High codimension links
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For a [[High_codimension_embeddings#Introduction|general introduction to embeddings]] as well as the [[High_codimension_embeddings#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S$1, $\S$2]{Skopenkov2016c}. | For a [[High_codimension_embeddings#Introduction|general introduction to embeddings]] as well as the [[High_codimension_embeddings#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S$1, $\S$2]{Skopenkov2016c}. | ||
− | A componentwise version of [[High_codimension_embeddings#Embedded_connected_sum|embedded connected sum]] \cite[$\S$5]{Skopenkov2016c} defines a commutative group structure on $E^m(S^p\sqcup S^q)$ for $m-3\ge p,q$ | + | A componentwise version of [[High_codimension_embeddings#Embedded_connected_sum|embedded connected sum]] \cite[$\S$5]{Skopenkov2016c} defines a commutative group structure on $E^m(S^p\sqcup S^q)$ for $m-3\ge p,q$; |
− | + | see \cite[Figure 3.3]{Skopenkov2006}, \cite{Haefliger1966}, \cite{Haefliger1966a} and \cite[Group Structure Lemma 2.2 and Remark 2.3.a]{Skopenkov2015}. | |
The following table was obtained by Zeeman around 1960: | The following table was obtained by Zeeman around 1960: |
Revision as of 20:10, 16 November 2016
This page has been accepted for publication in the Bulletin of the Manifold Atlas. |
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 2].
A componentwise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on for ; see [Skopenkov2006, Figure 3.3], [Haefliger1966], [Haefliger1966a] and [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a].
The following table was obtained by Zeeman around 1960:
See explanation below.
2 Examples
Recall that for each -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (the Hopf linking). For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf linking is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary (including ) the image of the Hopf Linking is the union of two -spheres
Alternatively, these spheres are given by equations:
This embedding is distinguished from the standard embedding by the linking coefficient.
Analogously for each one constructs an embedding which is not isotopic to the standard embedding.
Definition 2.2 of the Zeeman map for . Take Define embedding on to be the standard embedding into . Take any map . Define embedding on to be the composition
where is the equatorial inclusion and the latter inclusion is the standard [Skopenkov2006, Figure 3.2].
Clearly, is well-defined and is a homomorphism.
3 Invariants
Definition 3.1 of linking coefficient for . Fix orientations of , , and . Take an embedding . Take an embedding such that intersects transversally at exactly one point with positive sign [Skopenkov2006, Figure 3.1]. Then the restriction of to is a homotopy equivalence.
(Indeed, since , the complement is simply-connected. By Alexander duality induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is indeed independent of . Clearly, is a homomorphism.
(b) For or there are simpler alternative `homological' definitions, in which components are any closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for .
(d) This definition works for if is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for . So is surjective and is injective.
By Freudenthal Suspension Theorem is an isomorphism for . The stable suspension of the linking coefficient can be described alternatively as follows.
Definition 3.3 of the -invariant . For an embedding define a map [Skopenkov2006, Figure 3.1]
For define the -invariant by
The second isomorphism in this formula is given by the Freudenthal Suspension Theorem. The map is the quotient map. See [Skopenkov2006, Figure 3.4]. The map is an isomorphism for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
We have by [Kervaire1959a, Lemma 5.1]. Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the `metastable' range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for and for , in the PL and smooth cases, respectively.
The surjectivity of (=the injectivity of ) follows from . The injectivity of (=the surjectivity of ) is proved in [Haefliger1962t], [Zeeman1962] (or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a]: the collection of pairwise linking coefficients is bijective for and -dimensional links in .
5 Examples below the metastable range
Let us present an example of non-injectivity of the collection of pairwise linking coefficients.
Borromean rings example 5.1. The Borromean rings is a non-trivial embedding whose restrictions to 2-componented sublinks are trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by . The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6] are the three spheres given by the following three systems of equations:
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because the connected sum of the three components yields a non-trivial knot [Haefliger1962]).
Let us present an example of non-injectivity of the linking coefficient.
Whitehead link example 5.2. The Whitehead link is a non-trivial embedding whose linking coefficient is trivial.
The (higher-dimensional) Whitehead link is obtained from Borromean rings by joining two components with a tube.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in [Skopenkov2006a]). For the Whitehead link is distinguished from the standard embedding by . This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18]. For the Whitehead link is is distinguished from the standard embedding by more complicated invariants [Skopenkov2006a], [Haefliger1962t, 3].
