4-manifolds in 7-space
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.
2 Examples of knotted tori
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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 .
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 ).
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.
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We conjecture that .
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
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
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 .
For each there is an injective invariant called the Kreck invariant,
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
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 .
- [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
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- [Crowley&Skopenkov2016a] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, II. Smooth classification.
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- [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, http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, submitted to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, submitted to Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, submitted to Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.