4-manifolds in 7-space
This page has been accepted for publication in the Bulletin of the Manifold Atlas. |
This page has not been refereed. The information given here might be incomplete or provisional. |
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.
See general introduction on embeddings, notation and conventions in [Skopenkov2016c, 1,
2].
2 Examples
The Hudson tori and
are defined in Remark 3.5.d of [Skopenkov2016e].
For an orientable 4-manifold , an embedding
and a class
one can construct an embedding
by linked connected sum analogously to embeddings into
[Skopenkov2016e].
If
is simply-connected and CAT=PL, this gives a
free transitive action of
on
[Skopenkov2016e].
We also have
for the Whitney invariant [Skopenkov2016e].
Denote by the Hopf fibration.
2.1 The Lambrechts torus and the Hudson torus
These two embeddings are defined as compositions
Tex syntax error
![i=1,2](/images/math/d/b/0/db0d627749cd37aa9ea271980b62bd37.png)
![pr_2](/images/math/e/f/9/ef9ef27f85488ac8d909b7de6fba5b63.png)
Tex syntax erroris the standard embedding (which is defined in [Skopenkov2015a,
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![t^i:S^1\times S^3\to S^3](/images/math/a/b/1/ab1ba09fe44411788bc0c22b4ded2eec.png)
![t^i|_{S^1\times y}](/images/math/d/5/6/d5654ad0d6eb21fa079e77862071c2fd.png)
![y\in S^3](/images/math/e/4/8/e483e81db822c88e0e07c26efaee539c.png)
![i=1,2](/images/math/d/b/0/db0d627749cd37aa9ea271980b62bd37.png)
![\tau^1](/images/math/e/8/c/e8c64e7b7d45d5f09b5209d475613998.png)
![\tau^2](/images/math/3/d/d/3dd287ca08cbaa37726de6cc785cca8c.png)
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
.
These examples appear in [Skopenkov2006] but could be known earlier.
Note that is PL isotopic to the Hudson torus
defined in Remark 3.5.d of [Skopenkov2016e].
![S^1\times S^3\to S^7](/images/math/c/c/3/cc3ecefdf0ba30b94cab7de1bcd673ab.png)
Tex syntax error(which is defined in [Skopenkov2015a,
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
Take the Hopf fibration . Take the standard embeding
. Its complement has the homotopy type of
. Then
.
We have
Tex syntax error
We conjecture that .
Example 2.1 ([Boechat&Haefliger1970, p.164].
There is an 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 is the only embedding
(up to isotopy and a hyperplane reflection of
).
Theorem 2.2.
(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 . The inverse is defined using linked connected sum [Skopenkov2016e].
Part (c) follows by [Boechat&Haefliger1970]. Parts (a) and (b) are proved in [Skopenkov2005, Triviality Theorem (a)] or follow by Theorem 4.2 below.
Example 2.3 ([Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]).
This 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 The Boechat-Haefliger invariant
Let be a closed connected orientable 4-manifold and
an embedding.
Fix an orientation on
and an orientation on
.
Definition 3.1. 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 homology Alexander Duality isomorphism; it equals to the composition
of the cohomology Alexander and Poincar\'e duality isomorphisms.
Definition 3.2.
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 3.3.
(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 3.2 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].
4 Classification
See a classification of for a closed connected 4-manifold
such that
in [Skopenkov2016e]. Here we work in the smooth category.
Theorem 4.1 ([Haefliger1966], see also [Skopenkov2005], [Crowley&Skopenkov2008]). .
Theorem 4.2 ([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 4.3.
(a) There are exactly twelve isotopy classes of embeddings if
is an integral homology 4-sphere (cf. Theorem 4.1).
(b) 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
. (We take the standard basis in
.)
Addendum 4.4.
Under assumptions of Theorem 4.2 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 corollaries are examples of the effectiveness and the triviality of the embedded connected sum action of on
.
Corollary 4.5.
(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
. (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 embeddings
and
the embedding
is isotopic to
; in other words,
is injective.
(First proved in [Skopenkov2005] independently of Theorem 4.2.)
