4-manifolds in 7-space
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Contents |
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.
Basic information on embeddings of 4-manifolds in 7-space is a particular case of embeddings of n-manifolds in (2n-1)-space for n=4, see [SkopenkovE]. In this page we concentrate on phenomena peculiar for n=4.
See general introduction on embeddings, notation and conventions in [SkopenkovC, 1, 3].
2 Examples
The Hudson tori and are defined in Remark 3.5.d of [SkopenkovE].
For an orientable 4-manifold , an embedding and a class one can construct an embedding by linked connected sum analogously to embeddings into [SkopenkovE]. If is simply-connected and CAT=PL, this gives a free transitive action of on [SkopenkovE]. We also have for the Whitney invariant [SkopenkovE].
Denote by is the Hopf map.
2.1 The Lambrechts torus and the Hudson torus
These two embeddings are defined as compositions , where , is the projection onto the second factor, is the standard inclusion and maps are defined below. We shall see that are embeddings for each , hence and are embeddings.
Define , where is identified with the set of unit length quaternions and with the set of unit length complex numbers.
Define , where is identified with the 2-sphere formed by unit length quaternions of the form .
These examples appear in [Skopenkov2006] but could be known earlier. Note that is PL isotopic to the Hudson torus defined in Remark 3.5.d of [SkopenkovE].
Take the Hopf fibration . Take the standard embeding . Its complement has the homotopy type of . Then . This is the construction of P. Lambrechts (but could be known earlier). We have
Tex syntax error
Tex syntax erroris the standard embedding.
2.2 Embeddings of CP2 into R7
Tex syntax errorjoining to . Clearly, the boundary 3-sphere of is standardly embedded into
Tex syntax error. Hence extends to an embedding .
Apriori this extension need not be unique (because it can be changed by a connected sum with an embedding ). Surprisingly, it is unique, and is the only embedding (up to isotopy and a hyperplane reflection of ).
Theorem 2.1.
- There are exactly two smooth isotopy classes of smooth embeddings (differing by a hyperplane reflection of ).
- For each smooth embeddings and the embedding is smoothly isotopic to .
- The Whitney invariant is a 1--1 correspondence . The inverse is given by linked connected sum.
This follows by [Boechat&Haefliger1970], [Skopenkov2005, Triviality Theorem (a)] or by Theorem 4.2 below.
2.3 The Haefliger torus
This is a PL embedding which is (locally flat but) not PL isotopic to a smooth embedding [Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]. Take the Haefliger trefoil knot . Extend it to a conical embedding . By [Haefliger1962], the trefoil knot also extends to a smooth embedding [Skopenkov2006, Figure 3.7.a]. These two extensions together form the Haefliger torus [Skopenkov2006, Figure 3.7.b].
3 The Boechat-Haefliger invariant
Let be a closed connected orientable 4-manifold and an embedding. Fix an orientation on and an orientation on .
Definition 3.1. The composition
of the boundary map and the projection is an isomorphism, cf. [Skopenkov2008, the Alexander Duality Lemma]. The inverse to this composition is homology Alexander Duality isomorphism; it equals to the composition of the cohomology Alexander and Poincar\'e duality isomorphisms.
Definition 3.2. A homology Seifert surface for is the image of the fundamental class . Define
Remark 3.3. (a) Take a small oriented disk whose intersection with consists of exactly one point of sign and such that . A meridian of is . A homology Seifert surface for is uniquely defined by the condition .
(b) We have for the Whitney invariant [SkopenkovE]. This is proved analogously to [Skopenkov2008, 2, The Boechat-Haefliger Invariant Lemma].
(c) Definition 3.2 is equivalent to the original one [Boechat&Haefliger1970] by [Crowley&Skopenkov2008, Section Lemma 3.1]. Hence .
(d) Earlier notation was [Boechat&Haefliger1970], [Skopenkov2005] and [Crowley&Skopenkov2008].
