# 3-manifolds in 6-space

This page is being refereed under the supervision of the editorial board. Hence the page may not be edited at present. As always, the discussion page remains open for observations and comments. |

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 connected 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, Theorem 4.3], but is PS isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 of [Skopenkov2016c] and [Skopenkov2016f, Remark 1.1]).

This embedding represents a generator of [Haefliger1966, Theorem 5.16].

Let us construct the Haefliger (higher-dimensional) trefoil knot [Haefliger1962, 4.1]. 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:

Take any orientations the 3-spheres [Haefliger1962, 4.1] (the isotopy class of the Haefliger trefoil knot could potentially depend on these orientations, but this isotopy class would generate for any such choice). These orientations define an embedding up to isotopy. The Haefliger trefoil is the embedded connected sum of the components of 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 generalizations 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 [Skopenkov2006, Theorem 2.2.a]. For the classical classification in the PL category see [Skopenkov2016e, Theorem 2.1], [Skopenkov2006, Theorem 2.13].

**Theorem 3.1** [Haefliger1966, Theorem 5.16]**.**
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, 5] 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, 4], [Skopenkov2016e, Example 3.1].

See a higher-dimensional generalization [Skopenkov2016e, Theorem 6.3].

**Corollary 3.3.** (a) 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.

Part (a) was announced with a short outline of proof in [Hausmann1972], and proved in [Takase2006, Proof of Theorem 4.2 in p. 9 of the arxiv version]. For an alternative proof see [Skopenkov2008, Corollary (1) in p. 2 of the arxiv version].

**Addendum 3.4.**
If and are embeddings, then

E. g. for the embedded connected sum action of on 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

The Kreck invariant was invented by M. Kreck and appeared in [Skopenkov2008]. 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 [Smale1959, Theorem A] 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, 5], [Skopenkov2008, 1, definition of the Whitney invariant]). 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. \cite[ 4 ]{Skopenkov2016f}. 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, The Kreck Invariant Lemma in p. 7 of the arxiv version]**.**
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, submitted to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, submitted to Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, 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.

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 connected 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, Theorem 4.3], but is PS isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 of [Skopenkov2016c] and [Skopenkov2016f, Remark 1.1]).

This embedding represents a generator of [Haefliger1966, Theorem 5.16].

Let us construct the Haefliger (higher-dimensional) trefoil knot [Haefliger1962, 4.1]. 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:

Take any orientations the 3-spheres [Haefliger1962, 4.1] (the isotopy class of the Haefliger trefoil knot could potentially depend on these orientations, but this isotopy class would generate for any such choice). These orientations define an embedding up to isotopy. The Haefliger trefoil is the embedded connected sum of the components of 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 generalizations 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 [Skopenkov2006, Theorem 2.2.a]. For the classical classification in the PL category see [Skopenkov2016e, Theorem 2.1], [Skopenkov2006, Theorem 2.13].

**Theorem 3.1** [Haefliger1966, Theorem 5.16]**.**
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, 5] 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, 4], [Skopenkov2016e, Example 3.1].

See a higher-dimensional generalization [Skopenkov2016e, Theorem 6.3].

**Corollary 3.3.** (a) 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.

Part (a) was announced with a short outline of proof in [Hausmann1972], and proved in [Takase2006, Proof of Theorem 4.2 in p. 9 of the arxiv version]. For an alternative proof see [Skopenkov2008, Corollary (1) in p. 2 of the arxiv version].

**Addendum 3.4.**
If and are embeddings, then

E. g. for the embedded connected sum action of on 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

The Kreck invariant was invented by M. Kreck and appeared in [Skopenkov2008]. 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 [Smale1959, Theorem A] 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, 5], [Skopenkov2008, 1, definition of the Whitney invariant]). 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. \cite[ 4 ]{Skopenkov2016f}. 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, The Kreck Invariant Lemma in p. 7 of the arxiv version]**.**
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, submitted to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, submitted to Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, 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.

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 connected 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, Theorem 4.3], but is PS isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 of [Skopenkov2016c] and [Skopenkov2016f, Remark 1.1]).

This embedding represents a generator of [Haefliger1966, Theorem 5.16].

Let us construct the Haefliger (higher-dimensional) trefoil knot [Haefliger1962, 4.1]. 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:

Take any orientations the 3-spheres [Haefliger1962, 4.1] (the isotopy class of the Haefliger trefoil knot could potentially depend on these orientations, but this isotopy class would generate for any such choice). These orientations define an embedding up to isotopy. The Haefliger trefoil is the embedded connected sum of the components of 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 generalizations 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 [Skopenkov2006, Theorem 2.2.a]. For the classical classification in the PL category see [Skopenkov2016e, Theorem 2.1], [Skopenkov2006, Theorem 2.13].

**Theorem 3.1** [Haefliger1966, Theorem 5.16]**.**
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, 5] 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, 4], [Skopenkov2016e, Example 3.1].

See a higher-dimensional generalization [Skopenkov2016e, Theorem 6.3].

**Corollary 3.3.** (a) 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.

Part (a) was announced with a short outline of proof in [Hausmann1972], and proved in [Takase2006, Proof of Theorem 4.2 in p. 9 of the arxiv version]. For an alternative proof see [Skopenkov2008, Corollary (1) in p. 2 of the arxiv version].

**Addendum 3.4.**
If and are embeddings, then

E. g. for the embedded connected sum action of on 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

The Kreck invariant was invented by M. Kreck and appeared in [Skopenkov2008]. 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 [Smale1959, Theorem A] 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, 5], [Skopenkov2008, 1, definition of the Whitney invariant]). 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. \cite[ 4 ]{Skopenkov2016f}. 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, The Kreck Invariant Lemma in p. 7 of the arxiv version]**.**
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, submitted to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, submitted to Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, 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.

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).

## 2 Examples

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, Theorem 4.3], but is PS isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 of [Skopenkov2016c] and [Skopenkov2016f, Remark 1.1]).

This embedding represents a generator of [Haefliger1966, Theorem 5.16].

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

**Example 2.2** (The Hopf embedding of into )**.**
Represent Define

**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]:

## 3 Classification

**Theorem 3.1** [Haefliger1966, Theorem 5.16]**.**
There is an isomorphism .

**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 .

See a higher-dimensional generalization [Skopenkov2016e, Theorem 6.3].

**Corollary 3.3.** (a) If (i.e. is an integral homology sphere), then the Kreck invariant is a 1-1 correspondence.

**Addendum 3.4.**
If and are embeddings, then

**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 .

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

## 4 The Kreck invariant

*bundle isomorphism*. (By [Smale1959, Theorem A] 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.

**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*

Denote by and Poincaré duality (in any oriented manifold ).

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

For a joint Seifert class for and the classes

*the virtual signature of *
be

**Definition 4.4.** The `Kreck invariant' of two embeddings and such that is defined by

**The Kreck Invariant Lemma 4.5** [Skopenkov2008, The Kreck Invariant Lemma in p. 7 of the arxiv version]**.**
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 - [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 - [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

- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, 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.

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).