4-manifolds in 7-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 mathematician from other areas who want to apply or to learn the theory of embeddings.
Basic results on embeddings of closed connected 4-manifolds in 7-space are particular cases of results on
embeddings of -manifolds in
-space which is discussed in [Skopenkov2016e], [Skopenkov2006,
2.4 `The Whitney invariant'].
In this page we concentrate on more advanced classification results peculiar for
.
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 4-manifolds
in 7-space, and for the corresponding action of the group
on the set
, see e.g. [Skopenkov2016c,
5].
Remark 1.1 (PL and piecewise smooth embeddings). Any smooth manifold has a unique (up to PL homeomorphism) PL structure compatible with the given smooth structure [Milnor&Stasheff1974, Complement].
Since also any PL 4-manifold admits a unique smooth structure [Mandelbaum1980, 1.2], we may consider a smooth 4-manifold as a PL 4-manifold.
A map of a smooth manifold is piecewise smooth (PS) if it is smooth on every simplex of some triangulation of the manifold. Clearly, every smooth or PL map is PS.
For a smooth manifold let
be the set of PS embeddings
up to PS isotopy.
The forgetful map
is 1--1 [Haefliger1967, 2.4].
So a description of
is the same as a description of
.
2 Examples of knotted tori
The Hudson tori and
are defined for an integer
in Remark 3.5.d of [Skopenkov2016e] or in [Skopenkov2006, Example 2.10].
Tex syntax errorthe projection onto the
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Define by the equations
and
, respectively.
Example 2.1 (Spinning construction).
For an embedding denote by
the embedding
Tex syntax error
The restriction of to
is isotopic to (the restriction to
of) 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 embeddings are defined as compositions
Tex syntax error
where and maps
are defined below.
We shall see that
is an embedding for any
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 the Hopf fibration and
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
Tex syntax error
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 or
for any
.
The unpublished papers [Crowley&Skopenkov2016], [Crowley&Skopenkov2016a] prove that
- any PS embedding
represents
for some
.
- any smooth embedding
represents
for some
and
.
Example 2.4 (The Lambrechts torus). There is an 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
Tex syntax error
Tex syntax errorfor
![m\ge p+q+3](/images/math/3/f/7/3f791e68c3af31a14faaa173882052fa.png)
![p\ge0](/images/math/b/f/0/bf065379f797991fb7162986d2447de6.png)
![f:N\to S^n](/images/math/7/b/6/7b6dec8a999191cab715f1f12ff53b3b.png)
Tex syntax error.
(We conjecture that .)
Example 2.5 (the Haefliger torus [Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]).
There is a PL embedding which is not PS 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
, see the figure [Skopenkov2006, Figure 3.7.a]. These two extensions together form the Haefliger torus, see the figure [Skopenkov2006, Figure 3.7.b].
3 Embeddings of the complex projective plane
Example 3.1 [Boechat&Haefliger1970, p.164].
There is an embedding .
Recall that is the mapping cylinder of the Hopf fibration
. 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
.
Theorem 3.2. (a) There is only one embedding up to isotopy and a hyperplane reflection of
.
In other words, there are exactly two isotopy classes of embeddings
(differing by
composition with a hyperplane reflection of
).
(b) For any pair of embeddings and
the embedding
is isotopic to
.
(c) The Whitney invariant
(defined in [Skopenkov2016e], [Skopenkov2006, 2]) is a 1-1 correspondence
.
However, any PL embedding whose Whitney invariant w.r.t. the embedding of Example 3.1 is different from
is not smoothable.
Parts (a) and (b) are proved in [Skopenkov2005, Triviality Theorem (a)] or follow by Theorem 5.3 below. Part (c) follows by [Boechat&Haefliger1970], cf. a generalization presented in [Skopenkov2016e].
4 The Boechat-Haefliger invariant
We give definitions in more generality because this is natural and is required for 3-manifolds in 6-space [Skopenkov2016t].
Let be a closed connected orientable
-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.
This is well-known, see [Skopenkov2008,
2, 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.
This is not to be confused with another well-known homological Alexander duality isomorphism [Skopenkov2005, Alexander Duality Lemma 4.6].
Definition 4.2.
A `homology Seifert surface' for is the image
of the fundamental class
.
Denote by the intersection products
and
.
Remark 4.3.
Take a small oriented disk whose intersection with
consists of exactly one point of sign
and such that
.
A homology Seifert surface
for
is uniquely defined by the condition
.
Definition 4.4.
Define `the Boechat-Haefliger invariant' of
![\displaystyle \varkappa(f):=A_{f,2n+1-m}^{-1}\left(A_{f,n}[N]\cap A_{f,n}[N]\right)\in H_{2n+1-m}(N).](/images/math/6/9/1/691237d46ac79e4465fe5e708e568f66.png)
Clearly, a map is well-defined by
.
Remark 4.5.
