High codimension links
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(→Classification below the metastable range) |
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Let $C_q^{m-q}:=E^m_D(S^q)$. | Let $C_q^{m-q}:=E^m_D(S^q)$. | ||
− | For [[Knots,_i.e._embeddings_of_spheres|some information | + | For [[Knots,_i.e._embeddings_of_spheres|some information about this group]] see \cite[$\S$3.3]{Skopenkov2006}. |
{{beginthm|The Haefliger Theorem}}\label{belmethae} | {{beginthm|The Haefliger Theorem}}\label{belmethae} |
Revision as of 22:27, 19 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].
Although is not a manifold for , embeddings and isotopy between embeddings are defined analogously to the case of manifolds. Denote by the set of embeddings up to isotopy.
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 the Haefliger-Zeeman Theorem 4.1 below):
2 Examples
Recall that for each -manifold and , any 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 Link). For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary (including ) the image of the Hopf link is the union of two -spheres:
- either and in ;
- or given by equations:
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding which is not isotopic to the standard embedding. The image is the union of two spheres:
- either and in .
- or given by equations:
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
Denote by the equatorial inclusion. For a map representing an element of let
where is the standard embedding [Skopenkov2006, Figure 3.2]. We have . Let .
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely 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 the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is indeed independent of . One can check that is a homomorphism.
(b) For or there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for , by exchanging and in the above definition.
(d) This definition extends to the case when is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for (see Definition 2.2). So is surjective and is injective.
By the Freudenthal Suspension Theorem is an isomorphism for . The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . 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 .)
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have .
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. (D) If , then both and are isomorphisms for in the smooth category.
(PL) If , then both and are isomorphisms for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or 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].
Theorem 4.2. The collection of pairwise linking coefficients is bijective for and -dimensional links in .
5 Examples below the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1. The embedding defined below is a non-trivial embedding (called the Borromean rings) whose restrictions to each 2-component sublink is 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]).
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding (called the Whitehead link) whose linking coefficient is trivial.
The (higher-dimensional) Whitehead link is obtained from the 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 (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
Cf. the Haefliger Trefoil knot [Skopenkov2016t].
6 Classification below the metastable range
Let . For some information about 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 homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
Remark 6.2. (a) The Haefliger Theorem 6.1(b) implies that for
(b) This isomorphism is defined for , by injection
whose image is [Haefliger1962t].
(c) The map is not injective for each [Haefliger1962t] (and also for which was known earlier).
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
Although is not a manifold for , embeddings and isotopy between embeddings are defined analogously to the case of manifolds. Denote by the set of embeddings up to isotopy.
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 the Haefliger-Zeeman Theorem 4.1 below):
2 Examples
Recall that for each -manifold and , any 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 Link). For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary (including ) the image of the Hopf link is the union of two -spheres:
- either and in ;
- or given by equations:
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding which is not isotopic to the standard embedding. The image is the union of two spheres:
- either and in .
- or given by equations:
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
Denote by the equatorial inclusion. For a map representing an element of let
where is the standard embedding [Skopenkov2006, Figure 3.2]. We have . Let .
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely 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 the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is indeed independent of . One can check that is a homomorphism.
(b) For or there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for , by exchanging and in the above definition.
(d) This definition extends to the case when is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for (see Definition 2.2). So is surjective and is injective.
By the Freudenthal Suspension Theorem is an isomorphism for . The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . 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 .)
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have .
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. (D) If , then both and are isomorphisms for in the smooth category.
(PL) If , then both and are isomorphisms for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or 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].
Theorem 4.2. The collection of pairwise linking coefficients is bijective for and -dimensional links in .
5 Examples below the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1. The embedding defined below is a non-trivial embedding (called the Borromean rings) whose restrictions to each 2-component sublink is 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]).
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding (called the Whitehead link) whose linking coefficient is trivial.
