3-manifolds in 6-space
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1 Introduction
This page is intended not only for specialists in embeddings, but also for mathematicians from other areas who want to apply or to learn the theory of embeddings.
Basic information on embeddings of 3-manifolds in 6-space is a particular case of embeddings of n-manifolds in 2n-space for n=3, see [SkopenkovE]. In this page we concentrate on phenomena peculiar for n=3.
See general introduction on embeddings, notation and conventions in [SkopenkovC, 1, 3].
2 Examples
Simple examples are the Hudson tori .
2.1 The Haefliger trefoil knot
Let us construct a smooth embedding (which is a generator of ) [Haefliger1962], 4.1. A miraculous property of this embedding is that it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to the standard embedding.
Denote coordinates in by . The higher-dimensional trefoil knot is obtained by joining with two tubes the higher-dimensional Borromean rings, i.e. the three spheres given by the following three systems of equations:
See [Skopenkov2006, Figures 3.5 and 3.6] and higher-dimensional generalization [SkopenkovK].
The Hopf embedding of RP3 into S5
Represent Define
It is easy to check that is an embedding. (The image of this embedding is given by the equations , .)
It would be interesting to obtain an explicit construction of an embedding which is not isotopic to the composition of the Hopf embedding with the standard inclusion . (Such an embedding is unique up to PL isotopy by classification just below the stable range.)
2.2 Algebraic embeddings from the theory of integrable systems
Some 3-manifolds appear in the theory of integrable systems together with their embeddings into (given by a system of algebraic equations) [Bolsinov&Fomenko2004, Chapter 14]. E.g. the following system of equations corresponds to the Euler integrability case [Bolsinov&Fomenko2004, Chapter 14]:
where and are real variables while and are constants. This system of equations for some choices of and defines an embedding of , or of or of into [Bolsinov&Fomenko2004, Chapter 14].
3 Classification
The results of this subsection are proved in [Skopenkov2008] unless other references are given. Let be a closed connected orientable 3-manifold. We work in the smooth category. For a classification in the PL category see [SkopenkovE, Theorem 2.1]. The Whitney invariant is defined in [SkopenkovE, 4].
Theorem 3.1. The Whitney invariant
is surjective. For each the Kreck invariant
is bijective, where is the divisibility of the projection of to the free part of .
Recall that for an abelian group the divisibility of zero is zero and the divisibility of is .
Cf. higher-dimensional generalization [SkopenkovE, 5.2].
All isotopy classes of embeddings can be constructed from a certain given embedding using unlinked and linked embedded connected sum with embeddings [SkopenkovC], [SkopenkovE].
Corollary 3.2 ([Haefliger1966], [Hausmann1972], [Takase2006]). (a) The Kreck invariant is a 1--1 correspondence if is or an integral homology sphere. (For the Kreck invariant is also a group isomorphism; this follows not from Theorem 3.1 but from [Haefliger1966].)
(b) If (i.e. is a rational homology sphere, e.g. ), then is in (non-canonical) 1-1 correspondence with .
(c) Isotopy classes of embeddings with zero Whitney invariant are in 1-1 correspondence with , and for each integer there are exactly isotopy classes of embeddings with the Whitney invariant , cf. Corollary 3.4 below.
(d) The Whitney invariant is surjective and
Addendum 3.3. Let is an embedding, the generator of and is a connected sum of copies of . Then .
E. g. for the embedded connected sum action of on [SkopenkovC] is free while for we have the following corollary.
Corollary 3.4. (a) There is an embedding such that for each knot the embedding is isotopic to .
(b) For each embedding such that (e.g. for the standard embedding ) and each non-trivial knot the embedding is not isotopic to .
(We believe that this very corollary or the case of Theorem 3.1 are as non-trivial as the general case of Theorem 3.1.)
4 The Kreck invariant
We work in the smooth category. Let be a closed connected orientable 3-manifold and embeddings. Fix orientations on and on .
An orientation-preserving diffeomorphism such that is called a bundle isomorphism.
Definition 4.1. For a bundle isomorphism denote
A bundle isomorphism is called spin, if is spin.