(a) If , then
for any two embeddings
[Skopenkov2008,
2, The Boechat-Haefliger Invariant Lemma].
Here
is the Whitney invariant [Skopenkov2016e], [Skopenkov2006,
2].
We conjecture that this holds for any
, in particular, that
when
is even.
(b) Definition 4.4 is equivalent to the original one for [Boechat&Haefliger1970] by [Crowley&Skopenkov2008, Section Lemma 3.1].
Earlier notation for
was
[Boechat&Haefliger1970],
[Skopenkov2005] and
[Crowley&Skopenkov2008].
5 Classification
We use Stiefel-Whitney characteristic classes and
.
Theorem 5.1. (a) Any orientable (or compact nonclosed) 4-manifold embeds into .
(b) A closed 4-manifold embeds into
of and only if
.
The nonclosed case of (a) is simple [Hirsch1961]. The closed case of (a) easily follows either from (b) by [Massey1960] or from Theorem 5.3.a below by [Donaldson1987]. The latter was noticed in [Fuquan1994] together with a proof of (b).
The PL version of the closed case of (a) was proved in [Hirsch1965].
The PL version of (b) follows from [Fuquan1994, Main Theorem A] by the second paragraph of Remark 1.1.
A simpler proof of the PL versions of (b) is given as the proof of [Skopenkov1997, Corollary 1.3.a] (for specialists recall that for a closed 4-manifold
).
Theorem 5.2 ([Haefliger1966], see also [Skopenkov2005], [Crowley&Skopenkov2008]). There is an isomorphism .
Theorem 5.3.
Let be a closed connected oriented 4-manifold.
Tex syntax errorof the Boéchat-Haefliger invariant
![\displaystyle \varkappa:E^7_D(N)\to H_2(N)](/images/math/9/f/e/9fe91d2bb98db9a2da3cad5c2348a690.png)
![\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.](/images/math/5/3/e/53e20ce7966b65b7fcdc0fe7eddecc62.png)
![H_1(N)=0](/images/math/5/0/0/500d0aa2c8b9d280b3c5e4e86cb00666.png)
Tex syntax errorthere 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 Poincaré isomorphism.
-
is the intersection form.
-
is the signature (of the intersection form) of
.
-
is the maximal integer
such that both
and 24 are divisible by
.
-
is defined in [Crowley&Skopenkov2008].
Thus is surjective if
is not divisible by 2.
Tex syntax erroris divisible by 2 (for some
![u](/images/math/c/4/a/c4a5b5e310ed4c323e04d72afae39f53.png)
![u](/images/math/c/4/a/c4a5b5e310ed4c323e04d72afae39f53.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
For a classification when see [Crowley&Skopenkov2016] and [Crowley&Skopenkov2016a].
Known classification results in the PL category for closed connected 4-manifolds are particular cases of
higher-dimensional results presented in [Skopenkov2016e], [Skopenkov2006, Theorem 2.13].
Corollary 5.4 ([Crowley&Skopenkov2008, Corollary 1.2]).
(a) There are exactly twelve isotopy classes of embeddings if
is an integral homology 4-sphere (cf. Theorem 5.2).
(b) Identify using the standard basis.
For any 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 5.5 ([Crowley&Skopenkov2008, Addendum 1.3]).
Under the assumptions of Theorem 5.3 for any 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 5.6 ([Crowley&Skopenkov2008, Corollary 1.4]).
(a) Take an integer and the Hudson torus
defined in Remark 3.5.d of [Skopenkov2016e], [Skopenkov2006, Example 2.10]. If
, then for any embedding
the embedding
is isotopic to
. Moreover, for any 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 any 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 5.6(b) was first proved in [Skopenkov2005] independently of Theorem 5.3.
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
- [Donaldson1987] S. K. Donaldson, The orientation of Yang-Mills moduli spaces and
-manifold topology, J. Differential Geom. 26 (1987), no.3, 397–428. MR910015 (88j:57020) Zbl 0683.57005
- [Fuquan1994] F. Fuquan, Embedding four manifolds in
, Topology 33 (1994), 447-454.
- [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
- [Haefliger1967] A. Haefliger, Lissage des immersions-I, Topology, 6 (1967) 221--240.
- [Hirsch1961] M. W. Hirsch, The embedding of bounding manifold in euclidean space, Ann. of Math. 74 (1961), 494-497.
- [Hirsch1965] M. W. Hirsch, On embedding 4-manifolds in
, Proc. Camb. Phil. Soc. 61 (1965).
- [Mandelbaum1980] R. Mandelbaum, Four-Dimensional Topology: An introduction, Bull. Amer. Math. Soc. (N.S.) 2 (1980) 1-159.
- [Massey1960] W. S. Massey, On the Stiefel--Whitney classes of a manifold, I, Amer. J. Math. 82 (1960), 92-102.
