Embeddings just below the stable range: classification
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1 Introduction
This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn theory of embeddings.
Recall the unknotting theorem that if is a connected manifold of dimension , and , then every two embeddings are isotopic. In this page we summarise the situation for and some more general situations.
See general introduction on embeddings, notation and conventions.
2 Classification
Theorem 2.1. Let be a closed connected -manifold. The Whitney invariant
is bijective if either or and CAT=PL.
This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).
The classification of smooth embeddings of 3-manifolds in is more complicated.
For embeddings of -manifolds in see the case of 4-manifolds, [Yasui1984], for and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.
Theorem 2.1 is generalized to a description of for closed -connected -manifolds .
3 Examples
Together with the Haefliger knotted sphere, examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction [Hudson1963] was not as explicit as those below.)
For we define the standard embedding as the composition of standard embeddings.
3.1 Hudson tori
In this subsection we construct, for and , an embedding
The reader might first consider the case .
Definition 3.1. (This construction, as opposed to Definition 3.2, works for .) Take the standard embeddings (where means homothety with coefficient 2) and . Let . Take 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 this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)
Definition 3.2. For instead of we take copies () of -sphere outside `parallel' to , with standard orientation for or the opposite orientation for . Then we make embedded connected sum by tubes joining each -th copy to -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 .
Proposition 3.3 follows by calculation of the Whitney invariant (the fifth part of Remark 4.1 below) and, for even, by Theorem 2.1. 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], \S6).
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 Figure 2.2 of [Skopenkov2006]. 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 by two points lying in , i.e., by the image of an embedding . Extend the latter embedding to an embedding . (See Figure 2.3 of [Skopenkov2006].) 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 analogous proof.
(b) Embeddings and are smoothly isotopic for and are PL isotopic for [Skopenkov2006a]. It would be interesting to know if they are isotopic for , or 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
(d) Analogously one constructs the (left or right) Hudson torus for or, more generally, for . This is further generalized by generalized Zeeman construction. There are versions of these constructions corresponding to Definition 3.4.
3.2 Action by linked embedded connected sum
In this subsection, for and for a closed connected orientable -manifold with , we construct an embedding from an embedding .
Represent by an embedding . Since any orientable bundle over is trivial, . Identify with . It remains to make an embedded surgery of to obtain an -sphere , and then we set .
Take a vector field on normal to . Extend along this vector field to a smooth 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
This construction generalizes the construction of (from ).
Clearly, is or . Thus unless and CAT=DIFF
- all isotopy classes of embedings can be obtained (from a certain given embedding ) by the above construction;
- the above construction defines an action .
4 The Whitney invariant (for either n even or N orientable)
Fix orientations on and, if is odd, on . Fix an embedding . For an embedding the restrictions of and to are regular homotopic [Hirsch1959]. Since has an -dimensional spine, it follows that 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 . Then (i.e. `the intersection of this homotopy with ') is a 1-manifold (possibly non-compact) without boundary. Define to be the homology class of the closure of this 1-manifold:
The orientation on is defined for orientable as follows. (This orientation is defined for each but used only for odd .) 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 this 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 , it follows that this condition indeed defines an orientation on .
Remark 4.1.
- The Whitney invariant is well-defined, i.e. independent of the choice of and of the isotopy making outside . This is so because the above definition is clearly equivalent to an alternative one. It is for being well-defined that we need -coefficients when is even.
- Clearly, . The definition of depends on the choice of , but we write not for brevity.
- 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 ).
- The above definition makes sense for each , not only for .
- Clearly, is or for for the Hudson tori.
- for each embeddings and .
5 A generalization to highly-connected manifolds
Let be a closed -connected -manifold. We present description of generalizing Theorem 2.1, and its generalization to for .
Examples are Hudson tori .
5.1 Classification
Theorem 5.1. Let be a closed orientable homologically -connected -manifold, . Then the Whitney invariant
is a bijection, provided or in the PL or DIFF categories, respectively.
Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically -connected manifolds. The proof works for homologically -connected manifolds.
For this is covered by Theorem 2.1; for it is not. The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for .
E.g. by Theorem 5.1 the Whitney invariant is bijective for . It is in fact a group isomorphism; the generator is the Hudson torus.
Because of the existence of knots the analogues of Theorem 5.1 for in the PL case, and for in the smooth case are false. So for the smooth category and a classification is much harder: for 40 years the only known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].
Theorem 5.2. [Skopenkov2008] Let be a closed homologically -connected -manifold. Then the Whitney invariant
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 5.2 the Whitney invariant is surjective and for each there is a 1-1 correspondence .
Theorem 5.3. [Becker&Glover1971] Let be a closed -connected -manifold embeddable into , and . Then there is a 1-1 correspondence
For this is covered by Theorem 5.1; for it is not.
E.g. by Theorem 5.3 there is a 1-1 correspondence , , for , and . For a generalization see Knotted tori [Skopenkov2002].
Some estimations of for a closed -connected -manifold are presented in [Skopenkov2010].
5.2 The Whitney invariant
Let be an -manifold and embeddings. Roughly speaking, is defined as the homology class of the self-intersection set of a general position homotopy between and . We present an accurate definition in the smooth category for when either is even or is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.
Fix orientations on and on . Take embeddings . Take a general position homotopy between and . By general position the closure of the self-intersection set has codimension 2 singularities and so carries a homology class with coefficients. (Note that can be assumed to be a submanifold for .) For odd it has a natural orientation and so carries a homology class with coefficients. Define the Whitney invariant
by Analogously to [Skopenkov2006], \S2.4, this is well-defined.
6 An orientation on the self-intersection set
Let be a general position smooth map of an orientable -manifold . Assume that so that the closure of the self-intersection set of has codimension 2 singularities. Then
- (1) has a natural orientation.
- (2) the natural orientation on need not extend to .
- (3) the natural orientation on extend to if is odd [Hudson1969], Lemma 11.4.
- (4) has a natural orientation if is even.
Fix an orientation on and on .
Let us prove (1). Take points outside 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 . (Note that a change of the orientation of forces changes of the signs of and and so does not change the orientation of .)
We can see that (2) holds by considering the cone over a general position map having only one self-intersection point.
Let us prove (4). Take a -base at a point outside 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 . If is even, then 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 . (Note 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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