3-manifolds in 6-space
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.
The classification of 3-manifolds in 6-space is of course a particular case of the classification of n-manifolds in 2n-space which is discussed in [Skopenkov2016e]. In this page we recall the general results as they apply when and we discuss examples and invariants peculiar to the case .
Example 2.1 (The Haefliger trefoil knot). There is a smooth embedding with a surprising property that it is not smoothly isotopic to the standard embedding [Haefliger1962], but is piecewise smoothly isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 [Skopenkov2016c]). (This embedding is a generator of ) [Haefliger1962, 4.1].
Denote coordinates in by . The Haefliger (higher-dimensional) trefoil knot is obtained by joining with two tubes the higher-dimensional Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i.e. the three spheres given by the following three systems of equations:
Example 2.2 (The Hopf embedding of into ). 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 results just below the stable range, see [Skopenkov2016e, Theorem 2.1].)
Example 2.3 (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. For various choices of and this system of equations defines embeddings of either , or into [Bolsinov&Fomenko2004, Chapter 14].
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 [Skopenkov2016e, Theorem 2.1].
For the next theorem, the Whitney invariant is defined in [Skopenkov2016e]. For an abelian group the divisibility of the identity element is zero and the divisibility of is . The Kreck invariant is defined in Section 4 below.
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 .
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 .
Corollary 3.4. (a) The Hudson torus Hud(1) 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 .
See also [Avvakumov2016].
4 The Kreck invariant
An orientation-preserving diffeomorphism such that is called a bundle isomorphism. (By the Smale Theorem [Smale1959] this is equivalent to being isotopic to the restriction of a vector bundle isomorphism to the spherical bundle.)
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 of the Whitney invariant [Skopenkov2016e]). 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 Poincaré 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 Poincaré 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
Definition 4.4. The Kreck invariant of two embeddings and such that is defined by
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 ,
- be a spin bundle isomorphism,
- be a closed connected oriented 4-submanifold representing a joint Seifert class for and
- , be the Pontryagin number and Poincaré dual of the Euler classes of the normal bundle of in .
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [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, http://arxiv.org/abs/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
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, submitted to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, submitted to Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, submitted to Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Smale1959] S. Smale, Diffeomorphisms of the -sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626. MR0112149 (22 #3004) Zbl 0118.39103
- [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. 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).