3-manifolds in 6-space
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Most of 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 results on embeddings of closed connected 3-manifolds in 6-space are particular cases of results on embeddings of -manifolds in -space, which are discussed in [Skopenkov2016e], [Skopenkov2006, 2.4 `The Whitney invariant']. In this page we concentrate on more advanced classification results peculiar to the case .
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3]. Unless specified otherwise, we work in the smooth category. For definition of the embedded connected sum of embeddings of closed 3-manifolds in 6-space, and for the corresponding action of the group on the set , see e.g. [Skopenkov2016c, 4].
Example 2.1 (The Haefliger trefoil knot). There is a smooth embedding which is not smoothly isotopic to the standard embedding [Haefliger1962], but is PS isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 [Skopenkov2016c] and [Skopenkov2016f, Remark 1.1]).
This embedding is a generator of [Haefliger1962, 4.1].
To construct the Haefliger (higher-dimensional) trefoil knot , we start from the 3-dimensional Borromean rings (see Figure 6 of [Skopenkov2016h]), which are three disjoint 3-spheres in defined as follows. For coordinates in defined by , the three 3-spheres are given by the following three systems of equations:
The 3-spheres have a natural orientation described in [Haefliger1962]. These orientations define an embedding up to isotopy. The Haefliger trefoil is the embedded connected sum of the components if this embedding.
The construction of the Haefliger trefoil is analogous to the construction of the trefoil from the 1-dimensional Borromean rings, depicted in Figure 1. An intermediate stage of the construction, when two components are connected but the third remains disjoint, yields the Whitehead link . The 1-dimensional picture can also be regarded as a schematic picture for the construction of the Haefliger trefoil from the 3-dimensional Borromean rings.
Example 2.2 (The Hopf embedding of into ). Represent Define
It is easy to check that is an embedding. (The image of this embedding in 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 the classical classification results just below the stable range, see [Skopenkov2016e, Theorem 2.1], [Skopenkov2006, Theorem 2.13].)
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 following results of this subsection are proved in [Skopenkov2008] unless other references are given. Let be a closed connected oriented 3-manifold.
For the next theorem, the Whitney invariant and and the Kreck invariant are defined in [Skopenkov2016e], [Skopenkov2008], and in 4 below. For an abelian group the divisibility of the zero element is zero, and the divisibility of is .
Theorem 3.2. (a) The Whitney invariant
(b) For any the Kreck invariant
is bijective, where is the divisibility of the projection of to the free part of .
Although part (a) first appeared in [Skopenkov2008], it is (as opposed to (b)) a simple corollary of results by Hudson and Haefliger.
(b) If (i.e. is a rational homology sphere, e.g. ), then is in (non-canonical) 1-1 correspondence with . More precisely, the Whitney invariant is surjective, and every its preimage is in canonical 1-1 correspondence (given by the Kreck invariant) with .
(c) Isotopy classes of embeddings with zero Whitney invariant are in 1-1 correspondence with , and for any integer there are exactly isotopy classes of embeddings with the Whitney invariant , cf. Corollary 3.5 below.
Addendum 3.4. If and are embeddings, then
E. g. for the embedded connected sum action of on [Skopenkov2016c] is free while for we have part (a) of the following corollary.
Corollary 3.5. (a) There is an embedding such that for any knot the embedding is isotopic to . (We can take as the Hudson torus .)(b) For any embedding such that (e.g. for the standard embedding
Tex syntax error) and any non-trivial knot the embedding is not isotopic to .
For a related classification of some disconnected 3-manifolds in 6-space see [Skopenkov2016h, 6].
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 [Skopenkov2008, Spin Lemma]. Indeed, the restrictions to of and are isotopic (this is proved in definition of the Whitney invariant [Skopenkov2016e], [Skopenkov2008]). Define over using an isotopy between the restrictions to of and . Since , extends to . Then is spin.
Identify with the zero-dimensional homology group of a closed connected oriented manifold. The symbol of the intersection product in homology of 6-manifolds will be omitted.
Definition 4.2. Take a small oriented disk whose intersection with consists of exactly one point of sign and such that . 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 ).
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 the classes
are `homology Seifert surfaces' for , cf. [Skopenkov2016f, Remark 4.3]. 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'.
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.) Then the map is well-defined by .
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 .
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