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 the theory of embeddings.
Recall the unknotting theorem that if is a connected manifold of dimension , and , then every two embeddings are isotopic [Skopenkov2016c, 3]. In this page we summarise the situation for and some more general situations.
See general introduction on embeddings, notation and conventions in [Skopenkov2016c, 1, 2].
2 Classification
Theorem 2.1. Let be a closed connected -manifold. The Whitney invariant [Skopenkov2016e]
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 [Skopenkov2016t].
For embeddings of -manifolds in see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.
Theorem 2.1 is generalized in 5 to a description of for closed -connected -manifolds .
3 Examples
Together with the Haefliger knotted sphere [Skopenkov2016t], 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.
Let .
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 . 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 [Skopenkov2016c] 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 (Remark 4.3.e 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, 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 by two points lying in , i.e., by 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 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 4.3.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 links are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions could be further generalized [Skopenkov2016k].
3.2 Action by linked embedded connected sum
In this subsection we generalize the construction of 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 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 of which yields an embedding . Then we define to be the (linked) embedded connected sum of and (along 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 embedding by setting
with natural orientation.
By Definition 4.1 of the Whitney invariant, is or . Thus
- all isotopy classes of embeddings can be obtained from any chosen embedding by the above construction;
- linked embedded connected sum defines a free transitive action of on , unless and CAT=DIFF.
Parametric connected sum also defines a free transitive action of on for [Skopenkov2014, Remark 18.a].
4 The Whitney invariant
Let be a closed -manifold and embeddings. 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 4.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, 12], [Vrabec1977, p. 145], [Skopenkov2006, 2.4]. Before we present a simpler Definition 4.1 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 4.1. Assume that is -connected and . Then restrictions of and to are regular homotopic [Hirsch1959]. Since is -connected, has an -dimensional spine. 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 4.2. 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 (6) and so carries a homology class with coefficients. Define the Whitney invariant to be the homology class:
Clearly, (for both definitions).
The definition of depends on the choice of , but we write not for brevity.
Remark 4.3. (a) The Whitney invariant is well-defined by Definition 4.2, i.e. independent of the choice of , analogously to [Skopenkov2006, 2.4]. (The orientation is defined for each but used only for odd . When is even, for being well-defined we need -coefficients.)
(b) Definition 4.1 is equivalent to Definition 4.2. (Indeed, if on , we can take to be fixed on .) Hence the Whitney invariant is well-defined by Definition 4.1, i.e. independent of the choice of and of the isotopy making outside .
(c) Clearly, is not changed throughout isotopy of . Hence it gives a map .
(d) 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 ).
(e) For the Hudson tori is or for , is for .
(f) for each embeddings and .
(e) For the Whitney invariant is the collection of pairwise linking coefficients of the components of , cf. Definition 3.2 for and .
5 A generalization to highly-connected manifolds
Let be a closed -connected -manifold. Recall the unknotting theorem [Skopenkov2016c] that every two embeddings are isotopic when and . In this section we present description of generalizing Theorem 2.1, and its generalization to for .
Examples are Hudson tori . An action of on for is defined either by analogue of linked embedded connected sum [Skopenkov2016e], cf. [Skopenkov2010, Definition 1.4], or by parametric connected sum [Skopenkov2014, Remark 18.a]. Both these actions are free transitive by Theorem 5.1 below. If , then analogue of linked embedded connected sum gives only a construction of embeddings for each but not a well-defined action of on .
5.1 Classification just below the stable range
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, 11], [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 [Skopenkov2016c] for .
By Theorem 5.1 the Whitney invariant is bijective for . It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015]). The generator is Hudson torus . Also, for by Theorem 5.1 the Whitney invariants
are bijective. In the smooth category for even is not injective (see the next subsection), and is not surjective [Boechat1971], and is not injective [Skopenkov2016f].
5.2 Classification in the presence of smoothly knotted spheres
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 closed connected, a classification of is much harder: for 40 years the only known complete readily calculable classification results were for homology sphere . The following result for 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 analogue of 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 .
5.3 Classification further below the stable range
How to describe for ? See remarks on in . Some estimations of for a closed -connected -manifold are presented in [Skopenkov2010]. For one can go even further:
Theorem 5.3 [Becker&Glover1971]. Let be a closed -connected -manifold embeddable into , and . Then there is a 1-1 correspondence
For this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because is -connected). 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 [Skopenkov2016k] (to knotted tori) and [Skopenkov2002].
