3-manifolds in 6-space
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{{beginthm|The Kreck Invariant Lemma}}\label{th11}\cite{Skopenkov2008} | {{beginthm|The Kreck Invariant Lemma}}\label{th11}\cite{Skopenkov2008} | ||
− | Let $f,f':N\to\Rr^6$ be two embeddings such that $W(f)=W(f')$, $\varphi:\partial C_f\to\partial C_{f'}$ a spin isomorphism, $Y\subset M_\varphi$ a closed connected oriented 4-submanifold representing a joint | + | Let $f,f':N\to\Rr^6$ be two embeddings such that $W(f)=W(f')$, $\varphi:\partial C_f\to\partial C_{f'}$ a spin isomorphism, $Y\subset M_\varphi$ a closed connected oriented 4-submanifold representing a joint Seifert class for $f,f',\varphi$ and $\overline p_1\in\Zz$, $\overline e\in H_2(Y)$ are the Pontryagin number and Poincar\'e dual of the Euler classes of the normal bundle of $Y$ in $M_\varphi$. Then |
$$\frac{\sigma_{2[Y]}(M_\varphi)}{16}=\frac{\sigma(Y)-\overline p_1}8= | $$\frac{\sigma_{2[Y]}(M_\varphi)}{16}=\frac{\sigma(Y)-\overline p_1}8= | ||
\frac{\sigma(Y)-\overline e\cap\overline e}8.$$ | \frac{\sigma(Y)-\overline e\cap\overline e}8.$$ |
Revision as of 11:28, 26 April 2016
This page has been accepted for publication in the Bulletin of the Manifold Atlas. |
This page has not been refereed. The information given here might be incomplete or provisional. |
For notation and conventions throughout this page see high codimension embeddings. Many information on embeddings of 3-manifolds in 6-space is a particular case of embeddings of n-manifolds in 2n-space. In this page we concentrate on phenomena peculiar for n=3 and give references to general results.
Contents |
1 Examples
Simple examples are the Hudson torus and the Haefliger trefoil knot .
1.1 The Hopf 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.)
1.2 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:
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. A classification in the PL category.
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 .
Cf. higher-dimensional generalization.
All isotopy classes of embeddings can be constructed from a certain given embedding using unlinked and linked embedded connected sum with 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 follows not from Theorem 2.1 but from [Haefliger1966].)
- If (i.e. is a rational homology sphere, e.g. ), then is in (non-canonical) 1-1 correspondence with .
- Isotopy classes of embeddings with zero Whitney invariant are in 1-1 correspondence with , and for each integer there are exactly
Tex syntax error
isotopy classes of embeddings with the Whitney invariantTex syntax error
, cf. Corollary 2.4 below. - The Whitney invariant is surjective and
Addendum 2.3.
Let is an embedding, the generator of and is a connected sum ofTex syntax errorcopies of .
Then .
E. g. for the embedded connected sum action of on is free while for we have the following corollary.
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 .
(We believe that this very corollary or the case of Theorem 2.1 are as non-trivial as the general case of Theorem 2.1.)
3 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.
For a bundle isomorphism denote
A bundle isomorphism is called spin, if over is defined by an isotopy between the restrictions of and to . A spin bundle isomorphism exists because the restrictions to of and are isotopic (see Whitney invariant) and because . If is a spin bundle isomorphism, then is spin [Skopenkov2008], Spin Lemma.
Take a small oriented disk whose intersection with consists of exactly one point of sign and such that . Then is the meridian of .
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.
Remark 3.1.
For and aTex syntax error-submanifold (e.g. or ) denote
If is represented by a closed oriented 4-submanifold in general position to , then is represented by .
A homology Seifert surface 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 projection , cf. [Skopenkov2008], the Alexander Duality Lemma; this justifies the name `homology Seifert surface'.)
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'.
Denote by and Poincar\'e duality (in any manifold ).
We identify with the zero-dimensional homology groups and theTex syntax error-dimensional cohomology groups of closed connected oriented
Tex syntax error-manifolds. 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 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 3.2. The Kreck invariant of two embeddings and such that is defined by
where 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 3.3.[Skopenkov2008] Let be two embeddings such that , a spin 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
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
- [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.
- [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.
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).