This example shows that the dimension restriction is sharp in Theorem 4.1.
This example seems to be discovered by Whitehead, in connection with Whitehead product.
Cf. the Haefliger Trefoil knot [Skopenkov2016t].
6 Classification below the metastable range
Let . For some information on this group see [Skopenkov2006, 3.3].
The Haefliger Theorem 6.1. (a) [Haefliger1966a] If , then
(b) [Haefliger1966a, Theorem 10.7], [Skopenkov2009] If and , then there is a map for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
Part (b) implies that
This isomorphism is defined for , by map
This map is injective for , ; the image of this map is [Haefliger1962t]. Thus part (b) shows that is in general not injective.
The set for has been described in terms of exact sequences involving homotopy groups of spheres [Haefliger1966], [Haefliger1966a], cf. [Levine1965], [Habegger1986].
See [Skopenkov2009], [Crowley&Ferry&Skopenkov2011], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
7 References
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [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
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
- [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.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069
A componentwise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on for ; see [Skopenkov2006, Figure 3.3], [Haefliger1966], [Haefliger1966a] and [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a].
The following table was obtained by Zeeman around 1960:
See explanation below.
2 Examples
Recall that for each -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (the Hopf linking). For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf linking is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary (including ) the image of the Hopf Linking is the union of two -spheres
Alternatively, these spheres are given by equations:
This embedding is distinguished from the standard embedding by the linking coefficient.
Analogously for each one constructs an embedding which is not isotopic to the standard embedding.
Definition 2.2 of the Zeeman map for . Take Define embedding on to be the standard embedding into . Take any map . Define embedding on to be the composition
where is the equatorial inclusion and the latter inclusion is the standard [Skopenkov2006, Figure 3.2].
Clearly, is well-defined and is a homomorphism.
3 Invariants
Definition 3.1 of linking coefficient for . Fix orientations of , , and . Take an embedding . Take an embedding such that intersects transversally at exactly one point with positive sign [Skopenkov2006, Figure 3.1]. Then the restriction of to is a homotopy equivalence.
(Indeed, since , the complement is simply-connected. By Alexander duality induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is indeed independent of . Clearly, is a homomorphism.
(b) For or there are simpler alternative `homological' definitions, in which components are any closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for .
(d) This definition works for if is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for . So is surjective and is injective.
By Freudenthal Suspension Theorem is an isomorphism for . The stable suspension of the linking coefficient can be described alternatively as follows.
Definition 3.3 of the -invariant . For an embedding define a map [Skopenkov2006, Figure 3.1]
For define the -invariant by
The second isomorphism in this formula is given by the Freudenthal Suspension Theorem. The map is the quotient map. See [Skopenkov2006, Figure 3.4]. The map is an isomorphism for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
We have by [Kervaire1959a, Lemma 5.1]. Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the `metastable' range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for and for , in the PL and smooth cases, respectively.
The surjectivity of (=the injectivity of ) follows from . The injectivity of (=the surjectivity of ) is proved in [Haefliger1962t], [Zeeman1962] (or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a]: the collection of pairwise linking coefficients is bijective for and -dimensional links in .
5 Examples below the metastable range
Let us present an example of non-injectivity of the collection of pairwise linking coefficients.
Borromean rings example 5.1. The Borromean rings is a non-trivial embedding whose restrictions to 2-componented sublinks are trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by . The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6] are the three spheres given by the following three systems of equations:
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because the connected sum of the three components yields a non-trivial knot [Haefliger1962]).
Let us present an example of non-injectivity of the linking coefficient.
Whitehead link example 5.2. The Whitehead link is a non-trivial embedding whose linking coefficient is trivial.
The (higher-dimensional) Whitehead link is obtained from Borromean rings by joining two components with a tube.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in [Skopenkov2006a]). For the Whitehead link is distinguished from the standard embedding by . This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18]. For the Whitehead link is is distinguished from the standard embedding by more complicated invariants [Skopenkov2006a], [Haefliger1962t, 3].
This example shows that the dimension restriction is sharp in Theorem 4.1.
This example seems to be discovered by Whitehead, in connection with Whitehead product.
Cf. the Haefliger Trefoil knot [Skopenkov2016t].
6 Classification below the metastable range
Let . For some information on this group see [Skopenkov2006, 3.3].