Observe that in Theorem 5.3 can be replaced by for each .
6 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. Then
- (1) has a natural orientation.
- (2) the natural orientation on need not extend to : take the cone over a general position map having only one self-intersection point.
- (3) the natural orientation on extend to if is odd [Hudson1969], Lemma 11.4.
- (4) has a natural orientation if is even.
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 .)
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 . 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 . (Note that a change of the orientation of forces changes of the signs of and so does not change the orientation of .)
7 References
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- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension dans , (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
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- [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
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- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
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arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
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See general introduction on embeddings, notation and conventions in [Skopenkov2016c, 1, 2].
2 Classification
Theorem 2.1. Let be a closed connected -manifold. The Whitney invariant [Skopenkov2016e]
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 [Skopenkov2016t].
For embeddings of -manifolds in see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.
Theorem 2.1 is generalized in 5 to a description of for closed -connected -manifolds .
3 Examples
Together with the Haefliger knotted sphere [Skopenkov2016t], 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.
Let .
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 . 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 [Skopenkov2016c] 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 (Remark 4.3.e 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, 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 by two points lying in , i.e., by 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 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 4.3.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 links are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions could be further generalized [Skopenkov2016k].
3.2 Action by linked embedded connected sum
In this subsection we generalize the construction of 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 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 of which yields an embedding . Then we define to be the (linked) embedded connected sum of and (along 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 embedding by setting
with natural orientation.
By Definition 4.1 of the Whitney invariant, is or . Thus
- all isotopy classes of embeddings can be obtained from any chosen embedding by the above construction;
- linked embedded connected sum defines a free transitive action of on , unless and CAT=DIFF.
Parametric connected sum also defines a free transitive action of on for [Skopenkov2014, Remark 18.a].
4 The Whitney invariant
Let be a closed -manifold and embeddings. 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 4.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, 12], [Vrabec1977, p. 145], [Skopenkov2006, 2.4]. Before we present a simpler Definition 4.1 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 4.1. Assume that is -connected and . Then restrictions of and to are regular homotopic [Hirsch1959]. Since is -connected, has an -dimensional spine. 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 4.2. 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 (6) and so carries a homology class with coefficients. Define the Whitney invariant to be the homology class:
Clearly, (for both definitions).
The definition of depends on the choice of , but we write not for brevity.
Remark 4.3. (a) The Whitney invariant is well-defined by Definition 4.2, i.e. independent of the choice of , analogously to [Skopenkov2006, 2.4]. (The orientation is defined for each but used only for odd . When is even, for being well-defined we need -coefficients.)
(b) Definition 4.1 is equivalent to Definition 4.2. (Indeed, if on , we can take to be fixed on .) Hence the Whitney invariant is well-defined by Definition 4.1, i.e. independent of the choice of and of the isotopy making outside .
(c) Clearly, is not changed throughout isotopy of . Hence it gives a map .
(d) 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 ).
(e) For the Hudson tori is or for , is for .
(f) for each embeddings and .
(e) For the Whitney invariant is the collection of pairwise linking coefficients of the components of , cf. Definition 3.2 for and .
5 A generalization to highly-connected manifolds
Let be a closed -connected -manifold. Recall the unknotting theorem [Skopenkov2016c] that every two embeddings are isotopic when and . In this section we present description of generalizing Theorem 2.1, and its generalization to for .
Examples are Hudson tori . An action of on for is defined either by analogue of linked embedded connected sum [Skopenkov2016e], cf. [Skopenkov2010, Definition 1.4], or by parametric connected sum [Skopenkov2014, Remark 18.a]. Both these actions are free transitive by Theorem 5.1 below. If , then analogue of linked embedded connected sum gives only a construction of embeddings for each but not a well-defined action of on .
5.1 Classification just below the stable range
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, 11], [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 [Skopenkov2016c] for .
By Theorem 5.1 the Whitney invariant is bijective for . It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015]). The generator is Hudson torus . Also, for by Theorem 5.1 the Whitney invariants
are bijective. In the smooth category for even is not injective (see the next subsection), and is not surjective [Boechat1971], and is not injective [Skopenkov2016f].
5.2 Classification in the presence of smoothly knotted spheres
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 closed connected, a classification of is much harder: for 40 years the only known complete readily calculable classification results were for homology sphere . The following result for 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 analogue of 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 .
