4-manifolds in 7-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 the theory of embeddings.
Basic results on 4-manifolds in 7-space are particular cases of results on n-manifolds in (2n-1)-space for n=4 [Skopenkov2016e]. In this page we concentrate on more advanced results peculiar for n=4.
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 of knotted tori
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![p,q>0](/images/math/a/a/c/aac0c938c9f2e91f1239594c25967c69.png)
![p+q\le6](/images/math/5/f/7/5f7972ce82717d80ce4f0be133e3476b.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
The Hudson tori and
are defined for an integer
in Remark 3.5.d of [Skopenkov2016e].
Denote by the Hopf fibration and by
the projection onto the
-th factor of a Cartesian product.
Define
by the equations
and
, respectively.
Example 2.1 (Spinning construction).
For an embedding denote by
the embedding
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The restriction of to
is isotopic to the standard embedding.
We conjecture that if
is the Haefliger trefoil knot, then
is not smoothly isotopic to the connected sum of the standard embedding and any embedding
.
The following Examples 2.2 and 2.3 appear in [Skopenkov2006] but could be known earlier.
Example 2.2.
Two sembeddings are defined as compositions
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where and maps
are defined below. We shall see that
is an embedding for each
and
, hence
and
are embeddings.
Define , where
is identified with the set of unit length quaternions and
with the set of unit length complex numbers.
Define , where
is identified with the 2-sphere formed by unit
length quaternions of the form
.
It would be interesting to know if is PL or smoothly isotopic to the Hudson torus
.
Example 2.2 can be generalized as follows.
Example 2.3.
Define a map
Take a smooth map
.
Assuming that
, we have
.
Define the adjunction map
by
.
(Assuming that
, this map is obtained from
by the exponential law.)
Denote by
the restriction of the adjunction map.
We define the embedding
to be the composition
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We define the map by
, where
represents
(for the standard identification
).
Clearly, and
.
See a generalization in [Skopenkov2016k].
It would be interesting to know if is smoothly or piecewise smoothly (PS) isotopic to
for each
.
We conjecture that
- every PS embedding
is PS isotopic to
for some
.
- every smooth embedding
is smoothly isotopic to
for some
and embedding
.
Example 2.4 (the Lambrechts torus).
There is a smooth embedding whose complement is not homotopy equivalent to the complement of the standard embedding.
I learned this simple construction from P. Lambrechts. Take the Hopf fibration . Take the Hopf linking
[Skopenkov2016h]. Then
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We conjecture that .
Example 2.5 (the Haefliger torus [Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]).
There is a PL embedding which is (locally flat but) not PL isotopic to a smooth embedding.
Take the Haefliger trefoil knot . Extend it to a conical embedding
. By [Haefliger1962], the trefoil knot also extends to a smooth embedding
[Skopenkov2006, Figure 3.7.a]. These two extensions together form the Haefliger torus [Skopenkov2006, Figure 3.7.b].
3 Embeddings of the complex projective plabe
Example 3.1 ([Boechat&Haefliger1970, p.164].
There is a smooth embedding .
Recall that is the mapping cylinder of
. Recall that
. Define an embedding
by
, where
. In other words, the segment joining
and
is mapped onto the arc in
joining
to
. Clearly, the boundary 3-sphere of
is standardly embedded into
. Hence
extends to an embedding
.
Apriori the above extension need not be unique (because it can be changed by embedded connected sum with an embedding ). Surprisingly, it is unique, and in the smooth category is the only embedding
(up to isotopy and a hyperplane reflection of
).
Theorem 3.2.
(a) There are exactly two smooth isotopy classes of smooth embeddings (differing by a hyperplane reflection of
).
(b) For each pair of smooth embeddings and
the embedding
is smoothly isotopic to
.
(c) The Whitney invariant is a 1-1 correspondence .
Parts (a) and (b) are proved in [Skopenkov2005, Triviality Theorem (a)] or follow by Theorem 5.2 below. Part (c) follows by [Boechat&Haefliger1970], cf. a generalization presented in [Skopenkov2016e].
4 The Boechat-Haefliger invariant
Let be a closed connected orientable 4-manifold and
an embedding.
Fix an orientation on
and an orientation on
.
Definition 4.1. The composition
![\displaystyle H_{s+1}(C_f,\partial)\overset\partial\to H_s(\partial C_f)\overset{\nu_f}\to H_s(N)](/images/math/5/4/1/54170c6044501f4348a1ad0ec6f914ae.png)
of the boundary map and the projection
is an isomorphism, cf. [Skopenkov2008, the Alexander Duality Lemma].
