# 3-manifolds in 6-space

Askopenkov (Talk | contribs) (→Examples) |
(→Examples) |
||

Line 33: | Line 33: | ||

\left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1. \end{array}\right.$$ | \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1. \end{array}\right.$$ | ||

[[Image:HT.jpg|thumb|400px|Figure 1: The construction of the trefoil from the Borromean rings]] | [[Image:HT.jpg|thumb|400px|Figure 1: The construction of the trefoil from the Borromean rings]] | ||

− | The 3-spheres have a natural orientation described in \cite{Haefliger1962}. These orientations define embedding $S^3 \sqcup S^3 \sqcup S^3 \to \R^6$ up to isotopy. The Haefliger trefoil $t$ is the embedded connected sum of the components if this embedding. | + | The 3-spheres have a natural orientation described in \cite{Haefliger1962}. These orientations define an embedding $S^3 \sqcup S^3 \sqcup S^3 \to \R^6$ up to isotopy. The Haefliger trefoil $t$ is the embedded connected sum of the components if this embedding. |

− | <!-- with small embedded tubes to form an embedded 3-sphere and taking any | + | <!-- with small embedded tubes to form an embedded 3-sphere and taking any embedding with image this sphere. , and the tubes are chosen so that the connected sum has an orientation compatible to the orientations of the components.--> |

− | The construction of the Haefliger trefoil $t$ is analogous to the construction of the trefoil from the 1-dimensional Borromean rings, depicted in Figure 1. | + | The construction of the Haefliger trefoil $t$ 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 [[High_codimension_links#Examples beyond the metastable range|Whitehead link]] $w$. 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. |

<!--[[Media:35.pdf|figure 6]] of \cite{Skopenkov2016h} and [[Media:36.pdf|figure 1]] (cf. [[High_codimension_links|figure 7]] of \cite{Skopenkov2016h}).\cite[Figures 3.5 and 3.6]{Skopenkov2006}.--> | <!--[[Media:35.pdf|figure 6]] of \cite{Skopenkov2016h} and [[Media:36.pdf|figure 1]] (cf. [[High_codimension_links|figure 7]] of \cite{Skopenkov2016h}).\cite[Figures 3.5 and 3.6]{Skopenkov2006}.--> | ||

## Revision as of 00:57, 18 May 2019

This page has been accepted for publication in the Bulletin of the Manifold Atlas. |

The user responsible for this page is Askopenkov. No other user may edit this page at present. |

## Contents |

## 1 Introduction

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].

## 2 Examples

For any integer there is an embedding called the Hudson torus, , see [Skopenkov2016e, 3], [Skopenkov2006, Example 2.10].

Piecewise smooth (PS) embedding and isotopy are defined in [Skopenkov2016f, Remark 1.1].

**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.

For higher-dimensional generalization see [Skopenkov2016h, 5] and [Skopenkov2016k].

**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].

## 3 Classification

Recall that any 3-manifold embeds into by the strong Whitney embedding theorem. For the classical classification in the PL category see [Skopenkov2016e, Theorem 2.1], [Skopenkov2006, Theorem 2.13].

**Theorem 3.1** ([Haefliger1966], see also [Skopenkov2008])**.**
There is an isomorphism .

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

is surjective.

(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.

All isotopy classes of embeddings can be constructed from a certain given embedding using unlinked and linked embedded connected sum with embeddings [Skopenkov2016c], [Skopenkov2016e].

See a higher-dimensional generalization [Skopenkov2016e].

**Corollary 3.3.** (a) ([Hausmann1972], see also [Takase2006], [Skopenkov2008])
If (i.e. is an integral homology sphere), then the Kreck invariant is a 1-1 correspondence.

(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 .)

Tex syntax error) and any non-trivial knot the embedding is not isotopic to .

(We believe that this very corollary or the case of Theorem 3.2 are as hard to prove as the general case of Theorem 3.2.)

For a related classification of some disconnected 3-manifolds in 6-space see [Skopenkov2016h, 6].

## 4 The Kreck invariant

We work in the smooth category and use notation and conventions [Skopenkov2016c, 3]. Let be a closed connected oriented 3-manifold and embeddings. Fix orientation on , and so on .

