Embeddings just below the stable range: classification
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.
Recall the Whitney-Wu Unknotting Theorem: if is a connected manifold of dimension , and , then every two embeddings are isotopic [Skopenkov2016c, Theorem 3.2]. In this page we summarize the situation for and some more general situations.
Theorem 2.1. Assume that is a closed connected -manifold, either is odd or is orientable, and either or and we are in the PL category. The Whitney invariant,
is a 1-1 correspondence.
This is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorem 1.1] in the PL category.
If is even and is a closed connected non-orientable -manifold, then [Bausum1975, Theorem 43] asserts that there is a 1-1 correspondence
(because is given by multiplication with ). This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used the Haefliger-Wu invariant whose definition can be found in [Skopenkov2006, 5]. It would be interesting to check if this description of is equivalent to its different forms [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].
3 Hudson tori
Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)
For we define the standard embedding as the composition of the standard embeddings . (Note that in [Skopenkov2015], [Skopenkov2015a] a different embedding is called a standard embedding. Since these two embeddings are isotopic, no confusion would appear.)
Let us construct, for each and , an embedding
The reader might first consider the case .
Definition 3.1. Take the standard embeddings (where means homothety with coefficient 2) and . Take the embedded sphere and embedded torus,
Join them by an arc whose interior misses the two embedded manifolds. The Hudson torus is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the unlinked embedded connected sum [Skopenkov2016c] this is a linked embedded connected sum, i.e. the connected sum of two embeddings whose images are not contained in disjoint cubes.)
Definition 3.2. For we repeat the construction of Definition 3.1 replacing by copies () of . The copies are outside and are `parallel' to . The copies have the standard orientation for or the opposite orientation for . Then we make embedded connected sum by tubes joining each -th copy to the -th copy. We obtain an embedding . Let be the linked embedded connected sum of with the embedding from Definition 3.1.
Clearly, is isotopic to the standard embedding.
The original motivation for Hudson was that is not isotopic to for each (this is a particular case of Proposition 3.3 below).
One guesses that is not isotopic to for . And that a -valued invariant exists and is `realized' by the homotopy class of the map
However, this is only true for odd.
Proposition 3.3. For odd is isotopic to if and only if .
For even is isotopic to if and only if .
Analogously, is not isotopic to if . It would be interesting to know if the converse holds, e.g. is isotopic to ? It would also be interesting to find an explicit construction of an isotopy between and , cf. [Vrabec1977, 5].
Definition 3.4. Let us give, for and , another construction of embeddings
Define a map to be the constant on one component and the `standard inclusion' on the other component. This map gives an embedding
(See [Skopenkov2006, Figure 2.2]. The image of this embedding is the union of the standard and the graph of the identity map in .)
Take any . The disk intersects the image of this embedding at two points lying in , i.e., at the image of an embedding . Extend the latter embedding to an embedding . See [Skopenkov2006, Figure 2.3].) Thus we obtain the Hudson torus
Here , where is identified with .
The embedding is obtained in the same way starting from a map of degree instead of the `standard inclusion'.
Remark 3.5. (a) The analogue of Proposition 3.3 for replaced to holds, with an analogous proof.
(b) The embeddings and are smoothly isotopic for and are PL isotopic for [Skopenkov2006a]. It would be interesting to know if they are isotopic for , or if they are smoothly isotopic for .
(c) For these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of . The right and the left Hudson tori are not isotopic by Remark 5.3.e below.
(d) Analogously one constructs the Hudson torus for or, more generally, for . There are versions of these constructions corresponding to Definition 3.4. For this corresponds to the Zeeman construction [Skopenkov2016h] and its composition with the second unframed Kirby move. It would be interesting to know if the links are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions can be further generalized [Skopenkov2016k].
In this section we generalize the construction of the Hudson torus . For , a closed connected orientable -manifold , an embedding and , we construct an embedding . This embedding is obtained by linked embedded connected sum of with an -sphere representing the homology Alexander dual of .
More precisely, represent by an embedding . Since any orientable bundle over is trivial, . Identify with . In the next paragraph we recall definition of embedded surgery on , which yields an embedding . Then we define to be the (linked) embedded connected sum of and (along a certain arc joining their images).
Take a vector field on normal to . Extend along this vector field to a map . Since and , by general position we may assume that is an embedding and misses . Since , we have . Hence the standard framing of in extends to an -framing on in . Thus extends to an embedding
Define an embedding by setting
with natural orientation.
Proposition 4.1. Linked embedded connected sum, or parametric connected sum, define free transitive actions of on , unless in the smooth category.
5 The Whitney invariant
Let be a closed -manifold and fix an embedding . For any other embedding , there is an invariant , called the Whitney invariant, which we define in this section. Roughly speaking, is defined as the homology class of the self-intersection set of a general position homotopy between and . This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, 12], [Vrabec1977, p. 145], [Skopenkov2006, 2.4]. We begin by presenting a simpler definition of the Whitney invariant, Definition 5.1, for a particular case. For Theorem 2.1 only the case is required.
Fix an orientation on . Assume that either is even or is oriented.
