Knots, i.e. embeddings of spheres
Askopenkov (Talk | contribs) (→Classification) |
Askopenkov (Talk | contribs) m (→Classification) |
||
Line 19: | Line 19: | ||
For $m\ge n+3$ the group $E^m(S^n)$ has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs $(SG_n,SO_n)$ for $n = n_i+1$ and $n_i$ \cite{Haefliger1966a}, cf. \cite{Levine1965}, \cite{Miller1972}, \cite{Habegger1986}. | For $m\ge n+3$ the group $E^m(S^n)$ has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs $(SG_n,SO_n)$ for $n = n_i+1$ and $n_i$ \cite{Haefliger1966a}, cf. \cite{Levine1965}, \cite{Miller1972}, \cite{Habegger1986}. | ||
− | Here $SG_n$ is the space of maps $f \colon S^{n-1} \to S^{n-1}$ of degree $1$. Restricting an element of $SO_n$ to $S^{n-1} \subset \Rr^n$ identifies $SO_n$ as a subspace of $G_n$. | + | Here $SG_n$ is the space of maps $f \colon S^{n-1} \to S^{n-1}$ of degree $1$. Restricting an element of $SO_n$ to $S^{n-1} \subset \Rr^n$ identifies $SO_n$ as a subspace of $G_n$. |
Some readily calculable results are recalled in \cite[$\S$3.3]{Skopenkov2006}. | Some readily calculable results are recalled in \cite[$\S$3.3]{Skopenkov2006}. | ||
Revision as of 10:15, 31 August 2017
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
See general introduction on embeddings, notation and conventions in [Skopenkov2016c, 1, 2].
2 Examples
Analogously to the Haefliger trefoil knot for one constructs a smooth embedding . For even this embedding is a generator of ; it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962]. It would be interesting to know if for odd this embedding is a generator of . The last phrase of [Haefliger1962t] suggests that this is true for .
3 Classification
For the group has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs for and [Haefliger1966a], cf. [Levine1965], [Miller1972], [Habegger1986]. Here is the space of maps of degree . Restricting an element of to identifies as a subspace of . Some readily calculable results are recalled in [Skopenkov2006, 3.3].
See also [Kearton 1973], [Kearton 1973a], [Farber1983] for classification of simple codimension 2 knots and [Farber1981], [Farber1984] for classification of knots in the metastable range.
4 References
- [Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
- [Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).
- [Farber1984] M. Sh. Farber, An algebraic classification of some even-dimensional spherical knots I, II, Trans. Amer. Math. Soc. 281 (1984), 507-528; 529-570.
- [Habegger1986] N. Habegger, Knots and links in codimension greater than 2, Topology, 25:3 (1986) 253--260.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [Kearton 1973] Template:Kearton 1973
- [Kearton 1973a] Template:Kearton 1973a
- [Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
- [Miller1972] Template:Miller1972
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
2 Examples
Analogously to the Haefliger trefoil knot for one constructs a smooth embedding . For even this embedding is a generator of ; it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962]. It would be interesting to know if for odd this embedding is a generator of . The last phrase of [Haefliger1962t] suggests that this is true for .
3 Classification
For the group has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs for and [Haefliger1966a], cf. [Levine1965], [Miller1972], [Habegger1986]. Here is the space of maps of degree . Restricting an element of to identifies as a subspace of . Some readily calculable results are recalled in [Skopenkov2006, 3.3].
See also [Kearton 1973], [Kearton 1973a], [Farber1983] for classification of simple codimension 2 knots and [Farber1981], [Farber1984] for classification of knots in the metastable range.
4 References
- [Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
- [Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).
- [Farber1984] M. Sh. Farber, An algebraic classification of some even-dimensional spherical knots I, II, Trans. Amer. Math. Soc. 281 (1984), 507-528; 529-570.
- [Habegger1986] N. Habegger, Knots and links in codimension greater than 2, Topology, 25:3 (1986) 253--260.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [Kearton 1973] Template:Kearton 1973
- [Kearton 1973a] Template:Kearton 1973a
- [Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
- [Miller1972] Template:Miller1972
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
2 Examples
Analogously to the Haefliger trefoil knot for one constructs a smooth embedding . For even this embedding is a generator of ; it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962]. It would be interesting to know if for odd this embedding is a generator of . The last phrase of [Haefliger1962t] suggests that this is true for .
3 Classification
For the group has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs for and [Haefliger1966a], cf. [Levine1965], [Miller1972], [Habegger1986]. Here is the space of maps of degree . Restricting an element of to identifies as a subspace of . Some readily calculable results are recalled in [Skopenkov2006, 3.3].
See also [Kearton 1973], [Kearton 1973a], [Farber1983] for classification of simple codimension 2 knots and [Farber1981], [Farber1984] for classification of knots in the metastable range.
4 References
- [Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
- [Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).
- [Farber1984] M. Sh. Farber, An algebraic classification of some even-dimensional spherical knots I, II, Trans. Amer. Math. Soc. 281 (1984), 507-528; 529-570.
- [Habegger1986] N. Habegger, Knots and links in codimension greater than 2, Topology, 25:3 (1986) 253--260.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [Kearton 1973] Template:Kearton 1973
- [Kearton 1973a] Template:Kearton 1973a
- [Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
- [Miller1972] Template:Miller1972
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.