High codimension links
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 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 each -manifold and , any two embeddings are isotopic [Skopenkov2016c, 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 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:
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:
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.
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 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 .
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 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.
Denote coordinates in by . The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i are the three spheres given by the following three systems of equations:
The required embedding is any embedding whose image consists of Borromean rings.
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or 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.
Whitehead link example 5.2. 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].
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.
6 Classification beyond the metastable range
Theorem 6.1. [Haefliger1966a] If , then
Thus is isomorphic to the kernel of the restriction homomorphism and to the group of -Brunnian links (i.e. of links whose restrictions to the components are unknotted). For some information on the groups see [Skopenkov2006, 3.3].
Remark 6.3. (a) The Haefliger Theorem 6.2(b) implies that for each we have an isomorphism
(b) When but , the map
(c) For the map in (b) above is not injective [Haefliger1962t].
(d) [Haefliger1962t] For each we have an isomorphism
which is the sum of 3 pairwise linking coefficients, 3 pairwise -invariants and triplewise Massey invariant.
7 Classification in codimension 3
In this subsection we assume that is an -tuple such that .
Definition 7.1 (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 each 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 one component in the complement of the others gives then a map , see details in [Haefliger1966a, 1.4] (this 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 7.2. (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 link Haefliger sequence 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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