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Skopenkov1997] A. Skopenkov, On the deleted product criterion for embeddability of manifolds in
, Comment. Math. Helv. 72 (1997), 543-555.
- [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
- [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.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
![(2n-1)](/images/math/0/6/0/0600c5df3c8e820e54cdf05fe40f90da.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![n=4](/images/math/d/c/c/dcc689845e2da8f1c573cd4ff593a360.png)
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 4-manifolds
in 7-space, and for the corresponding action of the group
on the set
, see e.g. [Skopenkov2016c,
5].
Remark 1.1 (PL and piecewise smooth embeddings). Any smooth manifold has a unique (up to PL homeomorphism) PL structure compatible with the given smooth structure [Milnor&Stasheff1974, Complement].
Since also any PL 4-manifold admits a unique smooth structure [Mandelbaum1980, 1.2], we may consider a smooth 4-manifold as a PL 4-manifold.
A map of a smooth manifold is piecewise smooth (PS) if it is smooth on every simplex of some triangulation of the manifold. Clearly, every smooth or PL map is PS.
For a smooth manifold let
be the set of PS embeddings
up to PS isotopy.
The forgetful map
is 1--1 [Haefliger1967, 2.4].
So a description of
is the same as a description of
.
2 Examples of knotted tori
The Hudson tori and
are defined for an integer
in Remark 3.5.d of [Skopenkov2016e] or in [Skopenkov2006, Example 2.10].
Tex syntax errorthe projection onto the
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Define by the equations
and
, respectively.
Example 2.1 (Spinning construction).
For an embedding denote by
the embedding
Tex syntax error
The restriction of to
is isotopic to (the restriction to
of) 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 embeddings are defined as compositions
Tex syntax error
where and maps
are defined below.
We shall see that
is an embedding for any
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 the Hopf fibration and
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
Tex syntax error
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 or
for any
.
The unpublished papers [Crowley&Skopenkov2016], [Crowley&Skopenkov2016a] prove that
- any PS embedding
represents
for some
.
- any smooth embedding
represents
for some
and
.
Example 2.4 (The Lambrechts torus). There is an 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
Tex syntax error
Tex syntax errorfor
![m\ge p+q+3](/images/math/3/f/7/3f791e68c3af31a14faaa173882052fa.png)
![p\ge0](/images/math/b/f/0/bf065379f797991fb7162986d2447de6.png)
![f:N\to S^n](/images/math/7/b/6/7b6dec8a999191cab715f1f12ff53b3b.png)
Tex syntax error.
(We conjecture that .)
Example 2.5 (the Haefliger torus [Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]).
There is a PL embedding which is not PS 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
, see the figure [Skopenkov2006, Figure 3.7.a]. These two extensions together form the Haefliger torus, see the figure [Skopenkov2006, Figure 3.7.b].
3 Embeddings of the complex projective plane
Example 3.1 [Boechat&Haefliger1970, p.164].
There is an embedding .
Recall that is the mapping cylinder of the Hopf fibration
. 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
.
Theorem 3.2. (a) There is only one embedding up to isotopy and a hyperplane reflection of
.
In other words, there are exactly two isotopy classes of embeddings
(differing by
composition with a hyperplane reflection of
).
(b) For any pair of embeddings and
the embedding
is isotopic to
.
(c) The Whitney invariant
(defined in [Skopenkov2016e], [Skopenkov2006, 2]) is a 1-1 correspondence
.
However, any PL embedding whose Whitney invariant w.r.t. the embedding of Example 3.1 is different from
is not smoothable.
Parts (a) and (b) are proved in [Skopenkov2005, Triviality Theorem (a)] or follow by Theorem 5.3 below. Part (c) follows by [Boechat&Haefliger1970], cf. a generalization presented in [Skopenkov2016e].
4 The Boechat-Haefliger invariant
We give definitions in more generality because this is natural and is required for 3-manifolds in 6-space [Skopenkov2016t].
Let be a closed connected orientable
-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.
This is well-known, see [Skopenkov2008,
2, 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.
This is not to be confused with another well-known homological Alexander duality isomorphism [Skopenkov2005, Alexander Duality Lemma 4.6].
Definition 4.2.
A `homology Seifert surface' for is the image
of the fundamental class
.
Denote by the intersection products
and
.
Remark 4.3.
Take a small oriented disk whose intersection with
consists of exactly one point of sign
and such that
.
A homology Seifert surface
for
is uniquely defined by the condition
.
Definition 4.4.
Define `the Boechat-Haefliger invariant' of
![\displaystyle \varkappa(f):=A_{f,2n+1-m}^{-1}\left(A_{f,n}[N]\cap A_{f,n}[N]\right)\in H_{2n+1-m}(N).](/images/math/6/9/1/691237d46ac79e4465fe5e708e568f66.png)
Clearly, a map is well-defined by
.
Remark 4.5.
