Classification of knotted tori
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2015
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| _version_ | 1866914877754310656 |
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| author | Skopenkov, A. |
| author_facet | Skopenkov, A. |
| contents | For a smooth manifold $N$ denote by $E^m(N)$ the set of smooth isotopy classes of smooth embeddings $N\to\mathbb R^m$. A description of the set $E^m(S^p\times S^q)$ was known only for $p=q=0$ or for $p=0$, $m\ne q+2$ or for $2m\ge 2(p+q)+\max\{p,q\}+4$. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For $m\ge2p+q+3$ we introduce an abelian group structure on $E^m(S^p\times S^q)$ and describe this group `up to an extension problem'. This result has corollaries which, under stronger dimension restrictions, more explicitly describe $E^m(S^p\times S^q)$. The proof is based on relations between sets $E^m(N)$ for different $N$ and $m$, in particular, on a recent exact sequence of M. Skopenkov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1502_04470 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Classification of knotted tori Skopenkov, A. Geometric Topology Algebraic Topology 57R40, 57R52, 55Q40 For a smooth manifold $N$ denote by $E^m(N)$ the set of smooth isotopy classes of smooth embeddings $N\to\mathbb R^m$. A description of the set $E^m(S^p\times S^q)$ was known only for $p=q=0$ or for $p=0$, $m\ne q+2$ or for $2m\ge 2(p+q)+\max\{p,q\}+4$. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For $m\ge2p+q+3$ we introduce an abelian group structure on $E^m(S^p\times S^q)$ and describe this group `up to an extension problem'. This result has corollaries which, under stronger dimension restrictions, more explicitly describe $E^m(S^p\times S^q)$. The proof is based on relations between sets $E^m(N)$ for different $N$ and $m$, in particular, on a recent exact sequence of M. Skopenkov. |
| title | Classification of knotted tori |
| topic | Geometric Topology Algebraic Topology 57R40, 57R52, 55Q40 |
| url | https://arxiv.org/abs/1502.04470 |