A spin bundle isomorphism exists. Indeed, the restrictions to of and are isotopic (this is proved in Definition 4.1). Define over by an isotopy between the restrictions to of and . Since , extends to . Then is spin. Cf. [Skopenkov2008], Spin Lemma.
Definition 4.2. Take a small oriented disk whose intersection with consists of exactly one point of sign and such that . A meridian of is . A joint Seifert class for and a bundle isomorphism is a class
If and is a spin bundle isomorphism, then there is a joint Seifert class for and [Skopenkov2008], Agreement Lemma.
Denote by and Poincar\'e duality (in any oriented manifold ).
Remark 4.3. 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.
A homology Seifert surface for is the image of the fundamental class .
For denote If is represented by a closed oriented 4-submanifold in general position to , then is represented by .
For a joint Seifert class for and we have
This property provides an equivalent definition of a joint Seifert class which explains the name and which was used in [Skopenkov2008] together with the name `joint homology Seifert surface'.
Identify with the zero-dimensional homology group of closed connected oriented manifols. The intersection products in 6-manifolds are omitted from the notation. Denote by the signature of a 4-manifold . For a closed connected oriented 6-manifold and let the virtual signature of be
Since , there is a closed connected oriented 4-submanifold representing the class . Then by [Hirzebruch1966], end of 9.2 or else by [Skopenkov2008], Submanifold Lemma.
Definition 4.4. The Kreck invariant of two embeddings and such that is defined by
where , is the reduction modulo , is a spin bundle isomorphism and is a joint Seifert class for and . Cf. [Ekholm2001], 4.1, [Zhubr2009].
We have , so any closed connected oriented 4-submanifold of representing the class is spin, hence by the Rokhlin Theorem is indeed divisible by 16. The Kreck invariant is well-defined by [Skopenkov2008], Independence Lemma.
For fix an embedding such that and define . (We write not for simplicity.)
The choice of the other orientation for (resp. ) will in general give rise to different values for the Kreck invariant. But such a choice only permutes the bijection (resp. replaces it with the bijection ).
Let us present a formula for the Kreck invariant analogous to [Guillou&Marin1986], Remarks to the four articles of Rokhlin, II.2.7 and III.excercises.IV.3, [Takase2004], Corollary 6.5, [Takase2006], Proposition 4.1. This formula is useful when an embedding goes through or is given by a system of equations (because we can obtain a `Seifert surface' by changing the equality to the inequality in one of the equations). See also [Moriyama], [Moriyama2008].
The Kreck Invariant Lemma 4.5.[Skopenkov2008] Let
- be two embeddings such that ,
- a spin bundle isomorphism,
- a closed connected oriented 4-submanifold representing a joint Seifert class for and
- , are the Pontryagin number and Poincar\'e dual of the Euler classes of the normal bundle of in .
Then
5 References
- [Bolsinov&Fomenko2004] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems, Chapman \& Hall/CRC, Boca Raton, FL, 2004. MR2036760 (2004j:37106) Zbl 1056.37075
- [Ekholm2001] T. Ekholm, Differential 3-knots in 5-space with and without self-intersections, Topology 40 (2001), no.1, 157–196. MR1791271 (2001h:57033) Zbl 0964.57029
- [Guillou&Marin1986] L. Guillou and A.Marin, Eds., A la r\'echerche de la topologie perdue, 1986, Progress in Math., 62, Birkhauser, Basel
- [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
- [Hausmann1972] J. Hausmann, Plongements de sphères d'homologie, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A963–965. MR0315727 (47 #4276) Zbl 0244.57005
- [Hirzebruch1966] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, New York, 1966. MR0202713 (34 #2573) Zbl 0843.14009
- [Moriyama] T. Moriyama, Integral formula for an extension of Haefliger's embedding invariant, preprint.
- [Moriyama2008] T. Moriyama, An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv:0806.3733.
- [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
- [SkopenkovK] Template:SkopenkovK
- [Takase2004] M. Takase, A geometric formula for Haefliger knots, Topology 43 (2004), no.6, 1425–1447. MR2081431 (2005e:57032) Zbl 1060.57021
- [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
- [Zhubr2009] A. V. Zhubr, Exotic invariant for 6-manifolds: a direct construction, Algebra i Analiz, 21:3 (2009), 145–164. English translation: St.Petersburg Math. J., 21:3 (2009).