5.3 Classification further below the stable range
How to describe for ? See remarks on in . Some estimations of for a closed -connected -manifold are presented in [Skopenkov2010]. For one can go even further:
Theorem 5.3 [Becker&Glover1971]. Let be a closed -connected -manifold embeddable into , and . Then there is a 1-1 correspondence
For this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because is -connected). 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 [Skopenkov2016k] (to knotted tori) and [Skopenkov2002].
Observe that in Theorem 5.3 can be replaced by for each .
6 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. Then
- (1) has a natural orientation.
- (2) the natural orientation on need not extend to : take the cone over a general position map having only one self-intersection point.
- (3) the natural orientation on extend to if is odd [Hudson1969], Lemma 11.4.
- (4) has a natural orientation if is even.
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 .)
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 . 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 . (Note that a change of the orientation of forces changes of the signs of and so does not change the orientation of .)
7 References
- [Bausum1975] D. R. Bausum, Embeddings and immersions of manifolds in Euclidean space, Trans. Amer. Math. Soc. 213 (1975), 263–303. MR0474330 (57 #13976) Zbl 0323.57017
- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension dans , (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
- [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension dans , Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
- [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
- [Haefliger1962b] A. Haefliger, Plongements de variétés dans le domain stable, Séminaire Bourbaki, 245 (1962).
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.
arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1977] J. Vrabec, Knotting a -connected closed -manifold in , Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
- [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
- [Yasui1984] T. Yasui, Enumerating embeddings of -manifolds in Euclidean -space, J. Math. Soc. Japan 36 (1984), no.4, 555–576. MR759414 Zbl 0557.57019
See general introduction on embeddings, notation and conventions in [Skopenkov2016c, 1, 2].
2 Classification
Theorem 2.1. Let be a closed connected -manifold. The Whitney invariant [Skopenkov2016e]
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 [Skopenkov2016t].
For embeddings of -manifolds in see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.
Theorem 2.1 is generalized in 5 to a description of for closed -connected -manifolds .
3 Examples
Together with the Haefliger knotted sphere [Skopenkov2016t], 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.
Let .
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 . 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 [Skopenkov2016c] 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 (Remark 4.3.e 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, 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 by two points lying in , i.e., by 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 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 4.3.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 links are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions could be further generalized [Skopenkov2016k].
3.2 Action by linked embedded connected sum
In this subsection we generalize the construction of 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 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 of which yields an embedding . Then we define to be the (linked) embedded connected sum of and (along 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 embedding by setting
with natural orientation.
By Definition 4.1 of the Whitney invariant, is or . Thus
- all isotopy classes of embeddings can be obtained from any chosen embedding by the above construction;
- linked embedded connected sum defines a free transitive action of on , unless and CAT=DIFF.
Parametric connected sum also defines a free transitive action of on for [Skopenkov2014, Remark 18.a].
4 The Whitney invariant
Let be a closed -manifold and embeddings. 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 4.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, 12], [Vrabec1977, p. 145], [Skopenkov2006, 2.4]. Before we present a simpler Definition 4.1 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 4.1. Assume that is -connected and . Then restrictions of and to are regular homotopic [Hirsch1959]. Since is -connected, has an -dimensional spine. 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 4.2. 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 (6) and so carries a homology class with coefficients. Define the Whitney invariant to be the homology class:
Clearly, (for both definitions).
The definition of depends on the choice of , but we write not for brevity.
Remark 4.3. (a) The Whitney invariant is well-defined by Definition 4.2, i.e. independent of the choice of , analogously to [Skopenkov2006, 2.4]. (The orientation is defined for each but used only for odd . When is even, for being well-defined we need -coefficients.)
(b) Definition 4.1 is equivalent to Definition 4.2. (Indeed, if on , we can take to be fixed on .) Hence the Whitney invariant is well-defined by Definition 4.1, i.e. independent of the choice of and of the isotopy making outside .
(c) Clearly, is not changed throughout isotopy of . Hence it gives a map .
(d) 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 ).
(e) For the Hudson tori is or for , is for .
(f) for each embeddings and .
(e) For the Whitney invariant is the collection of pairwise linking coefficients of the components of , cf. Definition 3.2 for and .