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 ).

**Remark 4.3.** The homology Alexander Duality isomorphism is defined in [Skopenkov2016f, 4].

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'.

Denote by the signature of a 4-manifold .
We use characteristic classes and .
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.) 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 .

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, 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, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in 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.

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).

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].

## 2 Examples

For any integer there is an embedding called the Hudson torus, , see [Skopenkov2016e, 3], [Skopenkov2006, Example 2.10].

Piecewise smooth (PS) embedding and isotopy are defined in [Skopenkov2016f, Remark 1.1].

**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.

For higher-dimensional generalization see [Skopenkov2016h, 5] and [Skopenkov2016k].

**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].

## 3 Classification

Recall that any 3-manifold embeds into by the strong Whitney embedding theorem. For the classical classification in the PL category see [Skopenkov2016e, Theorem 2.1], [Skopenkov2006, Theorem 2.13].

**Theorem 3.1** ([Haefliger1966], see also [Skopenkov2008])**.**
There is an isomorphism .

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

is surjective.

(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.

All isotopy classes of embeddings can be constructed from a certain given embedding using unlinked and linked embedded connected sum with embeddings [Skopenkov2016c], [Skopenkov2016e].

See a higher-dimensional generalization [Skopenkov2016e].

**Corollary 3.3.** (a) ([Hausmann1972], see also [Takase2006], [Skopenkov2008])
If (i.e. is an integral homology sphere), then the Kreck invariant is a 1-1 correspondence.

(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 .)

Tex syntax error) and any non-trivial knot the embedding is not isotopic to .

(We believe that this very corollary or the case of Theorem 3.2 are as hard to prove as the general case of Theorem 3.2.)

For a related classification of some disconnected 3-manifolds in 6-space see [Skopenkov2016h, 6].

## 4 The Kreck invariant

We work in the smooth category and use notation and conventions [Skopenkov2016c, 3]. Let be a closed connected oriented 3-manifold and embeddings. Fix orientation on , and so on .

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 ).

**Remark 4.3.** The homology Alexander Duality isomorphism is defined in [Skopenkov2016f, 4].

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'.

Denote by the signature of a 4-manifold .
We use characteristic classes and .
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.) 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 .

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, 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, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in 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.

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).

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].

## 2 Examples

For any integer there is an embedding called the Hudson torus, , see [Skopenkov2016e, 3], [Skopenkov2006, Example 2.10].

Piecewise smooth (PS) embedding and isotopy are defined in [Skopenkov2016f, Remark 1.1].

**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.

For higher-dimensional generalization see [Skopenkov2016h, 5] and [Skopenkov2016k].

**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].

## 3 Classification

Recall that any 3-manifold embeds into by the strong Whitney embedding theorem. For the classical classification in the PL category see [Skopenkov2016e, Theorem 2.1], [Skopenkov2006, Theorem 2.13].

**Theorem 3.1** ([Haefliger1966], see also [Skopenkov2008])**.**
There is an isomorphism .

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

is surjective.

(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.

All isotopy classes of embeddings can be constructed from a certain given embedding using unlinked and linked embedded connected sum with embeddings [Skopenkov2016c], [Skopenkov2016e].

See a higher-dimensional generalization [Skopenkov2016e].

**Corollary 3.3.** (a) ([Hausmann1972], see also [Takase2006], [Skopenkov2008])
If (i.e. is an integral homology sphere), then the Kreck invariant is a 1-1 correspondence.

(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 .)

Tex syntax error) and any non-trivial knot the embedding is not isotopic to .

(We believe that this very corollary or the case of Theorem 3.2 are as hard to prove as the general case of Theorem 3.2.)

For a related classification of some disconnected 3-manifolds in 6-space see [Skopenkov2016h, 6].

## 4 The Kreck invariant

We work in the smooth category and use notation and conventions [Skopenkov2016c, 3]. Let be a closed connected oriented 3-manifold and embeddings. Fix orientation on , and so on .

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 ).

**Remark 4.3.** The homology Alexander Duality isomorphism is defined in [Skopenkov2016f, 4].

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'.

Denote by the signature of a 4-manifold .
We use characteristic classes and .
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.) 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 .

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, 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, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in 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.

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).