# 3-manifolds in 6-space

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## Contents |

## 1 Introduction

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.

The classification of 3-manifolds in 6-space is of course a particular case of the classification of n-manifolds in 2n-space which is discussed in [Skopenkov2016e]. In this page we recall the general results as they apply when and we discuss examples and invariants 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, 2].

## 2 Examples

For each integer there is an embedding called the Hudson torus, , see [Skopenkov2016e, 3].

**Example 2.1** (The Haefliger trefoil knot)**.**
There is a smooth embedding with a surprising property that it is not *smoothly* isotopic to the standard embedding [Haefliger1962], but is *piecewise smoothly* isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 [Skopenkov2016c]).
(This embedding is a generator of ) [Haefliger1962, 4.1].

Denote coordinates in by . The Haefliger (higher-dimensional) trefoil knot is obtained by joining with two tubes the higher-dimensional *Borromean rings* [Skopenkov2006, Figures 3.5 and 3.6], i.e. the three spheres given by the following three systems of equations:

See motivating examples of links [Skopenkov2016h] and a higher-dimensional generalization [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 is given by the equations , .)

It would be interesting to obtain an explicit construction of an embedding which is not isotopic to the composition of the Hopf embedding with the standard inclusion . (Such an embedding is unique up to PL isotopy by classification results just below the stable range, see [Skopenkov2016e, Theorem 2.1].)

**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

The results of this subsection are proved in [Skopenkov2008] unless other references are given. Let be a closed connected orientable 3-manifold. We work in the smooth category. For a classification in the PL category see [Skopenkov2016e, Theorem 2.1].

For the next theorem, the Whitney invariant is defined in [Skopenkov2016e]. For an abelian group the divisibility of the identity element is zero and the divisibility of is . The Kreck invariant is defined in Section 4 below.

**Theorem 3.1.** The Whitney invariant

is surjective. For each the Kreck invariant

is bijective, where is the divisibility of the projection of to the free part of .

Cf. a higher-dimensional generalization [Skopenkov2016e].

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

**Corollary 3.2** ([Haefliger1966], [Hausmann1972], [Takase2006])**.**
(a) The Kreck invariant is a 1--1 correspondence if is or an integral homology sphere. (For the Kreck invariant is also a group isomorphism; this follows not from Theorem 3.1 but from [Haefliger1966].)

(b) If (i.e. is a rational homology sphere, e.g. ), then is in (non-canonical) 1-1 correspondence with .

(c) Isotopy classes of embeddings with zero Whitney invariant are in 1-1 correspondence with , and for each integer there are exactly isotopy classes of embeddings with the Whitney invariant , cf. Corollary 3.4 below.

(d) The Whitney invariant is surjective and

**Addendum 3.3.**
Let is an embedding, the generator of and is a connected sum of copies of .
Then .

E. g. for the embedded connected sum action of on [Skopenkov2016c] is free while for we have the following corollary.

**Corollary 3.4.**
(a) The Hudson torus Hud(1) is an embedding such that for each knot the embedding is isotopic to .

(b) For each embedding such that (e.g. for the standard embedding ) and each non-trivial knot the embedding is not isotopic to .

(We believe that this very corollary or the case of Theorem 3.1 are as non-trivial as the general case of Theorem 3.1.)

See also [Avvakumov2016].

## 4 The Kreck invariant

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

An orientation-preserving diffeomorphism such that is called a *bundle isomorphism*. (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. Indeed, the restrictions to of and are isotopic (this is proved in definition of the Whitney invariant [Skopenkov2016e]). Define over by an isotopy between the restrictions to of and . Since , extends to . Then is spin. Cf. [Skopenkov2008, Spin Lemma].

**Definition 4.2.**
Take a small oriented disk whose intersection with consists of exactly one point
of sign and such that .
A *meridian of * is .
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 composition of the boundary map and the projection is an isomorphism, cf. [Skopenkov2008, the Alexander Duality Lemma].
The inverse to this composition is homology Alexander Duality isomorphism; it equals to the composition of the cohomology Alexander and Poincaré duality isomorphisms.

A *homology Seifert surface* for is the image of the fundamental class .

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 we have

This property provides an equivalent definition of a joint Seifert class which explains the name and which was used in [Skopenkov2008] together with the name `joint homology Seifert surface'.

Identify with the zero-dimensional homology group of closed connected oriented manifols.
The intersection products in 6-manifolds are omitted from the notation.
Denote by the signature of a 4-manifold .
For a closed connected oriented 6-manifold and 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.)

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

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- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, submitted to Bull. Man. Atl.
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