High codimension links
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Contents |
1 Introduction
For notation and conventions throughout this page see high codimension embeddings.
`Embedded connected sum' defines a commutative group structure on for . See Figure 3.3. of [Skopenkov2006], [Haefliger1966] [Haefliger1966a].
2 General position and the Hopf linking
General Position Theorem 2.1. For each -manifold and , every two embeddings are isotopic.
The restriction in Theorem 2.1 is sharp for non-connected manifolds.
Example: the Hopf linking 2.2. For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf Linking is shown in Figure~2.1.a of [Skopenkov2006]. For arbitrary (including ) the image of the Hopf Linking is the union of two -spheres:
3 The Zeeman construction and linking coefficient
The following table was obtained by Zeeman around 1960:
1 Construction of the Zeeman map
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. See Figure 3.2 of [Skopenkov2006]. Clearly, is well-defined and is a homomorphism.
2 Definition 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 (see Figure 3.1 of [Skopenkov2006]). 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 3.1. (a) Clearly, is indeed independent of . Clearly, is a homomorphism.
(b) For there is a simpler alternative `homological' definition. That definition works for as well.
(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.
3 Classification in the `metastable' range
The Haefliger-Zeeman Theorem 3.2. If , then both and are isomorphisms for and for , in the PL and DIFF cases, respectively.
The surjectivity of (=the injectivity of ) follows from . The injectivity of (=the surjectivity of ) is proved in [Haefliger1962T], [Zeeman1962] (or follows from the Haefliger-Weber Theorem 5.4 and Deleted Product Lemma 5.3.a of [Skopenkov2006]).
An analogue of this result holds for links with many components: the collection of pairwise linking coefficients is bijective for and -dimensional links in .
4 Alpha-invariant
By Freudenthal Suspension Theorem is an isomorphism for . The stable suspension of the linking coefficient can be described alternatively as follows. For an embedding define a map
See Figure 3.1 of [Skopenkov2006]. 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 Figure 3.4 of [Skopenkov2006]. 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 Lemma 5.1 of [Kervaire1959L].
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006].
4 Classification below the metastable range
1 Higher-dimensional Borromean rings
Let us present an example of non-injectivity of the collection of pairwise linking coefficients.
Borromean rings example 4.1. 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 Borromean rings are the three spheres given by the following three systems of equations:
See Figures 3.5 and 3.6 of [Skopenkov2006]. This embedding is distinguished from the standard embedding by the Massey invariant.
2 Higher-dimensional Whitehead link
Let us present an example of non-injectivity of the linking coefficient.
Whitehead link example 4.2. The Whitehead link is a non-trivial embedding whose linking coefficient is trivial.
The 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 . (It would be interesting to find or write a published proof of this fact.) For the Whitehead link is is distinguished from the standard embedding by more complicated invariants [Skopenkov2006a], [Haefliger1962T], \S3.
This example seems to be discovered by Whitehead, in connection with Whitehead product. It would be interesting to find a publication where it first appeared.
3 Classification
See The Haefliger Trefoil knot
5 Further discussion
6 References
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962T] Template:Haefliger1962T
- [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
- [Kervaire1959L] Template:Kervaire1959L
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [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.
- [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
2 General position and the Hopf linking
General Position Theorem 2.1. For each -manifold and , every two embeddings are isotopic.
The restriction in Theorem 2.1 is sharp for non-connected manifolds.
Example: the Hopf linking 2.2. For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf Linking is shown in Figure~2.1.a of [Skopenkov2006]. For arbitrary (including ) the image of the Hopf Linking is the union of two -spheres:
3 The Zeeman construction and linking coefficient
The following table was obtained by Zeeman around 1960:
1 Construction of the Zeeman map
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. See Figure 3.2 of [Skopenkov2006]. Clearly, is well-defined and is a homomorphism.
2 Definition 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 (see Figure 3.1 of [Skopenkov2006]). 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 3.1. (a) Clearly, is indeed independent of . Clearly, is a homomorphism.
(b) For there is a simpler alternative `homological' definition. That definition works for as well.
(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.
3 Classification in the `metastable' range
The Haefliger-Zeeman Theorem 3.2. If , then both and are isomorphisms for and for , in the PL and DIFF cases, respectively.
The surjectivity of (=the injectivity of ) follows from . The injectivity of (=the surjectivity of ) is proved in [Haefliger1962T], [Zeeman1962] (or follows from the Haefliger-Weber Theorem 5.4 and Deleted Product Lemma 5.3.a of [Skopenkov2006]).
An analogue of this result holds for links with many components: the collection of pairwise linking coefficients is bijective for and -dimensional links in .
4 Alpha-invariant
By Freudenthal Suspension Theorem is an isomorphism for . The stable suspension of the linking coefficient can be described alternatively as follows. For an embedding define a map
See Figure 3.1 of [Skopenkov2006]. 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 Figure 3.4 of [Skopenkov2006]. 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 Lemma 5.1 of [Kervaire1959L].
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006].
4 Classification below the metastable range
1 Higher-dimensional Borromean rings
Let us present an example of non-injectivity of the collection of pairwise linking coefficients.
Borromean rings example 4.1. 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 Borromean rings are the three spheres given by the following three systems of equations:
See Figures 3.5 and 3.6 of [Skopenkov2006]. This embedding is distinguished from the standard embedding by the Massey invariant.
2 Higher-dimensional Whitehead link
Let us present an example of non-injectivity of the linking coefficient.
Whitehead link example 4.2. The Whitehead link is a non-trivial embedding whose linking coefficient is trivial.
The 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 . (It would be interesting to find or write a published proof of this fact.) For the Whitehead link is is distinguished from the standard embedding by more complicated invariants [Skopenkov2006a], [Haefliger1962T], \S3.
This example seems to be discovered by Whitehead, in connection with Whitehead product. It would be interesting to find a publication where it first appeared.
3 Classification
See The Haefliger Trefoil knot
5 Further discussion
6 References
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962T] Template:Haefliger1962T
- [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
- [Kervaire1959L] Template:Kervaire1959L
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [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.
- [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