Embeddings just below the stable range: classification
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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.
Recall the Whitney-Wu Unknotting Theorem: if is a connected manifold of dimension , and , then every two embeddings are isotopic [Skopenkov2016c, Theorem 3.2], [Skopenkov2006, Theorem 2.5]. In this page we summarize the situation for and is a connected, as well as in some more general situations. For the classification of embeddings of some disconnected manifolds see [Skopenkov2016h].
Theorem 2.1. Assume that is a closed connected -manifold, and either or and we are in the PL category.
(a) If is oriented, the Whitney invariant,
is a 1-1 correspondence.
(b) If is non-orientable, then there is a 1-1 correspondence
Remark 2.2 (Comments on the proof). Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in 2], [Hudson1969, 11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].
In Part (b) we replaced the kernel from [Bausum1975, Theorem 43] by . This is possible because, as a specialist could see, is given by multiplication with the first Stefel-Whitney class (which equals to the first Wu class [Milnor&Stasheff1974, Theorem 11.4]). Since is non-orientable, . So by Poincaré duality, .
The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, 5]. It would be interesting to check if part (b) is equivalent to different forms of description of [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].
3 Hudson tori
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Example 3.1. Let us construct, for any and , a smooth embedding
We start with the cases .
Take the standard inclusion . The 'standard embedding' is given by the standard inclusions
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Tex syntax erroranalogously to
Tex syntax error, where means homothety with coefficient 2.
Take the embedding given by
Tex syntax error
Tex syntax errorjoins the images of and ; the interior of this segment misses the images. Let be the linked embedded connected sum of and along this segment, compatible with the orientation, cf. [Avvakumov2017, 1.5]. (Here 'linked' means that the images of the embeddings are not contained in disjoint cubes, unlike for the unlinked embedded connected sum [Skopenkov2016c, 5].)
For we repeat the above construction of 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 along natural segments joining every -th copy to the -th copy. We obtain an embedding which has disjoint images with . Let be the linked embedded connected sum of and .
The original motivation for Hudson was that is not isotopic to for any (this is a particular case of Proposition 3.2 below). One might guess that is not isotopic to for and that a -valued invariant of can be defined by the homotopy class of the map
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However, this is only true for odd.
Proposition 3.2. For odd is isotopic to if and only if .
For even is isotopic to if and only if .
Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for even, by Theorem 2.1. This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of for the PL category). It would be interesting to find an explicit construction of an isotopy between and , cf. [Vrabec1977, 5]. Analogously, is not isotopic to if . It would be interesting to know if the converse holds, e.g. is (PS or smoothly) isotopic to .
Example 3.3. Take any . Take a map of degree (so we can take ). Recall that . Define the smooth embedding to be the composition
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Let us present a geometric description of this embedding. Define a map by . This map gives an embedding
Tex syntax error
Tex syntax erroris the union of the graphs of the maps and . For any the disk
Tex syntax errorintersects the image at two points lying in
Tex syntax error, i.e., at the image of an embedding
Tex syntax error. The embedding is obtained by extending the latter embeddings to embeddings
Tex syntax errorfor all . See Figure 2.
Remark 3.4. (a) The analogue of Proposition 3.2 for replaced to holds, with an analogous proof.
(b) The embeddings and are smoothly isotopic for and are PS isotopic for [Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]). This follows by calculation of the Whitney invariant (Remark 5.3.d below). It would be interesting to know if they are smoothly isotopic for . It would be interesting to know if they are piecewise smoothly isotopic for .
(c) For Example 3.3 gives 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.d below.
(d) Analogously one constructs the Hudson torus for and or, more generally, for and . There are versions of these constructions corresponding to Definition 3.3. For this corresponds to the Zeeman map [Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a, 2.3]. It would be interesting to know if the links are isotopic, cf. [Skopenkov2015a, Remark 2.7.b]. These constructions can be further generalized [Skopenkov2016k].
In this section we generalize the construction of the Hudson torus . Let be a closed connected oriented -manifold. We work in the smooth category which we omit. Apparently analogous results hold for in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).
Example 4.1. For any , an embedding and , we shall construct an embedding . This embedding is said to be obtained by linked embedded connected sum of with an -sphere representing the `homology Alexander dual' of (defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).
Represent by an embedding . By definition, the class is represented by properly oriented . Since any orientable bundle over is trivial, . Take an embedding whose image is and which represents . By embedded surgery on we obtain an embedding representing (see details in Proposition 4.2 below). Define to be the linked embedded connected sum of and , along some arc joining their images.
Proposition 4.2 (Embedded surgery). For any , a neighborhood of a codimension at least 3 subpolyhedron in and an embedding there is an embedding homologous to .
Proof. Take a vector field on normal to . Extend along this vector field to a map .Since and is a neighborhood of a codimension at least 3 subpolyhedron, by general position we may assume that is an embedding and that
Tex syntax errormisses .
Since , we have . Hence the standard -framing of in extends to an -framing on in . Thus extends to an embedding
Take an embedding such that
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with proper orientation so that is homologous to . QED
The isotopy class of the embedding is independent of the choises in the construction. The independence of the arc and of the maps follows by and by Proposition 4.3 below, respectively.
5 The Whitney invariant
Let be a closed -manifold. Take an embedding . Fix an orientation on . For any other embedding we define the Whitney invariant
Here the coefficients are if is oriented and is odd, and are otherwise.Roughly speaking,
Tex syntax erroris defined as the homology class of the closure 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], see also [HaefligerHirsch1963]. The definition in the PL category is analogous [Hudson1969, 11], [Vrabec1977, p. 145], [Skopenkov2006, 2.4 `The Whitney invariant']. We begin by presenting a simpler definition, Definition 5.1, for a particular case.