5 A generalization to highly-connected manifolds
Let be a closed -connected -manifold. Recall the unknotting theorem [Skopenkov2016c] that every two embeddings are isotopic when and . In this section we present description of generalizing Theorem 2.1, and its generalization to for .
Examples are Hudson tori . An action of on for is defined either by analogue of linked embedded connected sum [Skopenkov2016e], cf. [Skopenkov2010, Definition 1.4], or by parametric connected sum [Skopenkov2014, Remark 18.a]. Both these actions are free transitive by Theorem 5.1 below. If , then analogue of linked embedded connected sum gives only a construction of embeddings for each but not a well-defined action of on .
5.1 Classification just below the stable range
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, 11], [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 [Skopenkov2016c] for .
By Theorem 5.1 the Whitney invariant is bijective for . It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015]). The generator is Hudson torus . Also, for by Theorem 5.1 the Whitney invariants
are bijective. In the smooth category for even is not injective (see the next subsection), and is not surjective [Boechat1971], and is not injective [Skopenkov2016f].
5.2 Classification in the presence of smoothly knotted spheres
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 closed connected, a classification of is much harder: for 40 years the only known complete readily calculable classification results were for homology sphere . The following result for 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 analogue of 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 .
5.3 Classification further below the stable range
How to describe for ? See remarks on in . Some estimations of for a closed -connected -manifold are presented in [Skopenkov2010]. For one can go even further:
Theorem 5.3 [Becker&Glover1971]. Let be a closed -connected -manifold embeddable into , and . Then there is a 1-1 correspondence
For this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because is -connected). 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 [Skopenkov2016k] (to knotted tori) and [Skopenkov2002].
Observe that in Theorem 5.3 can be replaced by for each .
6 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. Then
- (1) has a natural orientation.
- (2) the natural orientation on need not extend to : take the cone over a general position map having only one self-intersection point.
- (3) the natural orientation on extend to if is odd [Hudson1969], Lemma 11.4.
- (4) has a natural orientation if is even.
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 .)
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 . 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 . (Note that a change of the orientation of forces changes of the signs of and so does not change the orientation of .)
7 References
- [Bausum1975] D. R. Bausum, Embeddings and immersions of manifolds in Euclidean space, Trans. Amer. Math. Soc. 213 (1975), 263–303. MR0474330 (57 #13976) Zbl 0323.57017
- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension dans , (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
- [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension dans , Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
- [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
- [Haefliger1962b] A. Haefliger, Plongements de variétés dans le domain stable, Séminaire Bourbaki, 245 (1962).
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.
arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1977] J. Vrabec, Knotting a -connected closed -manifold in , Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
- [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
- [Yasui1984] T. Yasui, Enumerating embeddings of -manifolds in Euclidean -space, J. Math. Soc. Japan 36 (1984), no.4, 555–576. MR759414 Zbl 0557.57019
See general introduction on embeddings, notation and conventions in [Skopenkov2016c, 1, 2].
2 Classification
Theorem 2.1. Let be a closed connected -manifold. The Whitney invariant [Skopenkov2016e]
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 [Skopenkov2016t].
For embeddings of -manifolds in see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.
Theorem 2.1 is generalized in 5 to a description of for closed -connected -manifolds .
3 Examples
Together with the Haefliger knotted sphere [Skopenkov2016t], 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.
Let .
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 . 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 [Skopenkov2016c] 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 (Remark 4.3.e 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, 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 by two points lying in , i.e., by 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 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 4.3.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 links are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions could be further generalized [Skopenkov2016k].
3.2 Action by linked embedded connected sum
In this subsection we generalize the construction of 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 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 of which yields an embedding . Then we define to be the (linked) embedded connected sum of and (along 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 embedding by setting
with natural orientation.
By Definition 4.1 of the Whitney invariant, is or . Thus
- all isotopy classes of embeddings can be obtained from any chosen embedding by the above construction;
- linked embedded connected sum defines a free transitive action of on , unless and CAT=DIFF.
Parametric connected sum also defines a free transitive action of on for [Skopenkov2014, Remark 18.a].
4 The Whitney invariant
Let be a closed -manifold and embeddings. 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 4.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, 12], [Vrabec1977, p. 145], [Skopenkov2006, 2.4]. Before we present a simpler Definition 4.1 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 4.1. Assume that is -connected and . Then restrictions of and to are regular homotopic [Hirsch1959]. Since is -connected, has an -dimensional spine. 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 4.2. 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 (6) and so carries a homology class with coefficients. Define the Whitney invariant to be the homology class:
Clearly, (for both definitions).
