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.
On this page we describe readily calculable classifications of embeddings of closed disconnected manifolds into up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings , where are spheres (or even closed manifolds) and for every , under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].
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 [Skopenkov2016i]. Denote by the set of embeddings up to isotopy.
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set for [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, 1], [Avvakumov2017, 1.4], see Figure 1.
The standard embedding is defined by . Fix pairwise disjoint -discs . The standard embedding is defined by taking the union of the compositions of the standard embeddings with the fixed inclusions .
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 every positive integer there is an embedding , which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all the image of the Hopf link is the union of two -spheres which can be described as follows:
- either the spheres are and in ;
- or they are given as the sets of points in satisfying the following equations:
Analogously for any one constructs an embedding , which is not isotopic to the standard embedding. The image is the union of two spheres which can be described as follows:
- either the spheres are and in .
- or they are given as the points in satisfying the following equations:
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map
For a map representing an element of let
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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; see Figure 4.
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 choices of and of the representative of . One can check that is a homomorphism.
(b) Analogously one can define for , by exchanging and in the above definition.
(c) Clearly, for the Zeman map . So is surjective and is injective.
(d) For there is a simpler alternative definition using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for . The stable suspension 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 the Gauss map (see Figure 4)
For define the -invariant by
The second isomorphism in this formula is the suspension isomorphism. The map is the quotient map, see Figure 5.
The map is a 1--1 correspondence 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 easily check that is well-defined and for is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have for .
Hence , even though in general as we explain in 5.
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for in the smooth category, and 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, Theorem in 5], [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, each of the same dimension [Haefliger1962t, Theorem in 5]. Let be the -tuple consisting entirely of some positive integer .
Theorem 4.2. The collection of pairwise linking coefficients
is a 1-1 correspondence 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.
Denote coordinates in by . The (higher-dimensional) `Borromean rings' are the three spheres given by the following three systems of equations:
See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.
An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the 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, 4], yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].)
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). For every positive integer there is a non-trivial embedding whose linking coefficient is trivial.
Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3, 4]; see also the Wikipedia article on the Whitehead link. (For the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.)
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 equal to the Whitehead square of the generator [Haefliger1962t, end of 6] (I do not know a published proof of 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.
For some results on links related to the Whitehead link see [Skopenkov2015a, 2.5].
6 Linked manifolds
In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no connectivity assumptions are made on the source manifolds.
Theorem 6.1. Assume that are closed manifolds and for every . Then for an embedding and every one defines the linking coefficient , see Remark 3.2.e. We have if both and are orientable, and otherwise. Then the collection of pairwise linking coefficients
is well-defined and is a 1-1 correspondence, where of are orientable and are not.
We present the general case of the classification of embeddings of disjoint manifolds only for the case of two oriented components of the same dimension.
Theorem 6.2. Let and be closed -dimensional homologically -connected orientable manifolds. For an embedding one can define the invariant analogously to Definition 3.3. Then
is well-defined and is a 1-1 correspondence, provided .
Theorems 6.1 and 6.2 are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]; the calculations are analogous to the construction of the 1-1 correspondence in Definition 3.3. Apparently Theorem 6.1 has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.
Now we present an extension of Theorems 6.1 and 6.2 to a case where and . In particular, for the results below , and the manifolds are only -connected. An embedding is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers such that is even, Avvakumov has constructed a Brunnian embedding , which appears in the next result [Avvakumov2016, 1].
Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding is isotopic to for some integers such that is even. Two embeddings and are isotopic if and only if and both and are divisible by .
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for ). The following corollary shows that the relation between the embeddings and is not trivial.
Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings and such that the componentwise embedded connected sum is isotopic to but is not isotopic to .
7 Reduction to the case with unknotted components
In this section we assume that is an -tuple such that .
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
Take pairwise disjoint -discs in , i.e. take an embedding . Define
Then is a right inverse of the restriction homomorphism , i.e. . The unknotting homomorphism is defined to be the homomorphism
Informally, the homomorphism is obtained 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.
Theorem 7.1. [Haefliger1966a, Theorem 2.4] For , the homomorphism
is an isomorphism.
8 Classification beyond the metastable range
The map and its right inverse are constructed in [Haefliger1966a, 10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of see [Skopenkov2009, 3], [Skopenkov2006b, 5], cf. [Skopenkov2007, 2] and [Crowley&Skopenkov2016, 2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2. (a) Theorem 8.1 implies that for any we have an isomorphism
(b) For any , the map
is injective and its image is .
For see [Haefliger1962t, 6]. The following proof for and general remark are intended for specialists.
For there is an exact sequence , where is [Haefliger1966a, Corollary 10.3]. We have , and is the reduction modulo 2. By the exactness of the previous sequence, . By (a) . Hence is injective. We have by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009, 3], [Skopenkov2006b, 5] and geomeric interpretation of the EHP sequence [Koschorke&Sanderson1977, Main Theorem in 1] one can possibly prove that . Then (b) would follow.
(c) For any the map in (b) above is not injective [Haefliger1962t, 6].
(d) For any , we have an isomorphism
9 Classification in codimension at least 3
In this section we assume that is an -tuple such that . For this case a readily calculable classification of was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, 1.2] contains necessary and sufficient conditions on which determine when is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and 1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.
Theorem 9.1. There are algorithms which for integers(a) calculate
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(b) determine whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] Haefliger defined a long exact sequence of abelian groups
We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. We first note that in the sequence above the -tuples are the same for different terms. Denote . For any and positive integer denote by the homomorphism induced by the collapse map onto to the -component of the wedge. Denote and .
Analogous to Definition 3.1 there is a canonical homotopy equivalence . It can be shown that each component of a link has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement. It can also be shown that the homotopy class of a push off of the th component in the complement of the link gives a well-defined map . In fact, the map is a generalisation of the linking coefficient. Finally, define .
Taking the Whitehead product with the class of the identity in defines a homomorphism . Define .
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) 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 general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.
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