Knots, i.e. embeddings of spheres
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Proof for the first four columns, and for the fifth column when $k$ is even, are presented in \cite{Haefliger1966} (or are deduced from that paper using simple calculations, cf. \cite[$\S$3]{Skopenkov2005}). | Proof for the first four columns, and for the fifth column when $k$ is even, are presented in \cite{Haefliger1966} (or are deduced from that paper using simple calculations, cf. \cite[$\S$3]{Skopenkov2005}). | ||
− | The remaining results | + | The remaining results follow from \cite{Haefliger1966} and \cite{Milgram1972}. |
Alternative proofs for the case $(m,n)=(7,4)$ are given in \cite{Skopenkov2005}, \cite{Crowley&Skopenkov2008}. | Alternative proofs for the case $(m,n)=(7,4)$ are given in \cite{Skopenkov2005}, \cite{Crowley&Skopenkov2008}. | ||
{{beginthm|Theorem|\cite[Corollary G]{Milgram1972}}}\label{t:mi} | {{beginthm|Theorem|\cite[Corollary G]{Milgram1972}}}\label{t:mi} | ||
− | We have $E^m_D(S^n)=0$ if and only if either $2m\ge3n+4$, or $(m,n)=(6k+4,4k+2)$, or $(m,n)=(3k,2k)$ and $k\equiv3,11\mod12$, or $(m,n)=(3k+2,2k+2)$ and $k\equiv14,22\mod24$. | + | We have $E^m_D(S^n)=0$ if and only if either $2m\ge3n+4$, or $(m,n)=(6k+4,4k+2)$, |
+ | %$k\ne1$, | ||
+ | or $(m,n)=(3k,2k)$ and $k\equiv3,11\mod12$, | ||
+ | %$k\ne3$, | ||
+ | or $(m,n)=(3k+2,2k+2)$ and $k\equiv14,22\mod24$. | ||
{{endthm}} | {{endthm}} | ||
For a description of 2-components of $E^m_D(S^n)$ see \cite[Theorem F]{Milgram1972}. | For a description of 2-components of $E^m_D(S^n)$ see \cite[Theorem F]{Milgram1972}. | ||
<!--Some readily calculable corollaries of this classification are recalled in \cite[$\S$3.3]{Skopenkov2006}. involving the homotopy groups of spheres and of the pairs $(SG_n,SO_n),(SG,SO)$ --> | <!--Some readily calculable corollaries of this classification are recalled in \cite[$\S$3.3]{Skopenkov2006}. involving the homotopy groups of spheres and of the pairs $(SG_n,SO_n),(SG,SO)$ --> | ||
+ | |||
+ | Observe that no reliable reference (containing complete proofs) of results announced in \cite{Milgram1972} appeared. | ||
+ | Thus, strictly speaking, the corresponding results are conjectures. | ||
For $m\ge n+3$ the group $E^m_D(S^n)$ has been described as follows, in terms of exact sequences \cite{Haefliger1966}, cf. \cite{Levine1965}, \cite{Haefliger1966a}, \cite{Milgram1972}, \cite{Habegger1986}. | For $m\ge n+3$ the group $E^m_D(S^n)$ has been described as follows, in terms of exact sequences \cite{Haefliger1966}, cf. \cite{Levine1965}, \cite{Haefliger1966a}, \cite{Milgram1972}, \cite{Habegger1986}. |
Revision as of 09:51, 7 June 2021
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
2 Examples
Analogously to the Haefliger trefoil knot for one constructs a smooth embedding , see [Skopenkov2016h, 5]. 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 [Levine1965, Corollary in p. 44], [Haefliger1966]. For the group is finite unless and , when is the sum of and a finite group.