The definition of depends on the choice of , but we write not for brevity.
Remark 4.3. (a) The Whitney invariant is well-defined by Definition 4.2, i.e. independent of the choice of , analogously to [Skopenkov2006, 2.4]. (The orientation is defined for each but used only for odd . When is even, for being well-defined we need -coefficients.)
(b) Definition 4.1 is equivalent to Definition 4.2. (Indeed, if on , we can take to be fixed on .) Hence the Whitney invariant is well-defined by Definition 4.1, i.e. independent of the choice of and of the isotopy making outside .
(c) Clearly, is not changed throughout isotopy of . Hence it gives a map .
(d) 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 ).
(e) For the Hudson tori is or for , is for .
(f) for each embeddings and .
(e) For the Whitney invariant is the collection of pairwise linking coefficients of the components of , cf. Definition 3.2 for and .
5 A generalization to highly-connected manifolds
Let be a closed -connected -manifold. Recall the unknotting theorem [Skopenkov2016c] that every two embeddings are isotopic when and . In this section we present description of generalizing Theorem 2.1, and its generalization to for .
Examples are Hudson tori . An action of on for is defined either by analogue of linked embedded connected sum [Skopenkov2016e], cf. [Skopenkov2010, Definition 1.4], or by parametric connected sum [Skopenkov2014, Remark 18.a]. Both these actions are free transitive by Theorem 5.1 below. If , then analogue of linked embedded connected sum gives only a construction of embeddings for each but not a well-defined action of on .
5.1 Classification just below the stable range
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, 11], [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 [Skopenkov2016c] for .
By Theorem 5.1 the Whitney invariant is bijective for . It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015]). The generator is Hudson torus . Also, for by Theorem 5.1 the Whitney invariants
are bijective. In the smooth category for even is not injective (see the next subsection), and is not surjective [Boechat1971], and is not injective [Skopenkov2016f].
5.2 Classification in the presence of smoothly knotted spheres
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 closed connected, a classification of is much harder: for 40 years the only known complete readily calculable classification results were for homology sphere . The following result for 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 analogue of 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 .
5.3 Classification further below the stable range
How to describe for ? See remarks on in . Some estimations of for a closed -connected -manifold are presented in [Skopenkov2010]. For one can go even further:
Theorem 5.3 [Becker&Glover1971]. Let be a closed -connected -manifold embeddable into , and . Then there is a 1-1 correspondence
For this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because is -connected). 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 [Skopenkov2016k] (to knotted tori) and [Skopenkov2002].
Observe that in Theorem 5.3 can be replaced by for each .
6 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. Then
- (1) has a natural orientation.
- (2) the natural orientation on need not extend to : take the cone over a general position map having only one self-intersection point.
- (3) the natural orientation on extend to if is odd [Hudson1969], Lemma 11.4.
- (4) has a natural orientation if is even.
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 .)
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 . 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 . (Note that a change of the orientation of forces changes of the signs of and so does not change the orientation of .)
7 References
- [Bausum1975] D. R. Bausum, Embeddings and immersions of manifolds in Euclidean space, Trans. Amer. Math. Soc. 213 (1975), 263–303. MR0474330 (57 #13976) Zbl 0323.57017
- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension dans , (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
- [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension dans , Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
- [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
- [Haefliger1962b] A. Haefliger, Plongements de variétés dans le domain stable, Séminaire Bourbaki, 245 (1962).
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.
arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1977] J. Vrabec, Knotting a -connected closed -manifold in , Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
- [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
- [Yasui1984] T. Yasui, Enumerating embeddings of -manifolds in Euclidean -space, J. Math. Soc. Japan 36 (1984), no.4, 555–576. MR759414 Zbl 0557.57019
See general introduction on embeddings, notation and conventions in [Skopenkov2016c, 1, 2].
2 Classification
Theorem 2.1. Let be a closed connected -manifold. The Whitney invariant [Skopenkov2016e]
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 [Skopenkov2016t].
For embeddings of -manifolds in see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.
Theorem 2.1 is generalized in 5 to a description of for closed -connected -manifolds .
3 Examples
Together with the Haefliger knotted sphere [Skopenkov2016t], 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.
Let .
