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] [Haefliger1966C].
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].
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 .
(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 Borromean rings
The following example shows that the collection of pairwise linking coefficients is not injective for and -dimensional links with more than two components in .
2 The Whitehead link
3 The Haefliger Trefoil knot
4 Classification
5 Further discussion
6 References
- [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
- [Haefliger1966C] Template:Haefliger1966C
- [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.
- [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].
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 .
(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 Borromean rings
The following example shows that the collection of pairwise linking coefficients is not injective for and -dimensional links with more than two components in .
2 The Whitehead link
3 The Haefliger Trefoil knot
4 Classification
5 Further discussion
6 References
- [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
- [Haefliger1966C] Template:Haefliger1966C
- [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.
- [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