High codimension links
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Most of 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.
We describe classification of embeddings for .
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
For an -tuple denote . Although is not a manifold when are not all equal, embeddings and (ambient) isotopy between such embeddings are defined analogously to the case of manifolds. Denote by the set of embeddings up to isotopy.
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set for [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].
Recall that for any -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link). For any there is an embedding which is not isotopic to the standard embedding.
For the Hopf link is shown e.g. 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 the following equations in :
Analogously for any 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 the following equations in :
Definition 2.2 (The Zeeman map). We define a map
Denote by the equatorial inclusion. For a map representing an element of let
One can easily check that is well-defined and is a homomorphism.
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 , by general position 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 well-defined, i.e. is independent of the choises of and of representative of . One can check that is a homomorphism.
(b) Analogously one can define for , by exchanging and in the above definition.
(c) Clearly, (see Definition 2.2; this also holds for , see (d) and (e) below). So is surjective and is injective.
(d) This definition extends to the case , provided is simply-connected.
(e) 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].
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 the suspension isomorphism. 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 .)
One can check that is well-defined and is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have .
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]).
Theorem 4.2. The collection of pairwise linking coefficients
is bijective for .
5 Examples beyond 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.
The required embedding is any embedding whose image consists of Borromean rings.
This embedding is distinguished from the standard embedding by the well-known triple linking number called Massey number [Massey1968], for an elementary definition see [Skopenkov2017, 4.5 `Triple linking modulo 2' and 4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding because joining the three components with two tubes, i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c], yields a non-trivial knot [Haefliger1962], cf. the Haefliger Trefoil knot [Skopenkov2016t].)
For this and other results of this section are parts of low-dimensional link theory and so were known well before given references.
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.
Example 5.2 (Whitehead link). There is a non-trivial embedding whose linking coefficient is trivial.
The (higher-dimensional) `Whitehead link' is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of embedded connected sum of the components [Skopenkov2016c, 3, 4].
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. 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, the Whitehead link Lemma 2.14]. For the Whitehead link 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.
6 Reduction to unknotted components
Theorem 6.1. [Haefliger1966a] If , then
where is the subgroup of formed by links whose restrictions to the components are unknotted.
The isomorphism of Theorem 6.1 is the sum of the restriction and the unknotting homomorphisms
The restriction homomorphism is defined to be the sum of homomorphisms induced by the restrictions to components. The unknotting homomorphism is defined by taking embedded connected sums of components with knots representing the elements of inverse to the components, whose images are small and are close to the components. We have [Haefliger1966a, 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, 1.5].
For some information on the groups see [Skopenkov2006, 3.3]. (See also foundational paper [Haefliger1966] on where this information is less complete and harder to find, and foundational paper [Levine1965] on different but related group.)
7 Classification in codimension 3
Remark 7.2. (a) The Haefliger Theorem 7.1 implies that for any we have an isomorphism
(b) When but , the map
is injective and its image is . See [Haefliger1962t, 6] for ; for specialists note that this also follows from (a) above and , where is the map from the EHP sequence.
(c) For the map in (b) above is not injective [Haefliger1962t, 6].
(d) For any , we have an isomorphism
which is the sum of 3 pairwise linking coefficients, 3 pairwise -invariants and Massey number. This follows from [Haefliger1962t, 6] and . We conjecture that this result holds also for .
Below in this subsection we assume that is an -tuple such that .
Definition 7.3 (The Haefliger link sequence). We define the following long sequence of abelian groups
In the above sequence -tuples are the same for different terms. Denote . For any and integer denote by the homomorphisms induced by the projection to the -component of the wedge. Denote . Denote .
Analogously to Definition 3.1 there is a canonical homotopy equivalence . The homotopy class of a push off of one component in the complement of the entire link gives then a map , see details in [Haefliger1966a, 1.4]. (This map is a generalization of linking coefficient.) Define .
Taking the Whitehead product with the class of the identity in defines a homomorphism . Define .
The definition of the homomorphism is sketched in [Haefliger1966a, 1.5].
Theorem 7.4. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011] The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers splits into short exact sequences.
In [Crowley&Ferry&Skopenkov2011, 1.2, 1.3] one can find necessary and sufficient conditions on and when is finite, as well as an effective procedure for computing the rank of the group .
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