3-manifolds in 6-space
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For notation and conventions throughout this page see high codimension embeddings.
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1 Examples
1.1 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 isotopic to the standard embedding, but is isotopic to the standard embedding.
Denote coordinates in by . The higher-dimensional trefoil knot is obtained by joining with two tubes the higher-dimensional , i.e. the three spheres given by the following three systems of equations:
See Figures 3.5 and 3.6 of [Skopenkov2006].
1.2 The Hopf construction of an embedding
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.)
1.3 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:
where and are variables while and are constants. This defines embeddings of , or into .
2 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.
Classification Theorem 2.1. 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 .
All isotopy classes of embeddings can be constructed (from a certain given embedding) using connected sum with embeddings , see the construction in Embeddings just below the stable range and in High codimension embeddings.
Corollary 2.2.
- The Kreck invariant is a 1--1 correspondence if is or an integral homology sphere [Haefliger1966], [Hausmann1972], [Takase2006]. (For the Kreck invariant is also a group isomorphism; this does not follows from the Classification Theorem 2.1 but is proved in [Haefliger1966].)
- If (i.e. is a rational homology sphere, e.g. ), then is in (non-canonical) 1--1 correspondence with .
- Embeddings with zero Whitney invariant are in 1--1 correspondence with , and for each integer there are exactly distinct (i.e. non-isotopic) embeddings with the Whitney invariant , cf. Corollary 2.4 below.
- The Whitney invariant is surjective and .
Addendum 2.3. If is an embedding, is the generator of and is a connected sum of copies of , then
E. g. for the `connected sum' action of on is free while for we have the following corollary. (We believe that this very concrete corollary or the case are as non-trivial as the general case of the Classification Theorem.)
Corollary 2.4.
- There is an embedding such that for each knot the embedding is isotopic to .
- For each embedding such that (e.g. for the standard embedding ) and each non-trivial knot the embedding is not isotopic to .
3 The Kreck invariant
Let be a closed connected orientable 3-manifold. We work in the smooth category.
An orientation-preserving diffeomorphism such that is simply called an . For an isomorphism denote
An isomorphism is called , if over is defined by an isotopy between the restrictions of and to . A spin isomorphism exists because the restrictions to of and are isotopic (see the definition of the Whitney invariant) and because . If is a spin isomorphism, then is spin [Skopenkov2008], Spin Lemma.
Denote by the signature of a 4-manifold . Denote by and Poincar\'e duality (in any manifold ). For and a -submanifold (e.g. or ) denote
If is represented by a closed oriented 4-submanifold in general position to , then is represented by .
A for is the image of the fundamental class under the composition of the Alexander and Poincar\'e duality isomorphisms. (This composition is an inverse to the composition of the boundary map and the normal bundle map , cf. [Skopenkov2008], the Alexander Duality Lemma; this justifies the name `homology Seifert surface'.)
A for and is a class such that
If is a spin isomorphism and , then there is a joint homology Seifert surface for and [Skopenkov2008], Agreement Lemma.
We identify with the zero-dimensional homology groups and the -dimensional cohomology groups of closed connected oriented -manifolds. The intersection products in 6-manifolds are omitted from the notation. For a closed connected oriented 6-manifold and denote by
the virtual signature of . (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 3.1. The Kreck invariant of two embeddings and such that is defined by
where is a spin isomorphism and is a joint homology Seifert surface for and . Cf. [Ekholm2001], 4.1, [Zhubr].
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 T. Moriyama, Integral formula for an extension of Haeliger's embedding invariant.
The Kreck Invariant Lemma 3.2.[Skopenkov2008] Let be two embeddings such that , a spin isomorphism, a closed connected oriented 4-submanifold representing a joint homology Seifert surface and , are the Pontryagin number and Poincar\'e dual of the Euler classes of the normal bundle of in . Then
4 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
- [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
- [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.
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