Definition 5.1. Assume that is -connected and . Then restrictions of and to are regular homotopic [Hirsch1959]. Since is -connected, retracts to an -dimensional polyhedron. Therefore these restrictions are isotopic, cf. [Haefliger&Hirsch1963, 3.1.b], [Takase2006, Lemma 2.2]. So we can make an isotopy of and assume that on . Take a general position homotopy relative to between the restrictions of and to . Let (`the intersection of this homotopy with '). Since , by general position is a compact -manifold whose boundary is contained in . So carries a homology class with coefficients. For odd it has a natural orientation defined below, and so carries a homology class with coefficients. Define to be the homology class:
The orientation on (extendable to ) is defined for odd as follows. For each point take a vector at tangent to . Complete this vector to a positive base tangent to . Since , by general position there is a unique point such that . The tangent vector at thus gives a tangent vector at to . Complete this vector to a positive base tangent to , where the orientation on comes from . The union of the images of the constructed two bases is a base at of . If the latter base is positive, then call the initial vector of positive. Since a change of the orientation on forces a change of the orientation of the latter base of , this condition indeed defines an orientation on .
Definition 5.2. Assume that . Take a general position homotopy between and . Since , by general position the closure of the self-intersection set has codimension 2 singularities. So the closure carries a homology class with coefficients. (Note that can be assumed to be a submanifold for .) For odd it has a natural orientation (7) and so carries a homology class with coefficients. Define the Whitney invariant to be the homology class:
(The orientation is defined for each but used only for odd . When is even, for being well-defined we need -coefficients.)
Clearly, if is isotopic to . Hence the Whitney invariant defines a map
Clearly, (for both definitions).
The definition of depends on the choice of , but we write not for brevity.
(b) Definition 5.1 is a particular case of Definition 5.2. (Indeed, if on , we can take to be fixed on .) Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice of and of the isotopy making outside .
(c) Since a change of the orientation on forces a change of the orientation on , the class is independent of the choice of the orientation on . For the reflection with respect to a hyperplane we have (because we may assume that on and because a change of the orientation of forces a change of the orientation of ).
(d) For the Hudson tori is or for , is for .
(e) for each pair of embeddings and .
6 A generalization to highly-connected manifolds
In this section let be a closed orientable homologically -connected -manifold, . Recall the unknotting theorem [Skopenkov2016c] that all embeddings are isotopic when and . In this section we generalize Theorem 2.1 to a description of and further to for .
Some simple examples are the Hudson tori .
Example 6.1 (Linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]). If is -connected, then for an embedding and a class one can construct an embedding by linked connected sum analogously to the case .
We have for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of on (provided or in the PL or smooth categories, respectively). If , then this construction gives only a construction of embeddings for each but not a well-defined action of on .
6.2 Classification just below the stable range
Theorem 6.2. Let be a closed orientable homologically -connected -manifold, . Then the Whitney invariant
is a bijection, provided in the smooth category or in the categoriy.
This was proved for -connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, 11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof of [Haefliger&Hirsch1963], [Boechat&Haefliger1970], [Boechat1971] actually used the homological -connectedness assumption.
By Theorem 6.2 the Whitney invariant is bijective for . It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]). The Hudson torus generates for ; this holds by Theorem 6.2 because . Also, for by Theorem 6.2 the Whitney invariants
6.3 Classification in the presence of smoothly knotted spheres
Because of the existence of knotted spheres the analogues of Theorem 6.2 for in the PL case, and for in the smooth case are false.So for the smooth category, and closed connected, a classification of is much harder: for 40 years the only known complete readily calculable classification results were for homology spheres . E.g.
for each [Haefliger1966]. The following result for was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].
is surjective and for each the Kreck invariant
is a 1-1 correspondence, where is the divisibility of the projection of to the free part of .
Recall that the divisibility of zero is zero and the divisibility of is .
E.g. by Theorem 6.3 the Whitney invariant is surjective and for each there is a 1-1 correspondence .
6.4 Classification further below the stable range
How does one describe when is not -connected? For general see the remarks on in 2. We can say more as the connectivity of increases. Some estimations of for a closed -connected -manifold are presented in [Skopenkov2010]. For one can go even further:
Theorem 6.4 [Becker&Glover1971]. Let be a closed -connected -manifold embeddable into , and . Then there is a 1-1 correspondence
Observe that in Theorem 6.4 can be replaced by for each .
7 An orientation on the self-intersection set
Let be a general position smooth map of an oriented -manifold . Assume that so that the closure of the self-intersection set of has codimension 2 singularities.
Definition 7.1 (a natural orientation on ). Take points away from the singularities of and such that . Then a -base tangent to at gives a -base tangent to at . Since is orientable, we can take positive -bases and at and normal to and to . If the base of is positive, then call the base positive. This is well-defined because a change of the sign of forces changes of the signs of and .
We remark that
- a change of the orientation of forces changes of the signs of and and so does not change the orientation of .
- the natural orientation on need not extend to : take the cone over a general position map having only one self-intersection point.
- the natural orientation on extends to if is odd [Hudson1969, Lemma 11.4].
Definition 7.2 (a natural orientation on for even). Take a -base at a point away from the singularities of . Since is orientable, we can take a positive -base normal to in one sheet of . Analogously construct an -base for the other sheet of . Since is even, the orientation of the base of does not depend on choosing the first and the other sheet of . If the base is positive, then call the base positive. This is well-defined because a change of the sign of forces changes of the signs of and so of .
We remark that a change of the orientation of forces changes of the signs of and so does not change the orientation of .
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