Embeddings just below the stable range: classification
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Contents |
1 Introduction
Recall the unknotting theorem that if is a connected manifold of dimension , then there is just one isotopy class of embedding if . In this page we summarise the situation for , and give references to the case .
For notation and conventions see high codimension embeddings.
2 Classification
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 Whitney invariant is defined below.
The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. For a description of for a closed -connected -manifold see a generalization to highly-connected manifolds. Some estimations of for a closed -connected -manifold are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.
3 Examples
Together with the Haefliger knotted sphere , Hudson's examples 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 the one below).
For define the as the composition of standard embeddings.
3.1 Hudson tori 1
In this subsection, we recall for and . Hudson's construction of embedings
Take the standard embeddings (where means homothety with coefficient 2) and . Fix a point . The Hudson torus is the embedded connected sum of
(Unlike the connected sum mentioned in embedded conntected sum this is a `linked' connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)
For instead of an embedded -sphere we can take copies () of -sphere outside `parallel' to . Then we join these spheres by tubes so that the homotopy class of the resulting embedding
Let be the connected sum of this embedding with the above standard embedding .
Clearly, is isotopic to the standard embedding.
Proposition 3.1. For odd is isotopic to if and only if .
For even is isotopic to if and only if .
In particular, is not isotopic to for each (this was the original motivation for Hudson).
Proposition 3.1 follows by Remark 5.e and, for even, by Theorem 4, both results from classification just below the stable range.
It would be interesting to find an explicit construction of an isotopy between and (cf. [Vrabec1977], \S6) and to prove the analogue of Proposition 3.1 for .
3.2 Hudson tori 2
In this subsection we give, for and another construction of embeddings
Define a map to be the constant on one component and the standard embedding on the other component. This map gives an
(See Figure 2.2 of [Skopenkov2006].) Each disk intersects the image of this embedding at two points lying in . Extend this embedding for each to an embedding . (See Figure 2.3 of [Skopenkov2006].) Thus we obtain the Hudson torus
The embedding is obtained in the same way starting from a map of degree .
The same proposition as above holds with replaced to .
3.3 Remarks
We have is PL isotopic to [Skopenkov2006a]. It would be interesting to prove the smooth analogue of this result.
For these construction give what we call the Hudson torus. The Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of .
Analogously one constructs the (left) Hudson torus for or, more generally, for and for .
3.4 An action of the first homology group on embeddings
In this subsection, for and for orientable with , we construct an embedding from an embedding .
For , 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 odd or N orientable)
Fix orientations on and, if is even, 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.
- 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
Examples are the above Hudson tori . See also [Milgram&Rees1971].
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] -connected manifolds. The proof works for -connected manifolds.
E.g. by Theorem 5.1 we obtain that the Whitney invariant is bijective for . It is in fact a group isomorphism; the generator is the Hudson torus.
The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for .
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 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 there 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 we obtain that the Whitney invariant is surjective and for each there is a 1-1 correspondence .
5.2 The Whitney invariant
Roughly speaking, is the homology class of the self-intersection set of a general position homotopy between and . We present an accurate definition for an orientable -manifold and in the smooth category [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 is a closed -submanifold. (Note that is orientable, whereas need not be orientable.) For odd it has a natural orientation. Define the Whitney invariant
by Analogously to [Skopenkov2006], \S2.4, this is well-defined. (It is for being well-defined that we need -coefficients when is even.)
6 An orientation on the self-intersection set
7 References
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