(a) If , then
for any two embeddings
[Skopenkov2008,
2, The Boechat-Haefliger Invariant Lemma].
Here
is the Whitney invariant [Skopenkov2016e], [Skopenkov2006,
2].
We conjecture that this holds for any
, in particular, that
when
is even.
(b) Definition 4.4 is equivalent to the original one for [Boechat&Haefliger1970] by [Crowley&Skopenkov2008, Section Lemma 3.1].
Earlier notation for
was
[Boechat&Haefliger1970],
[Skopenkov2005] and
[Crowley&Skopenkov2008].
5 Classification
We use Stiefel-Whitney characteristic classes and
.
Theorem 5.1. (a) Any orientable (or compact nonclosed) 4-manifold embeds into .
(b) A closed 4-manifold embeds into
of and only if
.
The nonclosed case of (a) is simple [Hirsch1961]. The closed case of (a) easily follows either from (b) by [Massey1960] or from Theorem 5.3.a below by [Donaldson1987]. The latter was noticed in [Fuquan1994] together with a proof of (b).
The PL version of the closed case of (a) was proved in [Hirsch1965].
The PL version of (b) follows from [Fuquan1994, Main Theorem A] by the second paragraph of Remark 1.1.
A simpler proof of the PL versions of (b) is given as the proof of [Skopenkov1997, Corollary 1.3.a] (for specialists recall that for a closed 4-manifold
).
Theorem 5.2 ([Haefliger1966], see also [Skopenkov2005], [Crowley&Skopenkov2008]). There is an isomorphism .
Theorem 5.3.
Let be a closed connected oriented 4-manifold.
Tex syntax errorof the Boéchat-Haefliger invariant
![\displaystyle \varkappa:E^7_D(N)\to H_2(N)](/images/math/9/f/e/9fe91d2bb98db9a2da3cad5c2348a690.png)
![\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.](/images/math/5/3/e/53e20ce7966b65b7fcdc0fe7eddecc62.png)
![H_1(N)=0](/images/math/5/0/0/500d0aa2c8b9d280b3c5e4e86cb00666.png)
Tex syntax errorthere 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 Poincaré isomorphism.
-
is the intersection form.
-
is the signature (of the intersection form) of
.
-
is the maximal integer
such that both
and 24 are divisible by
.
-
is defined in [Crowley&Skopenkov2008].
Thus is surjective if
is not divisible by 2.
Tex syntax erroris divisible by 2 (for some
![u](/images/math/c/4/a/c4a5b5e310ed4c323e04d72afae39f53.png)
![u](/images/math/c/4/a/c4a5b5e310ed4c323e04d72afae39f53.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
For a classification when see [Crowley&Skopenkov2016] and [Crowley&Skopenkov2016a].
Known classification results in the PL category for closed connected 4-manifolds are particular cases of
higher-dimensional results presented in [Skopenkov2016e], [Skopenkov2006, Theorem 2.13].
Corollary 5.4 ([Crowley&Skopenkov2008, Corollary 1.2]).
(a) There are exactly twelve isotopy classes of embeddings if
is an integral homology 4-sphere (cf. Theorem 5.2).
(b) Identify using the standard basis.
For any 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 5.5 ([Crowley&Skopenkov2008, Addendum 1.3]).
Under the assumptions of Theorem 5.3 for any 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 5.6 ([Crowley&Skopenkov2008, Corollary 1.4]).
(a) Take an integer and the Hudson torus
defined in Remark 3.5.d of [Skopenkov2016e], [Skopenkov2006, Example 2.10]. If
, then for any embedding
the embedding
is isotopic to
. Moreover, for any 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 any 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 5.6(b) was first proved in [Skopenkov2005] independently of Theorem 5.3.
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
- [Donaldson1987] S. K. Donaldson, The orientation of Yang-Mills moduli spaces and
-manifold topology, J. Differential Geom. 26 (1987), no.3, 397–428. MR910015 (88j:57020) Zbl 0683.57005
- [Fuquan1994] F. Fuquan, Embedding four manifolds in
, Topology 33 (1994), 447-454.
- [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
- [Haefliger1967] A. Haefliger, Lissage des immersions-I, Topology, 6 (1967) 221--240.
- [Hirsch1961] M. W. Hirsch, The embedding of bounding manifold in euclidean space, Ann. of Math. 74 (1961), 494-497.
- [Hirsch1965] M. W. Hirsch, On embedding 4-manifolds in
, Proc. Camb. Phil. Soc. 61 (1965).
- [Mandelbaum1980] R. Mandelbaum, Four-Dimensional Topology: An introduction, Bull. Amer. Math. Soc. (N.S.) 2 (1980) 1-159.
- [Massey1960] W. S. Massey, On the Stiefel--Whitney classes of a manifold, I, Amer. J. Math. 82 (1960), 92-102.
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Skopenkov1997] A. Skopenkov, On the deleted product criterion for embeddability of manifolds in
, Comment. Math. Helv. 72 (1997), 543-555.