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 . 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 [Skopenkov2016c] 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 (Remark 4.3.e 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, 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 by two points lying in , i.e., by 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 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 4.3.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 links are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions could be further generalized [Skopenkov2016k].
3.2 Action by linked embedded connected sum
In this subsection we generalize the construction of 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 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 of which yields an embedding . Then we define to be the (linked) embedded connected sum of and (along 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 embedding by setting
with natural orientation.
By Definition 4.1 of the Whitney invariant, is or . Thus
- all isotopy classes of embeddings can be obtained from any chosen embedding by the above construction;
- linked embedded connected sum defines a free transitive action of on , unless and CAT=DIFF.
Parametric connected sum also defines a free transitive action of on for [Skopenkov2014, Remark 18.a].
4 The Whitney invariant
Let be a closed -manifold and embeddings. 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 4.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, 12], [Vrabec1977, p. 145], [Skopenkov2006, 2.4]. Before we present a simpler Definition 4.1 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 4.1. Assume that is -connected and . Then restrictions of and to are regular homotopic [Hirsch1959]. Since is -connected, has an -dimensional spine. 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 4.2. 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 (6) and so carries a homology class with coefficients. Define the Whitney invariant to be the homology class:
Clearly, (for both definitions).
The definition of depends on the choice of , but we write not for brevity.
Remark 4.3. (a) The Whitney invariant is well-defined by Definition 4.2, i.e. independent of the choice of , analogously to [Skopenkov2006, 2.4]. (The orientation is defined for each but used only for odd . When is even, for being well-defined we need -coefficients.)
(b) Definition 4.1 is equivalent to Definition 4.2. (Indeed, if on , we can take to be fixed on .) Hence the Whitney invariant is well-defined by Definition 4.1, i.e. independent of the choice of and of the isotopy making outside .
(c) Clearly, is not changed throughout isotopy of . Hence it gives a map .
(d) 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 ).
(e) For the Hudson tori is or for , is for .
(f) for each embeddings and .
(e) For the Whitney invariant is the collection of pairwise linking coefficients of the components of , cf. Definition 3.2 for and .
5 A generalization to highly-connected manifolds
Let be a closed -connected -manifold. Recall the unknotting theorem [Skopenkov2016c] that every two embeddings are isotopic when and . In this section we present description of generalizing Theorem 2.1, and its generalization to for .
Examples are Hudson tori . An action of on for is defined either by analogue of linked embedded connected sum [Skopenkov2016e], cf. [Skopenkov2010, Definition 1.4], or by parametric connected sum [Skopenkov2014, Remark 18.a]. Both these actions are free transitive by Theorem 5.1 below. If , then analogue of linked embedded connected sum gives only a construction of embeddings for each but not a well-defined action of on .
5.1 Classification just below the stable range
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, 11], [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 [Skopenkov2016c] for .
By Theorem 5.1 the Whitney invariant is bijective for . It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015]). The generator is Hudson torus . Also, for by Theorem 5.1 the Whitney invariants
are bijective. In the smooth category for even is not injective (see the next subsection), and is not surjective [Boechat1971], and is not injective [Skopenkov2016f].
5.2 Classification in the presence of smoothly knotted spheres
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 closed connected, a classification of is much harder: for 40 years the only known complete readily calculable classification results were for homology sphere . The following result for 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 analogue of 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 .
5.3 Classification further below the stable range
How to describe for ? See remarks on in . Some estimations of for a closed -connected -manifold are presented in [Skopenkov2010]. For one can go even further:
Theorem 5.3 [Becker&Glover1971]. Let be a closed -connected -manifold embeddable into , and . Then there is a 1-1 correspondence
For this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because is -connected). 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 [Skopenkov2016k] (to knotted tori) and [Skopenkov2002].
Observe that in Theorem 5.3 can be replaced by for each .
6 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. Then
- (1) has a natural orientation.
- (2) the natural orientation on need not extend to : take the cone over a general position map having only one self-intersection point.
- (3) the natural orientation on extend to if is odd [Hudson1969], Lemma 11.4.
- (4) has a natural orientation if is even.
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 .)
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 . 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 . (Note that a change of the orientation of forces changes of the signs of and so does not change the orientation of .)
7 References
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- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension dans , (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
- [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension dans , Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
- [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
- [Haefliger1962b] A. Haefliger, Plongements de variétés dans le domain stable, Séminaire Bourbaki, 245 (1962).
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