4-manifolds in 7-space
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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 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 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]. For definition of the embedded connected sum of embeddings of 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
Tex syntax errordefines the standard embedding
Tex syntax error. Denote by the same symbol
Tex syntax errorthe restrictions of
Tex syntax errorto for .
The Hudson tori and are defined for an integer in Remark 3.5.d of [Skopenkov2016e] or in [Skopenkov2006, Example 2.10].
Denote 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 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 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 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
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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 or for any .
The unpublished papers [Crowley&Skopenkov2016], [Crowley&Skopenkov2016a] prove that
- any PS embedding represents `PS isotopy class' of for some .
- any smooth embedding represents for some and .
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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Tex syntax errorfor by induction on using the following observation: if is an embedding, 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 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 [Skopenkov2006, Figure 3.7.a]. These two extensions together form the Haefliger torus [Skopenkov2006, Figure 3.7.b].
3 Embeddings of the complex projective plane
Example 3.1 [Boechat&Haefliger1970, p.164]. There is a smooth 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 .
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 composition with a hyperplane reflection of ).
(b) For any pair of smooth embeddings and the embedding is smoothly isotopic to .
(c) The Whitney invariant (defined in [Skopenkov2016e], [Skopenkov2006, 2]) 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
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.
A `homology Seifert surface' for is the image of the fundamental class .
Definition 4.2. Define `the Boechat-Haefliger invariant' of
Clearly, a map is well-defined by .
Remark 4.3. (a) 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 .
(b) We have for the Whitney invariant [Skopenkov2016e], [Skopenkov2006, 2]. 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 is Poincaré dual to [Boechat&Haefliger1970].
(d) Earlier notation for was [Boechat&Haefliger1970], [Skopenkov2005] and [Crowley&Skopenkov2008].
5 Classification
For the classification of for a closed connected 4-manifold with , see [Skopenkov2016e], [Skopenkov2006, Theorem 2.13]. Here we work in the smooth category.
Theorem 5.1 ([Haefliger1966], see also [Skopenkov2005], [Crowley&Skopenkov2008]). There is an isomorphism .
Theorem 5.2. Let be a closed connected orientable 4-manifold.
(a) [Boechat&Haefliger1970] The imageTex syntax errorof the Boéchat-Haefliger invariant
(b) [Crowley&Skopenkov2008]
If , then for anyTex syntax errorthere is an injective invariant called the Kreck invariant,
whose image is the subset of even elements.
Here
- is the Poincaré duality isomorphism and is the second Stiefel-Whitney class of , so that is the second Stiefel-Whitney class of as defined by Stiefel as the obstruction to being spin.
- 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.
Note thatTex syntax erroris divisible by 2 (for some or, equivalently, for any ) if and only if is spin.
For a classification when see [Crowley&Skopenkov2016] and [Crowley&Skopenkov2016a].
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 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 5.4. Under the assumptions of Theorem 5.2 for any pair of embeddings and
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], [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 5.5(b) was first proved in [Skopenkov2005] independently of Theorem 5.2.
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
- [Haefliger1967] A. Haefliger, Lissage des immersions-I, Topology, 6 (1967) 221--240.
- [Mandelbaum1980] R. Mandelbaum, Four-Dimensional Topology: An introduction, Bull. Amer. Math. Soc. (N.S.) 2 (1980) 1-159.
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [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.