- [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
- [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.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
![(2n-1)](/images/math/0/6/0/0600c5df3c8e820e54cdf05fe40f90da.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![n=4](/images/math/d/c/c/dcc689845e2da8f1c573cd4ff593a360.png)
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 4-manifolds
in 7-space, and for the corresponding action of the group
on the set
, see e.g. [Skopenkov2016c,
5].
Remark 1.1 (PL and piecewise smooth embeddings). Any smooth manifold has a unique (up to PL homeomorphism) PL structure compatible with the given smooth structure [Milnor&Stasheff1974, Complement].
Since also any PL 4-manifold admits a unique smooth structure [Mandelbaum1980, 1.2], we may consider a smooth 4-manifold as a PL 4-manifold.
A map of a smooth manifold is piecewise smooth (PS) if it is smooth on every simplex of some triangulation of the manifold. Clearly, every smooth or PL map is PS.
For a smooth manifold let
be the set of PS embeddings
up to PS isotopy.
The forgetful map
is 1--1 [Haefliger1967, 2.4].
So a description of
is the same as a description of
.
2 Examples of knotted tori
The Hudson tori and
are defined for an integer
in Remark 3.5.d of [Skopenkov2016e] or in [Skopenkov2006, Example 2.10].
Tex syntax errorthe projection onto the
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Define by the equations
and
, respectively.
Example 2.1 (Spinning construction).
For an embedding denote by
the embedding
Tex syntax error
The restriction of to
is isotopic to (the restriction to
of) 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 embeddings are defined as compositions
Tex syntax error
where and maps
are defined below.
We shall see that
is an embedding for any
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 the Hopf fibration and
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
Tex syntax error
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 or
for any
.
The unpublished papers [Crowley&Skopenkov2016], [Crowley&Skopenkov2016a] prove that
- any PS embedding
represents
for some
.
- any smooth embedding
represents
for some
and
.
Example 2.4 (The Lambrechts torus). There is an 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
Tex syntax error
Tex syntax errorfor
![m\ge p+q+3](/images/math/3/f/7/3f791e68c3af31a14faaa173882052fa.png)
![p\ge0](/images/math/b/f/0/bf065379f797991fb7162986d2447de6.png)
![f:N\to S^n](/images/math/7/b/6/7b6dec8a999191cab715f1f12ff53b3b.png)
Tex syntax error.
(We conjecture that .)
Example 2.5 (the Haefliger torus [Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]).
There is a PL embedding which is not PS 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
, see the figure [Skopenkov2006, Figure 3.7.a]. These two extensions together form the Haefliger torus, see the figure [Skopenkov2006, Figure 3.7.b].
3 Embeddings of the complex projective plane
Example 3.1 [Boechat&Haefliger1970, p.164].
There is an embedding .
Recall that is the mapping cylinder of the Hopf fibration
. 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
.
Theorem 3.2. (a) There is only one embedding up to isotopy and a hyperplane reflection of
.
In other words, there are exactly two isotopy classes of embeddings
(differing by
composition with a hyperplane reflection of
).
(b) For any pair of embeddings and
the embedding
is isotopic to
.
(c) The Whitney invariant
(defined in [Skopenkov2016e], [Skopenkov2006, 2]) is a 1-1 correspondence
.
However, any PL embedding whose Whitney invariant w.r.t. the embedding of Example 3.1 is different from
is not smoothable.
Parts (a) and (b) are proved in [Skopenkov2005, Triviality Theorem (a)] or follow by Theorem 5.3 below. Part (c) follows by [Boechat&Haefliger1970], cf. a generalization presented in [Skopenkov2016e].
4 The Boechat-Haefliger invariant
We give definitions in more generality because this is natural and is required for 3-manifolds in 6-space [Skopenkov2016t].
Let be a closed connected orientable
-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.
This is well-known, see [Skopenkov2008,
2, 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.
This is not to be confused with another well-known homological Alexander duality isomorphism [Skopenkov2005, Alexander Duality Lemma 4.6].
Definition 4.2.
A `homology Seifert surface' for is the image
of the fundamental class
.
Denote by the intersection products
and
.
Remark 4.3.
Take a small oriented disk whose intersection with
consists of exactly one point of sign
and such that
.
A homology Seifert surface
for
is uniquely defined by the condition
.
Definition 4.4.
Define `the Boechat-Haefliger invariant' of
![\displaystyle \varkappa(f):=A_{f,2n+1-m}^{-1}\left(A_{f,n}[N]\cap A_{f,n}[N]\right)\in H_{2n+1-m}(N).](/images/math/6/9/1/691237d46ac79e4465fe5e708e568f66.png)
Clearly, a map is well-defined by
.
Remark 4.5.
(a) If , then
for any two embeddings
[Skopenkov2008,
2, The Boechat-Haefliger Invariant Lemma].