4 Classification
See a classification of for a closed connected 4-manifold such that . Here we work in the smooth category.
Theorem 4.1. . [Haefliger1966], see also [Skopenkov2005], [Crowley&Skopenkov2008].
Theorem 4.2. [Crowley&Skopenkov2008] Let be a closed connected 4-manifold such that . Then the image of the Boéchat-Haefliger invariant
For each there is an injective invariant called the Kreck invariant,
whose image is the subset of even elements.
Here is the maximal integer such that both and 24 are divisible by . Thus is surjective if is not divisible by 2. Note that is divisible by 2 (for some or, equivalently, for each ) if and only if is spin.
For the definition of the Kreck invariant see [Crowley&Skopenkov2008].
Corollary 4.3. (a) There are exactly twelve isotopy classes of embeddings if is an integral homology 4-sphere (cf. Theorem 4.1).
(b) For each integer there are exactly isotopy classes of embeddings with , and the same holds for those with .
Other values ofTex syntax errorare not in the image of . (We take the standard basis in .)
Addendum 4.4. Under assumptions of Theorem 4.2 for each pair of embeddings and
The following corollaries are examples of the effectiveness and the triviality of the embedded connected sum action of on .
Corollary 4.5. (a) Take an integer and the Hudson torus defined in Remark 3.5.d. If , then for each embedding the embedding is isotopic to . (For a general integer the number of isotopy classes of embeddings is .)
(b) Let be a closed connected 4-manifold such that and the signature of is not divisible by the square of an integer . Then for each embeddings and the embedding is isotopic to ; in other words, is injective. (First proved in [Skopenkov2005].)
(c) If is a closed connected 4-manifold such that and for an embedding , then for each embedding the embedding is not isotopic to .
5 References
- [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
- [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
- [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.
- [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
- [SkopenkovC] A. Skopenkov, High codimension embeddings: classification, , submitted to Bull. MA
- [SkopenkovE] A. Skopenkov, Embeddings just below the stable range: classification, http://www.map.mpim-bonn.mpg.de/Embeddings_just_below_the_stable_range:_classification, submitted to Bull. MA
2 Examples
The Hudson tori and are defined in Remark 3.5.d of [SkopenkovE].
For an orientable 4-manifold , an embedding and a class one can construct an embedding by linked connected sum analogously to embeddings into [SkopenkovE]. If is simply-connected and CAT=PL, this gives a free transitive action of on [SkopenkovE]. We also have for the Whitney invariant [SkopenkovE].
Denote by is the Hopf map.
2.1 The Lambrechts torus and the Hudson torus
These two embeddings are defined as compositions , where , is the projection onto the second factor, is the standard inclusion and maps are defined below. We shall see that are embeddings for each , hence and are embeddings.
Define , where is identified with the set of unit length quaternions and with the set of unit length complex numbers.
Define , where is identified with the 2-sphere formed by unit length quaternions of the form .
These examples appear in [Skopenkov2006] but could be known earlier. Note that is PL isotopic to the Hudson torus defined in Remark 3.5.d of [SkopenkovE].
Take the Hopf fibration . Take the standard embeding . Its complement has the homotopy type of . Then . This is the construction of P. Lambrechts (but could be known earlier). We have
Tex syntax error
Tex syntax erroris the standard embedding.
2.2 Embeddings of CP2 into R7
Tex syntax errorjoining to . Clearly, the boundary 3-sphere of is standardly embedded into
Tex syntax error. Hence extends to an embedding .
Apriori this extension need not be unique (because it can be changed by a connected sum with an embedding ). Surprisingly, it is unique, and is the only embedding (up to isotopy and a hyperplane reflection of ).
Theorem 2.1.
- There are exactly two smooth isotopy classes of smooth embeddings (differing by a hyperplane reflection of ).
- For each smooth embeddings and the embedding is smoothly isotopic to .