The (higher-dimensional) Whitehead link is obtained from the 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 (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
Cf. the Haefliger Trefoil knot [Skopenkov2016t].
6 Classification below the metastable range
Let . For some information about 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 homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
Remark 6.2. (a) The Haefliger Theorem 6.1(b) implies that for
(b) This isomorphism is defined for , by injection
whose image is [Haefliger1962t].
(c) The map is not injective for each [Haefliger1962t] (and also for which was known earlier).
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
Although is not a manifold for , embeddings and isotopy between embeddings are defined analogously to the case of manifolds. Denote by the set of embeddings up to isotopy.
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 the Haefliger-Zeeman Theorem 4.1 below):
2 Examples
Recall that for each -manifold and , any 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 Link). For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary (including ) the image of the Hopf link is the union of two -spheres:
- either and in ;
- or given by equations:
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding which is not isotopic to the standard embedding. The image is the union of two spheres:
- either and in .
- or given by equations:
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
Denote by the equatorial inclusion. For a map representing an element of let
where is the standard embedding [Skopenkov2006, Figure 3.2]. We have . Let .
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely 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 the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is indeed independent of . One can check that is a homomorphism.
(b) For or there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for , by exchanging and in the above definition.
(d) This definition extends to the case when is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for (see Definition 2.2). So is surjective and is injective.
By the Freudenthal Suspension Theorem is an isomorphism for . The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . 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 .)
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have .
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. (D) If , then both and are isomorphisms for in the smooth category.
(PL) If , then both and are isomorphisms for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or 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].
Theorem 4.2. The collection of pairwise linking coefficients is bijective for and -dimensional links in .
5 Examples below the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1. The embedding defined below is a non-trivial embedding (called the Borromean rings) whose restrictions to each 2-component sublink is 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]).
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding (called the Whitehead link) whose linking coefficient is trivial.
The (higher-dimensional) Whitehead link is obtained from the 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 (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
Cf. the Haefliger Trefoil knot [Skopenkov2016t].
6 Classification below the metastable range
Let . For some information about 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 homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
Remark 6.2. (a) The Haefliger Theorem 6.1(b) implies that for
(b) This isomorphism is defined for , by injection
whose image is [Haefliger1962t].
(c) The map is not injective for each [Haefliger1962t] (and also for which was known earlier).
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
Although is not a manifold for , embeddings and isotopy between embeddings are defined analogously to the case of manifolds. Denote by the set of embeddings up to isotopy.
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 the Haefliger-Zeeman Theorem 4.1 below):
2 Examples
Recall that for each -manifold and , any 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 Link). For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary (including ) the image of the Hopf link is the union of two -spheres:
- either and in ;
- or given by equations:
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding which is not isotopic to the standard embedding. The image is the union of two spheres:
- either and in .
- or given by equations:
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
Denote by the equatorial inclusion. For a map representing an element of let
where is the standard embedding [Skopenkov2006, Figure 3.2]. We have . Let .
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely 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 the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is indeed independent of . One can check that is a homomorphism.
(b) For or there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for , by exchanging and in the above definition.
(d) This definition extends to the case when is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for (see Definition 2.2). So is surjective and is injective.
By the Freudenthal Suspension Theorem is an isomorphism for . The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . 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 .)
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have .
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. (D) If , then both and are isomorphisms for in the smooth category.
(PL) If , then both and are isomorphisms for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or 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].
Theorem 4.2. The collection of pairwise linking coefficients is bijective for and -dimensional links in .
5 Examples below the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1. The embedding defined below is a non-trivial embedding (called the Borromean rings) whose restrictions to each 2-component sublink is 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]).
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding (called the Whitehead link) whose linking coefficient is trivial.
The (higher-dimensional) Whitehead link is obtained from the 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 (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
Cf. the Haefliger Trefoil knot [Skopenkov2016t].
6 Classification below the metastable range
Let . For some information about 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 homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
Remark 6.2. (a) The Haefliger Theorem 6.1(b) implies that for
(b) This isomorphism is defined for , by injection
whose image is [Haefliger1962t].
