Knots, i.e. embeddings of spheres
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+ | {{beginthm|Theorem|(\cite{Haefliger1966}, \cite{Milgram1972})}}\label{t:hami} | ||
+ | We have the following table for the group $E^m_D(S^n)$, where $k,s\ge1$: | ||
+ | $$\begin{array}{c|c|c|c|c|c|c|c} | ||
+ | (m,n) &2m\ge3q+4 &(6k,4k-1) &(6k+3,4k+1) &(7,4) &(6k-2,4k-2) &(12s+7,8s+4) & (12s+1,8s+1)\\ | ||
+ | \hline | ||
+ | E^m_D(S^n)| &0 &\Z &\Z_2 &\Z_{12} &0 &\Z_4 &\Z_2\oplus\Z_2 | ||
+ | \end{array}$$ | ||
+ | {{endthm}} | ||
+ | |||
+ | <!--Some readily calculable corollaries of this classification are recalled in \cite[$\S$3.3]{Skopenkov2006}.--> | ||
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{Milgram1972}, \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{Milgram1972}, \cite{Habegger1986}. | ||
Here $SG_n$ is the space of maps $f \colon S^{n-1} \to S^{n-1}$ of degree $1$. | 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 $G_n$. | Restricting an element of $SO_n$ to $S^{n-1} \subset \Rr^n$ identifies $SO_n$ as a subspace of $G_n$. | ||
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Revision as of 11:52, 10 April 2019
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, 3].
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
Theorem 3.1 ([Haefliger1966], [Milgram1972]). We have the following table for the group , where :
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], [Milgram1972], [Habegger1986]. Here is the space of maps of degree . Restricting an element of to identifies as a subspace of .
4 Codimension 2 knots
For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in ), a complete readily calculable classification (in the sense of Remark 1.1 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification. See e.g. interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].
On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a] and [Ranicki1998].
5 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
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [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
- [Kearton1983] C. Kearton, An algebraic classification of certain simple even-dimensional knots, Trans. Amer. Math. Soc. 176 (1983), 1–53.
- [Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
- [Levine1969a] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR0246314 (39 #7618) Zbl 0176.22101
- [Milgram1972] R. J. Milgram, On the Haefliger knot groups, Bull. of the Amer. Math. Soc., 78:5 (1972) 861--865.
- [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
- [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
Theorem 3.1 ([Haefliger1966], [Milgram1972]). We have the following table for the group , where :
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], [Milgram1972], [Habegger1986]. Here is the space of maps of degree . Restricting an element of to identifies as a subspace of .
4 Codimension 2 knots
For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in ), a complete readily calculable classification (in the sense of Remark 1.1 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification. See e.g. interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].
On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a] and [Ranicki1998].
5 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
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [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
- [Kearton1983] C. Kearton, An algebraic classification of certain simple even-dimensional knots, Trans. Amer. Math. Soc. 176 (1983), 1–53.
- [Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
- [Levine1969a] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR0246314 (39 #7618) Zbl 0176.22101
- [Milgram1972] R. J. Milgram, On the Haefliger knot groups, Bull. of the Amer. Math. Soc., 78:5 (1972) 861--865.
- [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
- [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
Theorem 3.1 ([Haefliger1966], [Milgram1972]). We have the following table for the group , where :
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], [Milgram1972], [Habegger1986]. Here is the space of maps of degree . Restricting an element of to identifies as a subspace of .
4 Codimension 2 knots
For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in ), a complete readily calculable classification (in the sense of Remark 1.1 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification. See e.g. interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].
On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a] and [Ranicki1998].
5 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
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [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
- [Kearton1983] C. Kearton, An algebraic classification of certain simple even-dimensional knots, Trans. Amer. Math. Soc. 176 (1983), 1–53.
- [Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
- [Levine1969a] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR0246314 (39 #7618) Zbl 0176.22101
- [Milgram1972] R. J. Milgram, On the Haefliger knot groups, Bull. of the Amer. Math. Soc., 78:5 (1972) 861--865.
- [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.