Here
is the Whitney invariant [Skopenkov2016e], [Skopenkov2006,
2].
We conjecture that this holds for any
, in particular, that
when
is even.
(b) Definition 4.4 is equivalent to the original one for [Boechat&Haefliger1970] by [Crowley&Skopenkov2008, Section Lemma 3.1].
Earlier notation for
was
[Boechat&Haefliger1970],
[Skopenkov2005] and
[Crowley&Skopenkov2008].
5 Classification
We use Stiefel-Whitney characteristic classes and
.
Theorem 5.1. (a) Any orientable (or compact nonclosed) 4-manifold embeds into .
(b) A closed 4-manifold embeds into
of and only if
.
The nonclosed case of (a) is simple [Hirsch1961]. The closed case of (a) easily follows either from (b) by [Massey1960] or from Theorem 5.3.a below by [Donaldson1987]. The latter was noticed in [Fuquan1994] together with a proof of (b).
The PL version of the closed case of (a) was proved in [Hirsch1965].
The PL version of (b) follows from [Fuquan1994, Main Theorem A] by the second paragraph of Remark 1.1.
A simpler proof of the PL versions of (b) is given as the proof of [Skopenkov1997, Corollary 1.3.a] (for specialists recall that for a closed 4-manifold
).
Theorem 5.2 ([Haefliger1966], see also [Skopenkov2005], [Crowley&Skopenkov2008]). There is an isomorphism .
Theorem 5.3.
Let be a closed connected oriented 4-manifold.
Tex syntax errorof the Boéchat-Haefliger invariant
![\displaystyle \varkappa:E^7_D(N)\to H_2(N)](/images/math/9/f/e/9fe91d2bb98db9a2da3cad5c2348a690.png)
![\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.](/images/math/5/3/e/53e20ce7966b65b7fcdc0fe7eddecc62.png)
![H_1(N)=0](/images/math/5/0/0/500d0aa2c8b9d280b3c5e4e86cb00666.png)
Tex syntax errorthere 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 Poincaré isomorphism.
-
is the intersection form.
-
is the signature (of the intersection form) of
.
-
is the maximal integer
such that both
and 24 are divisible by
.
-
is defined in [Crowley&Skopenkov2008].
Thus is surjective if
is not divisible by 2.
Tex syntax erroris divisible by 2 (for some
![u](/images/math/c/4/a/c4a5b5e310ed4c323e04d72afae39f53.png)
![u](/images/math/c/4/a/c4a5b5e310ed4c323e04d72afae39f53.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
For a classification when see [Crowley&Skopenkov2016] and [Crowley&Skopenkov2016a].
Known classification results in the PL category for closed connected 4-manifolds are particular cases of
higher-dimensional results presented in [Skopenkov2016e], [Skopenkov2006, Theorem 2.13].
Corollary 5.4 ([Crowley&Skopenkov2008, Corollary 1.2]).
(a) There are exactly twelve isotopy classes of embeddings if
is an integral homology 4-sphere (cf. Theorem 5.2).
(b) Identify using the standard basis.
For any 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 5.5 ([Crowley&Skopenkov2008, Addendum 1.3]).
Under the assumptions of Theorem 5.3 for any 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 5.6 ([Crowley&Skopenkov2008, Corollary 1.4]).
(a) Take an integer and the Hudson torus
defined in Remark 3.5.d of [Skopenkov2016e], [Skopenkov2006, Example 2.10]. If
, then for any embedding
the embedding
is isotopic to
. Moreover, for any 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 any 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 5.6(b) was first proved in [Skopenkov2005] independently of Theorem 5.3.
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
- [Donaldson1987] S. K. Donaldson, The orientation of Yang-Mills moduli spaces and
-manifold topology, J. Differential Geom. 26 (1987), no.3, 397–428. MR910015 (88j:57020) Zbl 0683.57005
- [Fuquan1994] F. Fuquan, Embedding four manifolds in
, Topology 33 (1994), 447-454.
- [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
- [Haefliger1967] A. Haefliger, Lissage des immersions-I, Topology, 6 (1967) 221--240.
- [Hirsch1961] M. W. Hirsch, The embedding of bounding manifold in euclidean space, Ann. of Math. 74 (1961), 494-497.
- [Hirsch1965] M. W. Hirsch, On embedding 4-manifolds in
, Proc. Camb. Phil. Soc. 61 (1965).
- [Mandelbaum1980] R. Mandelbaum, Four-Dimensional Topology: An introduction, Bull. Amer. Math. Soc. (N.S.) 2 (1980) 1-159.
- [Massey1960] W. S. Massey, On the Stiefel--Whitney classes of a manifold, I, Amer. J. Math. 82 (1960), 92-102.
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Skopenkov1997] A. Skopenkov, On the deleted product criterion for embeddability of manifolds in
, Comment. Math. Helv. 72 (1997), 543-555.