The Haefliger Theorem 6.1. (a) [Haefliger1966a] If , then
(b) [Haefliger1966a, Theorem 10.7], [Skopenkov2009] If and , then there is a map for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
Part (b) implies that
This isomorphism is defined for , by map
This map is injective for , ; the image of this map is [Haefliger1962t]. Thus part (b) shows that is in general not injective.
The set for has been described in terms of exact sequences involving homotopy groups of spheres [Haefliger1966], [Haefliger1966a], cf. [Levine1965], [Habegger1986].
See [Skopenkov2009], [Crowley&Ferry&Skopenkov2011], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
7 References
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [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
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
- [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.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069
A componentwise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on for ; see [Skopenkov2006, Figure 3.3], [Haefliger1966], [Haefliger1966a] and [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a].
The following table was obtained by Zeeman around 1960:
See explanation below.
2 Examples
Recall that for each -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (the Hopf linking). For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf linking is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary (including ) the image of the Hopf Linking is the union of two -spheres
Alternatively, these spheres are given by equations:
This embedding is distinguished from the standard embedding by the linking coefficient.
Analogously for each one constructs an embedding which is not isotopic to the standard embedding.
Definition 2.2 of the Zeeman map for . Take Define embedding on to be the standard embedding into . Take any map . Define embedding on to be the composition
where is the equatorial inclusion and the latter inclusion is the standard [Skopenkov2006, Figure 3.2].
Clearly, is well-defined and is a homomorphism.
3 Invariants
Definition 3.1 of linking coefficient for . Fix orientations of , , and . Take an embedding . Take an embedding such that intersects transversally at exactly one point with positive sign [Skopenkov2006, Figure 3.1]. Then the restriction of to is a homotopy equivalence.
(Indeed, since , the complement is simply-connected. By Alexander duality induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is indeed independent of . Clearly, is a homomorphism.
(b) For or there are simpler alternative `homological' definitions, in which components are any closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for .
(d) This definition works for if is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for . So is surjective and is injective.
By Freudenthal Suspension Theorem is an isomorphism for . The stable suspension of the linking coefficient can be described alternatively as follows.
Definition 3.3 of the -invariant . For an embedding define a map [Skopenkov2006, Figure 3.1]
For define the -invariant by
The second isomorphism in this formula is given by the Freudenthal Suspension Theorem. The map is the quotient map. See [Skopenkov2006, Figure 3.4]. The map is an isomorphism for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
We have by [Kervaire1959a, Lemma 5.1]. Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the `metastable' range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for and for , in the PL and smooth cases, respectively.
The surjectivity of (=the injectivity of ) follows from . The injectivity of (=the surjectivity of ) is proved in [Haefliger1962t], [Zeeman1962] (or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a]: the collection of pairwise linking coefficients is bijective for and -dimensional links in .
5 Examples below the metastable range
Let us present an example of non-injectivity of the collection of pairwise linking coefficients.
Borromean rings example 5.1. The Borromean rings is a non-trivial embedding whose restrictions to 2-componented sublinks are trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by . The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6] are the three spheres given by the following three systems of equations:
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because the connected sum of the three components yields a non-trivial knot [Haefliger1962]).
Let us present an example of non-injectivity of the linking coefficient.
Whitehead link example 5.2. The Whitehead link is a non-trivial embedding whose linking coefficient is trivial.
The (higher-dimensional) Whitehead link is obtained from Borromean rings by joining two components with a tube.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in [Skopenkov2006a]). For the Whitehead link is distinguished from the standard embedding by . This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18]. For the Whitehead link is is distinguished from the standard embedding by more complicated invariants [Skopenkov2006a], [Haefliger1962t, 3].
This example shows that the dimension restriction is sharp in Theorem 4.1.
This example seems to be discovered by Whitehead, in connection with Whitehead product.
Cf. the Haefliger Trefoil knot [Skopenkov2016t].
6 Classification below the metastable range
Let . For some information on this group see [Skopenkov2006, 3.3].
The Haefliger Theorem 6.1. (a) [Haefliger1966a] If , then
(b) [Haefliger1966a, Theorem 10.7], [Skopenkov2009] If and , then there is a map for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
Part (b) implies that
This isomorphism is defined for , by map
This map is injective for , ; the image of this map is [Haefliger1962t]. Thus part (b) shows that is in general not injective.
The set for has been described in terms of exact sequences involving homotopy groups of spheres [Haefliger1966], [Haefliger1966a], cf. [Levine1965], [Habegger1986].
See [Skopenkov2009], [Crowley&Ferry&Skopenkov2011], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
7 References
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [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
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
- [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.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069