3-manifolds in 6-space
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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.
The classification of embeddings of 3-manifolds in 6-space is of course a particular case of
classification of embeddings of n-manifolds in 2n-space which is discussed in [Skopenkov2016e], [Skopenkov2006, 2.4 `The Whitney invariant'].
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,
3].
For definition of the
embedded connected sum
of embeddings of 3-manifolds
in 6-space, and for the corresponding action of the group
on the set
, see e.g. [Skopenkov2016c,
5].
2 Examples
For any integer there is an embedding called the Hudson torus,
, see [Skopenkov2016e,
3], [Skopenkov2006, Example 2.10].
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 [Skopenkov2016f, Remark 1.1] isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 [Skopenkov2016c]).
This embedding is a generator of [Haefliger1962,
4.1].
Here is the construction of .
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:
![\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1\end{array}\right., \qquad \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..](/images/math/b/3/f/b3f574425ee3eab2ba05dc7a32c1a793.png)
See motivating examples of links [Skopenkov2016h] and a higher-dimensional generalization [Skopenkov2016k].
Example 2.2 (The Hopf embedding of into
).
Represent
Define
![\displaystyle h:\Rr P^3\to S^5\subset\Cc^3\quad\text{by}\quad h[(x,y)]=(x^2,2xy,y^2).](/images/math/3/a/4/3a40725f0fba92894a95ccb55824db7e.png)
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]:
![\displaystyle R_1^2+R_2^2+R_3^2=c_1,\quad R_1S_1+R_2S_2+R_3S_3=c_2,\quad \frac{S_1^2}{A_1}+\frac{S_2^2}{A_2}+\frac{S_3^2}{A_3}=c_3,](/images/math/4/4/a/44af3531a5e0778314b830d776d91a4b.png)
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 the classical a classification in the PL category see [Skopenkov2016e, Theorem 2.1], [Skopenkov2006, Theorem 2.13].
For the next theorem,
the Whitney invariant
is defined in [Skopenkov2016e], [Skopenkov2008].
For an abelian group
the divisibility of the identity element is zero and the divisibility of
is
.
The Kreck invariant
is defined in
4 below.
Theorem 3.1. The Whitney invariant
![\displaystyle W:E^6(N)\to H_1(N)](/images/math/f/6/e/f6e8afcb5a6a2d467d59ec088f1cd436.png)
is surjective. For each the Kreck invariant
![\displaystyle \eta_a:W^{-1}(u)\to\Zz_{d(a)}](/images/math/8/4/5/8450e79f095ff4efd863acc62ade1092.png)
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
.
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 3.4 below.
(d) The Whitney invariant is surjective and
![\displaystyle |W^{-1}(a_1\oplus a_2)|=\begin{cases} \infty & d(a_1)=d(a_2)=0\\ \gcd(d(a_1),d(a_2)) &\text{otherwise} \end{cases}.](/images/math/3/1/2/31217387fc6d14a34e746e85424bcf8e.png)
Addendum 3.3.
Let be an isotopy class of an embedding
,
the generator of
and
the connected sum of
copies of
.
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 3.4.
(a) The Hudson torus Hud(1) is an embedding such that for any knot
the embedding
is isotopic to
.
(b) For any embedding such that
(e.g. for the standard embedding
) and any non-trivial knot
the embedding
is not isotopic to
.
(We believe that this very corollary or the case of Theorem 3.1 are as hard to prove as the general case of Theorem 3.1.)
See [Avvakumov2016] for classification of embeddings of some disconnected 3-manifolds in 6-space.
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.
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
![\displaystyle M_\varphi:=C_f\cup_\varphi(-C_{f'}).](/images/math/a/1/3/a133b333b68f4e4b2f3124eca69d7546.png)
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.
Fix orientations on and on
. Identify with
the zero-dimensional homology group of a closed connected oriented manifold. The sign of the intersection product
in homology of 6-manifolds
[Kirby1989, Chapter II] is omitted from the notation.
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
![\displaystyle Y\in H_4(M_\varphi)\quad\text{such that}\quad Y[\partial D^3_f]=1.](/images/math/3/d/c/3dc5178b68f6a78091cf6fc7750e7b87.png)
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 called 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
![\displaystyle Y\cap C_f=A_f[N]\quad\text{and}\quad Y\cap C_{f'}=A_{f'}[N].](/images/math/e/9/0/e90166bcfe56c6f6c07da06a6cb601a9.png)
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
.
For a closed connected oriented 6-manifold
and
let the virtual signature of
be
![\displaystyle \sigma_x(Q):=\frac{xPDp_1(Q)-x^3}3\in H_0(Q)=\Zz.](/images/math/e/2/a/e2a60a44a289d7dfc1e692c2544cf49f.png)
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
![\displaystyle \eta(f,f'):=\rho_d\frac{\sigma_{2Y}(M_\varphi)}{16}=\rho_d\frac{PDp_1(M_\varphi)Y-4Y^3}{24}\in\Zz_d,](/images/math/3/4/5/3459090e181f9a0cea225265566abe87.png)
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
![\displaystyle \frac{\sigma_{2[Y]}(M_\varphi)}{16}=\frac{\sigma(Y)-\overline p_1}8= \frac{\sigma(Y)-\overline e\cap\overline e}8.](/images/math/9/7/6/976d016617f48101c3311526cf28aa4a.png)
5 References
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