- [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
- [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.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
![(2n-1)](/images/math/0/6/0/0600c5df3c8e820e54cdf05fe40f90da.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![n=4](/images/math/d/c/c/dcc689845e2da8f1c573cd4ff593a360.png)
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 4-manifolds
in 7-space, and for the corresponding action of the group
on the set
, see e.g. [Skopenkov2016c,
5].
Remark 1.1 (PL and piecewise smooth embeddings). Any smooth manifold has a unique (up to PL homeomorphism) PL structure compatible with the given smooth structure [Milnor&Stasheff1974, Complement].
Since also any PL 4-manifold admits a unique smooth structure [Mandelbaum1980, 1.2], we may consider a smooth 4-manifold as a PL 4-manifold.
A map of a smooth manifold is piecewise smooth (PS) if it is smooth on every simplex of some triangulation of the manifold. Clearly, every smooth or PL map is PS.
For a smooth manifold let
be the set of PS embeddings
up to PS isotopy.
The forgetful map
is 1--1 [Haefliger1967, 2.4].
So a description of
is the same as a description of
.
2 Examples of knotted tori
The Hudson tori and
are defined for an integer
in Remark 3.5.d of [Skopenkov2016e] or in [Skopenkov2006, Example 2.10].
Tex syntax errorthe projection onto the
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Define by the equations
and
, respectively.
Example 2.1 (Spinning construction).
For an embedding denote by
the embedding
Tex syntax error
The restriction of to
is isotopic to (the restriction to
of) 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 embeddings are defined as compositions
Tex syntax error
where and maps
are defined below.
We shall see that
is an embedding for any
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 the Hopf fibration and
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
Tex syntax error
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 or
for any
.
The unpublished papers [Crowley&Skopenkov2016], [Crowley&Skopenkov2016a] prove that
- any PS embedding
represents
for some
.
- any smooth embedding
represents
for some
and
.
Example 2.4 (The Lambrechts torus). There is an 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
Tex syntax error
Tex syntax errorfor
![m\ge p+q+3](/images/math/3/f/7/3f791e68c3af31a14faaa173882052fa.png)
![p\ge0](/images/math/b/f/0/bf065379f797991fb7162986d2447de6.png)
![f:N\to S^n](/images/math/7/b/6/7b6dec8a999191cab715f1f12ff53b3b.png)
Tex syntax error.
(We conjecture that .)
Example 2.5 (the Haefliger torus [Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]).
There is a PL embedding which is not PS 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
, see the figure [Skopenkov2006, Figure 3.7.a]. These two extensions together form the Haefliger torus, see the figure [Skopenkov2006, Figure 3.7.b].
3 Embeddings of the complex projective plane
Example 3.1 [Boechat&Haefliger1970, p.164].
There is an embedding .
Recall that is the mapping cylinder of the Hopf fibration
. 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
.
Theorem 3.2. (a) There is only one embedding up to isotopy and a hyperplane reflection of
.
In other words, there are exactly two isotopy classes of embeddings
(differing by
composition with a hyperplane reflection of
).
(b) For any pair of embeddings and
the embedding
is isotopic to
.
(c) The Whitney invariant
(defined in [Skopenkov2016e], [Skopenkov2006, 2]) is a 1-1 correspondence
.
However, any PL embedding whose Whitney invariant w.r.t. the embedding of Example 3.1 is different from
is not smoothable.
Parts (a) and (b) are proved in [Skopenkov2005, Triviality Theorem (a)] or follow by Theorem 5.3 below. Part (c) follows by [Boechat&Haefliger1970], cf. a generalization presented in [Skopenkov2016e].
4 The Boechat-Haefliger invariant
We give definitions in more generality because this is natural and is required for 3-manifolds in 6-space [Skopenkov2016t].
Let be a closed connected orientable
-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.
This is well-known, see [Skopenkov2008,
2, 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.
This is not to be confused with another well-known homological Alexander duality isomorphism [Skopenkov2005, Alexander Duality Lemma 4.6].
Definition 4.2.
A `homology Seifert surface' for is the image
of the fundamental class
.
Denote by the intersection products
and
.
Remark 4.3.
Take a small oriented disk whose intersection with
consists of exactly one point of sign
and such that
.
A homology Seifert surface
for
is uniquely defined by the condition
.
Definition 4.4.
Define `the Boechat-Haefliger invariant' of
![\displaystyle \varkappa(f):=A_{f,2n+1-m}^{-1}\left(A_{f,n}[N]\cap A_{f,n}[N]\right)\in H_{2n+1-m}(N).](/images/math/6/9/1/691237d46ac79e4465fe5e708e568f66.png)
Clearly, a map is well-defined by
.
Remark 4.5.
(a) If , then
for any two embeddings
[Skopenkov2008,
2, The Boechat-Haefliger Invariant Lemma].