- The Whitney invariant is a 1--1 correspondence . The inverse is given by linked connected sum.
This follows by [Boechat&Haefliger1970], [Skopenkov2005, Triviality Theorem (a)] or by Theorem 4.2 below.
2.3 The Haefliger torus
This is a PL embedding which is (locally flat but) not PL isotopic to a smooth embedding [Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]. Take the Haefliger trefoil knot . Extend it to a conical embedding . By [Haefliger1962], the trefoil knot also extends to a smooth embedding [Skopenkov2006, Figure 3.7.a]. These two extensions together form the Haefliger torus [Skopenkov2006, Figure 3.7.b].
3 The Boechat-Haefliger invariant
Let be a closed connected orientable 4-manifold and an embedding. Fix an orientation on and an orientation on .
Definition 3.1. The composition
of the boundary map and the projection is an isomorphism, cf. [Skopenkov2008, the Alexander Duality Lemma]. The inverse to this composition is homology Alexander Duality isomorphism; it equals to the composition of the cohomology Alexander and Poincar\'e duality isomorphisms.
Definition 3.2. A homology Seifert surface for is the image of the fundamental class . Define
Remark 3.3. (a) Take a small oriented disk whose intersection with consists of exactly one point of sign and such that . A meridian of is . A homology Seifert surface for is uniquely defined by the condition .
(b) We have for the Whitney invariant [SkopenkovE]. This is proved analogously to [Skopenkov2008, 2, The Boechat-Haefliger Invariant Lemma].
(c) Definition 3.2 is equivalent to the original one [Boechat&Haefliger1970] by [Crowley&Skopenkov2008, Section Lemma 3.1]. Hence .
(d) Earlier notation was [Boechat&Haefliger1970], [Skopenkov2005] and [Crowley&Skopenkov2008].
4 Classification
See a classification of for a closed connected 4-manifold such that . Here we work in the smooth category.
Theorem 4.1. . [Haefliger1966], see also [Skopenkov2005], [Crowley&Skopenkov2008].
Theorem 4.2. [Crowley&Skopenkov2008] Let be a closed connected 4-manifold such that . Then the image of the Boéchat-Haefliger invariant
For each there is an injective invariant called the Kreck invariant,
whose image is the subset of even elements.
Here is the maximal integer such that both and 24 are divisible by . Thus is surjective if is not divisible by 2. Note that is divisible by 2 (for some or, equivalently, for each ) if and only if is spin.
For the definition of the Kreck invariant see [Crowley&Skopenkov2008].
Corollary 4.3. (a) There are exactly twelve isotopy classes of embeddings if is an integral homology 4-sphere (cf. Theorem 4.1).
(b) For each integer there are exactly isotopy classes of embeddings with , and the same holds for those with .
Other values ofTex syntax errorare not in the image of . (We take the standard basis in .)
Addendum 4.4. Under assumptions of Theorem 4.2 for each pair of embeddings and
The following corollaries are examples of the effectiveness and the triviality of the embedded connected sum action of on .
Corollary 4.5. (a) Take an integer and the Hudson torus defined in Remark 3.5.d. If , then for each embedding the embedding is isotopic to . (For a general integer the number of isotopy classes of embeddings is .)
(b) Let be a closed connected 4-manifold such that and the signature of is not divisible by the square of an integer . Then for each embeddings and the embedding is isotopic to ; in other words, is injective. (First proved in [Skopenkov2005].)
(c) If is a closed connected 4-manifold such that and for an embedding , then for each embedding the embedding is not isotopic to .
5 References
- [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
- [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
- [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.
- [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
- [SkopenkovC] A. Skopenkov, High codimension embeddings: classification, , submitted to Bull. MA
- [SkopenkovE] A. Skopenkov, Embeddings just below the stable range: classification, http://www.map.mpim-bonn.mpg.de/Embeddings_just_below_the_stable_range:_classification, submitted to Bull. MA