2 Examples
Simple examples are the Hudson tori .
2.1 The Haefliger trefoil knot
Let us construct a smooth embedding (which is a generator of ) [Haefliger1962], 4.1. A miraculous property of this embedding is that it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to the standard embedding.
Denote coordinates in by . The higher-dimensional trefoil knot is obtained by joining with two tubes the higher-dimensional Borromean rings, i.e. the three spheres given by the following three systems of equations:
See [Skopenkov2006, Figures 3.5 and 3.6] and higher-dimensional generalization [SkopenkovK].
The Hopf embedding of RP3 into S5
Represent Define
It is easy to check that is an embedding. (The image of this embedding is given by the equations , .)
It would be interesting to obtain an explicit construction of an embedding which is not isotopic to the composition of the Hopf embedding with the standard inclusion . (Such an embedding is unique up to PL isotopy by classification just below the stable range.)
2.2 Algebraic embeddings from the theory of integrable systems
Some 3-manifolds appear in the theory of integrable systems together with their embeddings into (given by a system of algebraic equations) [Bolsinov&Fomenko2004, Chapter 14]. E.g. the following system of equations corresponds to the Euler integrability case [Bolsinov&Fomenko2004, Chapter 14]:
where and are real variables while and are constants. This system of equations for some choices of and defines an embedding of , or of or of into [Bolsinov&Fomenko2004, Chapter 14].
3 Classification
The results of this subsection are proved in [Skopenkov2008] unless other references are given. Let be a closed connected orientable 3-manifold. We work in the smooth category. For a classification in the PL category see [SkopenkovE, Theorem 2.1]. The Whitney invariant is defined in [SkopenkovE, 4].
Theorem 3.1. The Whitney invariant
is surjective. For each the Kreck invariant
is bijective, where is the divisibility of the projection of to the free part of .
Recall that for an abelian group the divisibility of zero is zero and the divisibility of is .
Cf. higher-dimensional generalization [SkopenkovE, 5.2].
All isotopy classes of embeddings can be constructed from a certain given embedding using unlinked and linked embedded connected sum with embeddings [SkopenkovC], [SkopenkovE].
Corollary 3.2 ([Haefliger1966], [Hausmann1972], [Takase2006]). (a) The Kreck invariant is a 1--1 correspondence if is or an integral homology sphere. (For the Kreck invariant is also a group isomorphism; this follows not from Theorem 3.1 but from [Haefliger1966].)
(b) If (i.e. is a rational homology sphere, e.g. ), then is in (non-canonical) 1-1 correspondence with .
(c) Isotopy classes of embeddings with zero Whitney invariant are in 1-1 correspondence with , and for each integer there are exactly isotopy classes of embeddings with the Whitney invariant , cf. Corollary 3.4 below.
(d) The Whitney invariant is surjective and
Addendum 3.3. Let is an embedding, the generator of and is a connected sum of copies of . Then .
E. g. for the embedded connected sum action of on [SkopenkovC] is free while for we have the following corollary.
Corollary 3.4. (a) There is an embedding such that for each knot the embedding is isotopic to .
(b) For each embedding such that (e.g. for the standard embedding ) and each non-trivial knot the embedding is not isotopic to .
(We believe that this very corollary or the case of Theorem 3.1 are as non-trivial as the general case of Theorem 3.1.)
4 The Kreck invariant
We work in the smooth category. Let be a closed connected orientable 3-manifold and embeddings. Fix orientations on and on .
An orientation-preserving diffeomorphism such that is called a bundle isomorphism.
Definition 4.1. For a bundle isomorphism denote
A bundle isomorphism is called spin, if is spin.
A spin bundle isomorphism exists. Indeed, the restrictions to of and are isotopic (this is proved in Definition 4.1). Define over by an isotopy between the restrictions to of and . Since , extends to . Then is spin. Cf. [Skopenkov2008], Spin Lemma.