(c) If is a closed connected 4-manifold such that
and
for an embedding
, then for each embedding
the embedding
is not isotopic to
.
5 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.
- [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.
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
2 Examples
The Hudson tori and
are defined in Remark 3.5.d of [Skopenkov2016e].
For an orientable 4-manifold , an embedding
and a class
one can construct an embedding
by linked connected sum analogously to embeddings into
[Skopenkov2016e].
If
is simply-connected and CAT=PL, this gives a
free transitive action of
on
[Skopenkov2016e].
We also have
for the Whitney invariant [Skopenkov2016e].
Denote by the Hopf fibration.
2.1 The Lambrechts torus and the Hudson torus
These two embeddings are defined as compositions
Tex syntax error
![i=1,2](/images/math/d/b/0/db0d627749cd37aa9ea271980b62bd37.png)
![pr_2](/images/math/e/f/9/ef9ef27f85488ac8d909b7de6fba5b63.png)
Tex syntax erroris the standard embedding (which is defined in [Skopenkov2015a,
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![t^i:S^1\times S^3\to S^3](/images/math/a/b/1/ab1ba09fe44411788bc0c22b4ded2eec.png)
![t^i|_{S^1\times y}](/images/math/d/5/6/d5654ad0d6eb21fa079e77862071c2fd.png)
![y\in S^3](/images/math/e/4/8/e483e81db822c88e0e07c26efaee539c.png)
![i=1,2](/images/math/d/b/0/db0d627749cd37aa9ea271980b62bd37.png)
![\tau^1](/images/math/e/8/c/e8c64e7b7d45d5f09b5209d475613998.png)
![\tau^2](/images/math/3/d/d/3dd287ca08cbaa37726de6cc785cca8c.png)
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
.
These examples appear in [Skopenkov2006] but could be known earlier.
Note that is PL isotopic to the Hudson torus
defined in Remark 3.5.d of [Skopenkov2016e].
![S^1\times S^3\to S^7](/images/math/c/c/3/cc3ecefdf0ba30b94cab7de1bcd673ab.png)
Tex syntax error(which is defined in [Skopenkov2015a,
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
Take the Hopf fibration . Take the standard embeding
. Its complement has the homotopy type of
. Then
.
We have
Tex syntax error
We conjecture that .
Example 2.1 ([Boechat&Haefliger1970, p.164].
There is an 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 is the only embedding
(up to isotopy and a hyperplane reflection of
).
Theorem 2.2.
(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 . The inverse is defined using linked connected sum [Skopenkov2016e].
Part (c) follows by [Boechat&Haefliger1970]. Parts (a) and (b) are proved in [Skopenkov2005, Triviality Theorem (a)] or follow by Theorem 4.2 below.
Example 2.3 ([Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]).
This 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 The Boechat-Haefliger invariant
Let be a closed connected orientable 4-manifold and
an embedding.
Fix an orientation on
and an orientation on
.
Definition 3.1. 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 homology Alexander Duality isomorphism; it equals to the composition
of the cohomology Alexander and Poincar\'e duality isomorphisms.
Definition 3.2.
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 3.3.
(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 3.2 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].
4 Classification
See a classification of for a closed connected 4-manifold
such that
in [Skopenkov2016e]. Here we work in the smooth category.
Theorem 4.1 ([Haefliger1966], see also [Skopenkov2005], [Crowley&Skopenkov2008]). .
Theorem 4.2 ([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 4.3.
(a) There are exactly twelve isotopy classes of embeddings if
is an integral homology 4-sphere (cf. Theorem 4.1).
(b) 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
. (We take the standard basis in
.)
Addendum 4.4.
Under assumptions of Theorem 4.2 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 corollaries are examples of the effectiveness and the triviality of the embedded connected sum action of on
.
Corollary 4.5.
(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
. (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 embeddings
and
the embedding
is isotopic to
; in other words,
is injective.
(First proved in [Skopenkov2005] independently of Theorem 4.2.)
(c) If is a closed connected 4-manifold such that
and
for an embedding
, then for each embedding
the embedding
is not isotopic to
.
5 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.
- [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.