4-manifolds in 7-space
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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 are 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 the definition of the embedded connected sum of embeddings of closed connected 4-manifolds in 7-space and for the corresponding action of the group on the set , see e.g. [Skopenkov2016c, 4].
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, Theorems 10.5 and 10.6]. 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 equivalent to a description of .
2 Examples of knotted tori
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 [Skopenkov2016t, Example 2.1], then is not smoothly isotopic to the connected sum of the standard embedding and any embedding .
Example 2.2. Two embeddings are defined as compositions
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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 PS 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 or for any .
- 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.
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(I conjecture that .)
Example 2.5 (the Haefliger torus). There is a PL embedding which is not PS isotopic to a smooth embedding.
Take the Haefliger trefoil knot [Skopenkov2016t, Example 2.1]. Extend it to a PL conical embedding . By [Haefliger1962, 4.2] the trefoil knot also extends to a proper smooth embedding into of the punctured torus (or disk with handle), see Figure 1. These two extensions together form the required PL embedding , see Figure 2 for . By [Boechat&Haefliger1970, p.165] this PL embedding is not PS isotopic to a smooth embedding.
For a higher-dimensional generalization see [Boechat1971, 6.2].
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 .
Alternatively, define an embedding by
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 Boechat-Haefliger invariant (defined below) is an injection whose image is the set of odd integeres. However, any PL embedding whose Boechat-Haefliger is different from is not smoothable.
Parts (a) and (b) are proved in [Skopenkov2005, Triviality Theorem (a)] (they also follow by Theorem 5.3 below). Part (c) follows by [Boechat&Haefliger1970, Theorems 1.6 and 2.1] and Corollary 5.6(b) below.
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
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 homology 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
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, 5], [Skopenkov2006, 2]. We conjecture that this holds when is odd and 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].
We use Stiefel-Whitney characteristic classes and (for non-orientable 4-manifolds) .
Theorem 5.1. (a) Any closed orientable 4-manifold embeds into .
(b) A closed 4-manifold embeds into if and only if .
The PL version of (a) was proved in [Hirsch1965]. It was noticed in [Fuquan1994, p. 447] that the smooth version of (a) easily follows from Theorem 5.3.a below by [Donaldson1987]. (The smooth version of (a) also follows from (b) because for orientable 4-manifolds [Massey1960].) The smooth version of (b) is [Fuquan1994, Main Theorem A]. The PL version of (b) follows from the smooth version 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 ).
Any compact connected nonclosed 4-manifold embeds into . This follows by taking a 3-spine of , bringing a map to general position on and restricting the obtained map to sufficiently thin neighborhood of in ; this neighborhood is homeomorphic to .
Theorem 5.2. There is an isomorphism .
This is stated in [Haefliger1966, the last line] and follows by [Haefliger1966, 4.11] together with well-known fact [Skopenkov2005, Lemma 3.1]. For alternative proofs see [Skopenkov2005, 3, 4] and [Crowley&Skopenkov2008, Corollary 1.2.a].
Let be a closed connected oriented 4-manifold.
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whose image is the subset of even elements.
- is Poincaré isomorphism.
- is the maximal integer such that both and 24 are divisible by .
- is defined in [Crowley&Skopenkov2008, 2].
Tex syntax erroris divisible by 2 (for some or, equivalently, for any ) if and only if is spin.
If , then all isotopy classes of embeddings can be constructed from a certain given embedding using unlinked and linked embedded connected sum with embeddings [Skopenkov2016c, 4], [Skopenkov2016e, 3].
(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]). If and , are embeddings, then
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) If and 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 and for an embedding , then for every embedding the embedding is not isotopic to .
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