Definition 4.2. Take a small oriented disk whose intersection with consists of exactly one point of sign and such that . A meridian of is . A joint Seifert class for and a bundle isomorphism is a class
If and is a spin bundle isomorphism, then there is a joint Seifert class for and [Skopenkov2008], Agreement Lemma.
Denote by and Poincar\'e duality (in any oriented manifold ).
Remark 4.3. 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.
A homology Seifert surface for is the image of the fundamental class .
For denote If is represented by a closed oriented 4-submanifold in general position to , then is represented by .
For a joint Seifert class for and we have
This property provides an equivalent definition of a joint Seifert class which explains the name and which was used in [Skopenkov2008] together with the name `joint homology Seifert surface'.
Identify with the zero-dimensional homology group of closed connected oriented manifols. The intersection products in 6-manifolds are omitted from the notation. Denote by the signature of a 4-manifold . For a closed connected oriented 6-manifold and let the virtual signature of be
Since , there is a closed connected oriented 4-submanifold representing the class . Then by [Hirzebruch1966], end of 9.2 or else by [Skopenkov2008], Submanifold Lemma.
Definition 4.4. The Kreck invariant of two embeddings and such that is defined by
where , is the reduction modulo , is a spin bundle isomorphism and is a joint Seifert class for and . Cf. [Ekholm2001], 4.1, [Zhubr2009].
We have , so any closed connected oriented 4-submanifold of representing the class is spin, hence by the Rokhlin Theorem is indeed divisible by 16. The Kreck invariant is well-defined by [Skopenkov2008], Independence Lemma.
For fix an embedding such that and define . (We write not for simplicity.)
The choice of the other orientation for (resp. ) will in general give rise to different values for the Kreck invariant. But such a choice only permutes the bijection (resp. replaces it with the bijection ).
Let us present a formula for the Kreck invariant analogous to [Guillou&Marin1986], Remarks to the four articles of Rokhlin, II.2.7 and III.excercises.IV.3, [Takase2004], Corollary 6.5, [Takase2006], Proposition 4.1. This formula is useful when an embedding goes through or is given by a system of equations (because we can obtain a `Seifert surface' by changing the equality to the inequality in one of the equations). See also [Moriyama], [Moriyama2008].
The Kreck Invariant Lemma 4.5.[Skopenkov2008] Let
- be two embeddings such that ,
- a spin bundle isomorphism,
- a closed connected oriented 4-submanifold representing a joint Seifert class for and
- , are the Pontryagin number and Poincar\'e dual of the Euler classes of the normal bundle of in .
Then
5 References
- [Bolsinov&Fomenko2004] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems, Chapman \& Hall/CRC, Boca Raton, FL, 2004. MR2036760 (2004j:37106) Zbl 1056.37075
- [Ekholm2001] T. Ekholm, Differential 3-knots in 5-space with and without self-intersections, Topology 40 (2001), no.1, 157–196. MR1791271 (2001h:57033) Zbl 0964.57029
- [Guillou&Marin1986] L. Guillou and A.Marin, Eds., A la r\'echerche de la topologie perdue, 1986, Progress in Math., 62, Birkhauser, Basel
- [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
- [Hausmann1972] J. Hausmann, Plongements de sphères d'homologie, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A963–965. MR0315727 (47 #4276) Zbl 0244.57005
- [Hirzebruch1966] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, New York, 1966. MR0202713 (34 #2573) Zbl 0843.14009
- [Moriyama] T. Moriyama, Integral formula for an extension of Haefliger's embedding invariant, preprint.
- [Moriyama2008] T. Moriyama, An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv:0806.3733.
- [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
- [SkopenkovK] Template:SkopenkovK
- [Takase2004] M. Takase, A geometric formula for Haefliger knots, Topology 43 (2004), no.6, 1425–1447. MR2081431 (2005e:57032) Zbl 1060.57021
- [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
- [Zhubr2009] A. V. Zhubr, Exotic invariant for 6-manifolds: a direct construction, Algebra i Analiz, 21:3 (2009), 145–164. English translation: St.Petersburg Math. J., 21:3 (2009).