Here
is the Whitney invariant [Skopenkov2016e], [Skopenkov2006,
2].
We conjecture that this holds for any
, in particular, that
when
is even.
(b) Definition 4.4 is equivalent to the original one for [Boechat&Haefliger1970] by [Crowley&Skopenkov2008, Section Lemma 3.1].
Earlier notation for
was
[Boechat&Haefliger1970],
[Skopenkov2005] and
[Crowley&Skopenkov2008].
5 Classification
We use Stiefel-Whitney characteristic classes and
.
Theorem 5.1. (a) Any orientable (or compact nonclosed) 4-manifold embeds into .
(b) A closed 4-manifold embeds into
of and only if
.
The nonclosed case of (a) is simple [Hirsch1961]. The closed case of (a) easily follows either from (b) by [Massey1960] or from Theorem 5.3.a below by [Donaldson1987]. The latter was noticed in [Fuquan1994] together with a proof of (b).
The PL version of the closed case of (a) was proved in [Hirsch1965].
The PL version of (b) follows from [Fuquan1994, Main Theorem A] by the second paragraph of Remark 1.1.
A simpler proof of the PL versions of (b) is given as the proof of [Skopenkov1997, Corollary 1.3.a] (for specialists recall that for a closed 4-manifold
).
Theorem 5.2 ([Haefliger1966], see also [Skopenkov2005], [Crowley&Skopenkov2008]). There is an isomorphism .
Theorem 5.3.
Let be a closed connected oriented 4-manifold.
Tex syntax errorof the Boéchat-Haefliger invariant
![\displaystyle \varkappa:E^7_D(N)\to H_2(N)](/images/math/9/f/e/9fe91d2bb98db9a2da3cad5c2348a690.png)
![\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.](/images/math/5/3/e/53e20ce7966b65b7fcdc0fe7eddecc62.png)
![H_1(N)=0](/images/math/5/0/0/500d0aa2c8b9d280b3c5e4e86cb00666.png)
Tex syntax errorthere 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 Poincaré isomorphism.
-
is the intersection form.
-
is the signature (of the intersection form) of
.
-
is the maximal integer
such that both
and 24 are divisible by
.
-
is defined in [Crowley&Skopenkov2008].
Thus is surjective if
is not divisible by 2.
Tex syntax erroris divisible by 2 (for some
![u](/images/math/c/4/a/c4a5b5e310ed4c323e04d72afae39f53.png)
![u](/images/math/c/4/a/c4a5b5e310ed4c323e04d72afae39f53.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
For a classification when see [Crowley&Skopenkov2016] and [Crowley&Skopenkov2016a].
Known classification results in the PL category for closed connected 4-manifolds are particular cases of
higher-dimensional results presented in [Skopenkov2016e], [Skopenkov2006, Theorem 2.13].
Corollary 5.4 ([Crowley&Skopenkov2008, Corollary 1.2]).
(a) There are exactly twelve isotopy classes of embeddings if
is an integral homology 4-sphere (cf. Theorem 5.2).
(b) Identify using the standard basis.
For any 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 5.5 ([Crowley&Skopenkov2008, Addendum 1.3]).
Under the assumptions of Theorem 5.3 for any 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 5.6 ([Crowley&Skopenkov2008, Corollary 1.4]).
(a) Take an integer and the Hudson torus
defined in Remark 3.5.d of [Skopenkov2016e], [Skopenkov2006, Example 2.10]. If
, then for any embedding
the embedding
is isotopic to
. Moreover, for any 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 any 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 5.6(b) was first proved in [Skopenkov2005] independently of Theorem 5.3.
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
- [Donaldson1987] S. K. Donaldson, The orientation of Yang-Mills moduli spaces and
-manifold topology, J. Differential Geom. 26 (1987), no.3, 397–428. MR910015 (88j:57020) Zbl 0683.57005
- [Fuquan1994] F. Fuquan, Embedding four manifolds in
, Topology 33 (1994), 447-454.
- [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
- [Haefliger1967] A. Haefliger, Lissage des immersions-I, Topology, 6 (1967) 221--240.
- [Hirsch1961] M. W. Hirsch, The embedding of bounding manifold in euclidean space, Ann. of Math. 74 (1961), 494-497.
- [Hirsch1965] M. W. Hirsch, On embedding 4-manifolds in
, Proc. Camb. Phil. Soc. 61 (1965).
- [Mandelbaum1980] R. Mandelbaum, Four-Dimensional Topology: An introduction, Bull. Amer. Math. Soc. (N.S.) 2 (1980) 1-159.
- [Massey1960] W. S. Massey, On the Stiefel--Whitney classes of a manifold, I, Amer. J. Math. 82 (1960), 92-102.
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Skopenkov1997] A. Skopenkov, On the deleted product criterion for embeddability of manifolds in
, Comment. Math. Helv. 72 (1997), 543-555.
- [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
- [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.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.