For Theorem 2.1 only the case is required.
Definition 5.1. Assume that is -connected and . Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of and to are isotopic, cf. [Takase2006, Lemma 2.2]. (Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of and to are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since is -connected, retracts to an -dimensional polyhedron. Therefore these restrictions are isotopic.)
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
Tex syntax erroris a compact -manifold whose boundary is contained in .
So carries a homology class with coefficients. If is odd and is oriented, then has a natural orientation defined below, and so carries a homology class with coefficients. Define to be the homology class:
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Tex syntax error) is defined (for odd and is oriented) as follows (cf. Remark 7.3). For any point take a base at tangent to . Complete this base to a positive base tangent to . Since , by general position there is a unique point such that . The tangent base at thus gives a base at tangent to such that . Complete this base to a positive base tangent to , where the orientation on comes from . The union of the images of the constructed bases is a base at of . If is positive, then call the tangent base of `positive'. Since a change of the orientation on forces a change of the orientation of , this condition indeed defines an orientation on .
Definition 5.2. Assume that . Take a general position homotopy between and .The closure
Tex syntax errorof the self-intersection set carries a cycle mod 2. If is oriented and is odd, the closure also carries an integer cycle. See [Hudson1967, 11], [Skopenkov2006, 2.3 `The Whitney obstruction'].
Let us informally explain these facts. For by general position the closure
Tex syntax errorcan be assumed to be a submanifold. In general, since , by general position the closure has codimension 2 singularities, as defined in 7. So the closure carries a cycle mod 2. When is odd the closure also has a canonical orientation (see Definition 7.1 and Remark 7.2), so the closure carries an integer cycle.
Define the Whitney invariant to be the homology class:
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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.
Remark 5.3. (a) The Whitney invariant is well-defined by Definition 5.2, i.e. is independent of the choice of a general position homotopy from to .This follows from the equality
Tex syntax errorfor a general position homotopy between general position homotopies from to . See details in [Hudson1969, 11].
(b) Definition 5.1 is a particular case of Definition 5.2. (Indeed, if on , we can take to be fixed on . See details in [Skopenkov2010, Difference Lemma 2.4].) 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) The class is independent of the choice of the orientation on (because a change of the orientation on forces a change of the orientation on or on
Tex syntax error). For the reflection with respect to a hyperplane we have (because a change of the orientation on forces a change of the orientation on or on
Tex syntax error; for Definition 5.1 also observe that we may assume that on ).
(d) For the Hudson tori is or for , and .
For this is clear by Definition 5.1. For and this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For the proof is analogous.(e) for any pair of embeddings and . This is clear by Definition 5.1 because . Let us prove the latter equality. Take the identical isotopy of and a general position homotopy between and the standard embedding. Then the boundary connected sum is a general position homotopy between and an embedding isotopic to . The cycle
Tex syntax erroris null-homologous in and hence in ; cf. [Skopenkov2008, Addendum to the Classification Theorem].
(g) The Whitney invariant need not be a bijection for . This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking even, non-orientable, and applying by Theorem 2.1.b.
6 A generalization to highly-connected manifolds
In this section let be a closed orientable homologically -connected -manifold, . Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] 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 (cf. [Skopenkov2010, Definition 1.4]). Assume that is -connected and . Then for an embedding and a class one can construct an embedding by linked embedded connected sum analogously to the case presented in Example 4.1.
Theorem 6.2. Let be a closed oriented homologically -connected -manifold, . Then the Whitney invariant
is a bijection, provided in the smooth category or in the PL category.
This was proved for -connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in [Weber1967], [Hudson1969, 11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993, 7]. The proof actually used the homological -connectedness assumption (basically because the -connectedness was used to ensure high enough connectedness of the complement in to the image of , by Alexander duality and simple connectedness of the complement, so homological -connectedness of is sufficient).
For Theorem 6.2 is covered by Theorem 2.1; for it is not. For the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3]. For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].
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 any [Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2]. The following result for was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008, 4].
is surjective and for any 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 .
How does one describe when is not -connected? For general see the sentence on at the end of 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, Corollary 1.3]. Let be a closed -connected -manifold embeddable into , and . Then there is a 1-1 correspondence
The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, 5]. For Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because is -connected). For Theorem 6.4 is covered by Theorem 6.2; for it is not. For application to knotted tori see [Skopenkov2016k, Theorem 5.1]. For generalization to arbitrary manifolds see survey [Skopenkov2006, 5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4 can be replaced by for any .
7 An orientation on the self-intersection set
Tex syntax errorof the self-intersection set of has codimension 2 singularities, i.e., there is
Tex syntax errorsuch that
- both and
Tex syntax errorare subpolyhedra of some triangulation of ,
- we have and
- is an open manifold consisting of self-transverse double points of .
Definition 7.1 (A canonical orientation on ). Take points away from and such that . Then a -base tangent to at gives a -base tangent to at . Since is oriented, 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 .
Remark 7.2 (Properties of the orientation just defined on )..
- A change of the orientation of forces changes of the signs of and and so does not change the orientation of .
- The orientation on need not extend to
Tex syntax error: take the smooth cone over a general position map having only two transverse self-intersection points, where the smooth cone is defined by , for and .
- The orientation on extends to
Tex syntax errorif is odd [Hudson1969, Lemma 11.4].
Remark 7.3 (A canonical orientation on for even). This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.
Take a -base at a point . Since is oriented, 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 at . 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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