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 mathematician from other areas who want to apply or to learn theory of embeddings.
Many information on embeddings of 3-manifolds in 6-space is a particular case of embeddings of n-manifolds in 2n-space. In this page we concentrate on phenomena peculiar for n=3 and give references to general results.
See general introduction on embeddings, notation and conventions.
2 Examples
Simple examples are the Hudson tori and the Haefliger trefoil knot
.
2.1 The Hopf embedding of RP3 into S5
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 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 just below the stable range.)
2.2 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:
![\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 variables while
and
are constants.
This defines embeddings of
,
or
into
.
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. A classification in the PL category.
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
.
Recall that for an abelian group the divisibility of zero is zero and the divisibility of
is
.
Cf. higher-dimensional generalization.
All isotopy classes of embeddings can be constructed from a certain given embedding using
unlinked and
linked embedded connected sum with embeddings
.
Corollary 3.2.
(a) The Kreck invariant is a 1--1 correspondence if
is
or an integral homology sphere [Haefliger1966], [Hausmann1972], [Takase2006]. (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
![\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 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
is free while for
we have the following corollary.
Corollary 3.4.
(a) There 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.)
4 The Kreck invariant
We work in the smooth category. 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.
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.
Indeed, the restrictions to of
and
are isotopic (this is proved in Definition 4.1).
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
![\displaystyle Y\in H_5(M_\varphi)\quad\text{such that}\quad Y\cap [\partial D^3_f]=1.](/images/math/b/9/2/b92fed83797b13cdac2b6b9bc83c5ea8.png)
If and
is a spin bundle isomorphism, then there is a joint Seifert class for
and
[Skopenkov2008], Agreement Lemma.
Denote by and
Poincar\'e 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 Poincar\'e 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'.
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
![\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
,
-
a spin bundle isomorphism,
-
a closed connected oriented 4-submanifold representing a joint Seifert class for
and
-
,
are the Pontryagin number and Poincar\'e 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
- [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.
- [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
- [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).