The inverse
to this composition is the homology Alexander Duality isomorphism; it equals to the composition
of the cohomology Alexander and Poincaré duality isomorphisms.
Definition 4.2.
A homology Seifert surface for is the image
of the fundamental class
.
Define
![\displaystyle \varkappa(f):=A_{f,2}^{-1}\left(A_{f,4}[N]\cap A_{f,4}[N]\right)\in H_2(N).](/images/math/5/2/6/526b41e5f32beb29292745853899e8d0.png)
Remark 4.3.
(a) Take a small oriented disk whose intersection with
consists of exactly one point
of sign
and such that
.
A meridian of
is
.
A homology Seifert surface
for
is uniquely defined by the condition
.
(b) We have for the Whitney invariant
[Skopenkov2016e].
This is proved analogously to [Skopenkov2008,
2, The Boechat-Haefliger Invariant Lemma].
(c) Definition 4.2 is equivalent to the original one [Boechat&Haefliger1970] by [Crowley&Skopenkov2008, Section Lemma 3.1].
Hence .
(d) Earlier notation was [Boechat&Haefliger1970],
[Skopenkov2005] and
[Crowley&Skopenkov2008].
5 Classification
For the classification of for a closed connected 4-manifold
with
, see [Skopenkov2016e]. Here we work in the smooth category.
Theorem 5.1 ([Haefliger1966], see also [Skopenkov2005], [Crowley&Skopenkov2008]). There is an isomorphism .
Theorem 5.2 ([Crowley&Skopenkov2008]). Let be a closed connected 4-manifold such that
. Then the image of the Boéchat-Haefliger invariant
![\displaystyle \varkappa:E^7_D(N)\to H_2(N)](/images/math/2/c/a/2ca4b6cf32c84d90c13faf18e275dd2f.png)
![\displaystyle \text{is}\quad im \varkappa=\{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.](/images/math/8/2/2/822af13b07c8441f924ed0ee95fdbbed.png)
For each there is an injective invariant called the Kreck invariant,
![\displaystyle \eta_u:\varkappa^{-1}(u)\to\Zz_{\gcd(u,24)}](/images/math/b/a/0/ba03ff3e3f79a98c38a994022616b771.png)
whose image is the subset of even elements.
Here is the maximal integer
such that both
and 24 are divisible by
.
Thus
is surjective if
is not divisible by 2.
Note that
is divisible by 2 (for some
or, equivalently, for each
) if and only if
is spin.
For the definition of the Kreck invariant see [Crowley&Skopenkov2008].
Corollary 5.3.
(a) There are exactly twelve isotopy classes of embeddings if
is an integral homology 4-sphere (cf. Theorem 5.1).
(b) Identify using the standard basis.
For each integer
there are exactly
isotopy classes of embeddings
with
, and the same holds for those with
.
Other values of
are not in the image of
.
Addendum 5.4.
Under the assumptions of Theorem 5.2 for each pair of embeddings and
![\displaystyle \varkappa(f\#g)=\varkappa(f)\quad\text{and}\quad\eta_{\varkappa(f)}(f\#g)\equiv\eta_{\varkappa(f)}(f)+\eta_0(g)\mod\gcd(\varkappa(f),24).](/images/math/0/7/4/07401121d21e2edfe0fd25ea756daeb6.png)
The following corollary gives examples where the embedded connected sum action of on
is trivial and where it is effective.
Corollary 5.5.
(a) Take an integer and the Hudson torus
defined in Remark 3.5.d of [Skopenkov2016e]. If
, then for each embedding
the embedding
is isotopic to
. Moreover, for a general integer
the number of isotopy classes of embeddings
is
.
(b) Let be a closed connected 4-manifold such that
and the signature
of
is not divisible by the square of an integer
.
Then for each pair of embeddings
and
the embedding
is isotopic to
; in other words,
is injective.
(c) If is a closed connected 4-manifold such that
and
for an embedding
, then for every embedding
the embedding
is not isotopic to
.
We remark that Corollary 5.5(b) was first proved in [Skopenkov2005] independently of Theorem 5.2.
For classification when see [Crowley&Skopenkov2016] and [Crowley&Skopenkov2016a].
6 References
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension
dans
, (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
- [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension
dans
, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
- [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Crowley&Skopenkov2016a] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, II. Smooth classification. Proc. A of the Royal Soc. of Edinburgh, to appear. arXiv:1612.04776
- [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
- [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
- [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
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [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.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.