/2312.16905 |