3-manifolds in 6-space
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].
Here is the construction of .
Denote coordinates in
by
.
The Haefliger (higher-dimensional) trefoil knot can be constructed from the 3-dimensional Borromean rings, which an embedding
; see Figure 6 of [Skopenkov2016h]). The construction is analogous to the construction of the trefoil from the 1-dimensional Borromean rings, depicted in Figure 1. figure 1
<!by joining with two tubes the higher-dimensional Borromean rings,
see figure 6 of [Skopenkov2016h] and figure 1 (cf. figure 7 of [Skopenkov2016h]). -->
These are 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,\sqrt2xy,y^2).](/images/math/a/8/a/a8a67c9a1d736e528f511ad4ed420700.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
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
![\displaystyle W:E^6_D(N)\to H_1(N)](/images/math/b/c/4/bc401690743c9624de25d938c6ce71ef.png)
is surjective.
(b) For any the Kreck invariant
![\displaystyle \eta_u:W^{-1}(u)\to\Zz_{d(u)}](/images/math/2/d/6/2d6d76828884f217379208f1ee3a7526.png)
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
![\displaystyle W(f\#g)=W(f)\quad\text{and}\quad\eta_{W(f)}(f\#g)\equiv\eta_{W(f)}(f)+\eta_0(g)\mod d(W(f)).](/images/math/3/0/4/304e98bba20413a301e207552ca5af7c.png)
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
.)
![f:N\to\Rr^6](/images/math/7/c/d/7cd987e0bcb85fea61215f52dbc3dacf.png)
![f(N)\subset\Rr^5](/images/math/8/a/3/8a3d59bf470c8f866b3e7f2ee8b153b1.png)
Tex syntax error) and any non-trivial knot
![g:S^3\to\Rr^6](/images/math/4/f/7/4f75bd600355b5bc0cf21ead88e365ba.png)
![f\# g](/images/math/7/5/a/75a319ce5928547620d50020a8a57ced.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
(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
![\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.
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
![\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 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
![\displaystyle Y\cap C_f=A_f[N]\quad\text{and}\quad Y\cap C_{f'}=A_{f'}[N]](/images/math/6/9/0/6905fa47d8fa67614555dbc17455f507.png)
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
![\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.)
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
![\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
- [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, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv: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).