Theorem 3.2 (Haefliger-Milgram). We have the following table for the group ; in the table ; in the last three columns :
Proof for the first four columns, and for the fifth column when is even, are presented in [Haefliger1966] (or are deduced from that paper using simple calculations, cf. [Skopenkov2005, 3]). The remaining results follow from [Haefliger1966] and [Milgram1972]. Alternative proofs for the case are given in [Skopenkov2005], [Crowley&Skopenkov2008].
Theorem 3.3 [Milgram1972, Corollary G]. We have if and only if either , or , %, or and , %, or and .
For a description of 2-components of see [Milgram1972, Theorem F].
Observe that no reliable reference (containing complete proofs) of results announced in [Milgram1972] appeared. Thus, strictly speaking, the corresponding results are conjectures.
For the group has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].
Theorem 3.4 [Haefliger1966]. For there is the following exact sequence of abelian groups:
Here be the space of maps of degree . Restricting an element of to identifies as a subspace of . Let . Analogously define . Let be the stabilization homomorphism. The attaching invariant and the map are defined in [Haefliger1966], see also [Skopenkov2005, 3].
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 1.2 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots. See e.g. the 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
- [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
- [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
- [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
2 Examples
Analogously to the Haefliger trefoil knot for one constructs a smooth embedding , see [Skopenkov2016h, 5]. 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 [Levine1965, Corollary in p. 44], [Haefliger1966]. For the group is finite unless and , when is the sum of and a finite group.
Theorem 3.2 (Haefliger-Milgram). We have the following table for the group ; in the table ; in the last three columns :
Proof for the first four columns, and for the fifth column when is even, are presented in [Haefliger1966] (or are deduced from that paper using simple calculations, cf. [Skopenkov2005, 3]). The remaining results follow from [Haefliger1966] and [Milgram1972]. Alternative proofs for the case are given in [Skopenkov2005], [Crowley&Skopenkov2008].
Theorem 3.3 [Milgram1972, Corollary G]. We have if and only if either , or , %, or and , %, or and .
For a description of 2-components of see [Milgram1972, Theorem F].
Observe that no reliable reference (containing complete proofs) of results announced in [Milgram1972] appeared. Thus, strictly speaking, the corresponding results are conjectures.
For the group has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].
Theorem 3.4 [Haefliger1966]. For there is the following exact sequence of abelian groups:
Here be the space of maps of degree . Restricting an element of to identifies as a subspace of . Let . Analogously define . Let be the stabilization homomorphism. The attaching invariant and the map are defined in [Haefliger1966], see also [Skopenkov2005, 3].
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 1.2 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots. See e.g. the 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
- [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
- [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
- [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
2 Examples
Analogously to the Haefliger trefoil knot for one constructs a smooth embedding , see [Skopenkov2016h, 5]. 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 [Levine1965, Corollary in p. 44], [Haefliger1966]. For the group is finite unless and , when is the sum of and a finite group.
Theorem 3.2 (Haefliger-Milgram). We have the following table for the group ; in the table ; in the last three columns :
Proof for the first four columns, and for the fifth column when is even, are presented in [Haefliger1966] (or are deduced from that paper using simple calculations, cf. [Skopenkov2005, 3]). The remaining results follow from [Haefliger1966] and [Milgram1972]. Alternative proofs for the case are given in [Skopenkov2005], [Crowley&Skopenkov2008].
Theorem 3.3 [Milgram1972, Corollary G]. We have if and only if either , or , %, or and , %, or and .
For a description of 2-components of see [Milgram1972, Theorem F].
Observe that no reliable reference (containing complete proofs) of results announced in [Milgram1972] appeared. Thus, strictly speaking, the corresponding results are conjectures.
For the group has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].
Theorem 3.4 [Haefliger1966]. For there is the following exact sequence of abelian groups:
Here be the space of maps of degree . Restricting an element of to identifies as a subspace of . Let . Analogously define . Let be the stabilization homomorphism. The attaching invariant and the map are defined in [Haefliger1966], see also [Skopenkov2005, 3].
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 1.2 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots. See e.g. the 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
- [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
- [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
- [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.