(c) The map is not injective for each [Haefliger1962t] (and also for which was known earlier).
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
Although is not a manifold for , embeddings and isotopy between embeddings are defined analogously to the case of manifolds. Denote by the set of embeddings up to isotopy.
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 the Haefliger-Zeeman Theorem 4.1 below):
2 Examples
Recall that for each -manifold and , any 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 Link). For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary (including ) the image of the Hopf link is the union of two -spheres:
- either and in ;
- or given by equations:
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding which is not isotopic to the standard embedding. The image is the union of two spheres:
- either and in .
- or given by equations:
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
Denote by the equatorial inclusion. For a map representing an element of let
where is the standard embedding [Skopenkov2006, Figure 3.2]. We have . Let .
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely 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 the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is indeed independent of . One can check that is a homomorphism.
(b) For or there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for , by exchanging and in the above definition.
(d) This definition extends to the case when is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for (see Definition 2.2). So is surjective and is injective.
By the Freudenthal Suspension Theorem is an isomorphism for . The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . 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 .)
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have .
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. (D) If , then both and are isomorphisms for in the smooth category.
(PL) If , then both and are isomorphisms for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or 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].
Theorem 4.2. The collection of pairwise linking coefficients is bijective for and -dimensional links in .
5 Examples below the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1. The embedding defined below is a non-trivial embedding (called the Borromean rings) whose restrictions to each 2-component sublink is 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]).
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding (called the Whitehead link) whose linking coefficient is trivial.
The (higher-dimensional) Whitehead link is obtained from the 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 (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
Cf. the Haefliger Trefoil knot [Skopenkov2016t].
6 Classification below the metastable range
Let . For some information about 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 homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
Remark 6.2. (a) The Haefliger Theorem 6.1(b) implies that for
(b) This isomorphism is defined for , by injection
whose image is [Haefliger1962t].
(c) The map is not injective for each [Haefliger1962t] (and also for which was known earlier).
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
Although is not a manifold for , embeddings and isotopy between embeddings are defined analogously to the case of manifolds. Denote by the set of embeddings up to isotopy.
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 the Haefliger-Zeeman Theorem 4.1 below):
2 Examples
Recall that for each -manifold and , any 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 Link). For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary (including ) the image of the Hopf link is the union of two -spheres:
- either and in ;
- or given by equations:
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding which is not isotopic to the standard embedding. The image is the union of two spheres:
- either and in .
- or given by equations:
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
Denote by the equatorial inclusion. For a map representing an element of let
where is the standard embedding [Skopenkov2006, Figure 3.2]. We have . Let .
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely 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 the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is indeed independent of . One can check that is a homomorphism.
(b) For or there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for , by exchanging and in the above definition.
(d) This definition extends to the case when is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for (see Definition 2.2). So is surjective and is injective.
By the Freudenthal Suspension Theorem is an isomorphism for . The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . 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 .)
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have .
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. (D) If , then both and are isomorphisms for in the smooth category.
(PL) If , then both and are isomorphisms for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or 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].
Theorem 4.2. The collection of pairwise linking coefficients is bijective for and -dimensional links in .
5 Examples below the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1. The embedding defined below is a non-trivial embedding (called the Borromean rings) whose restrictions to each 2-component sublink is 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]).
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding (called the Whitehead link) whose linking coefficient is trivial.
The (higher-dimensional) Whitehead link is obtained from the 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 (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
Cf. the Haefliger Trefoil knot [Skopenkov2016t].
6 Classification below the metastable range
Let . For some information about 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 homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
Remark 6.2. (a) The Haefliger Theorem 6.1(b) implies that for
(b) This isomorphism is defined for , by injection
whose image is [Haefliger1962t].
(c) The map is not injective for each [Haefliger1962t] (and also for which was known earlier).
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