Embeddings of manifolds with boundary: classification
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In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In 4 we introduce an invariant of embedding of a -manifold in -space for even . In 6 which is independent from 3, 4 and 5 we state generalisations of theorems from 2 to highly-connected manifolds.
If the category is omitted, then we assume the smooth (DIFF) category. Denote the set of all embeddings up to isotopy. We denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1] for the DIFF case and [Skopenkov2002, Theorem 1.3] for the PL case. Usually there exist easier direct proofs than deduction from this criterion.
We do not claim the references we give are references to original proofs.
2 Embedding and unknotting theorems
Theorem 2.1. Assume that is a compact connected -manifold.
(a) Then embeds into .
(b) If has non-empty boundary, then embeds into .
Part (a) is well-known strong Whitney embedding theorem.
Theorem 2.2. Assume that is a compact connected -manifold and either
(b) has non-empty boundary and .
Then any two embeddings of into are isotopic.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
These basic results can be generalized to the highly-connected manifolds (see 6). All stated theorems of 2 and 6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example of non-isotopic embeddings
The following example is folklore.
Example 3.1. Let be the cylinder over .
(a) Then there exist non-isotopic embeddings of into .
(b) Then for each there exist an embedding such that .
(c) Then defined by the formula is well-defined and is a bijection for .
Define by the formula , where .
Let , where is the standard embedding.Thus .
Proof of part (c). Clearly is well-defined. By (b) is surjective. Now take any two embeddings such that . Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields. Since Unknotting Spheres Theorem implies that there exists an isotopy of and . Thus we can assume . Since it follows that normal fields on and are homotopic in class of normal fields. This implies and are isotopic.
Example 3.2. Let . Assume . Then there exists a bijection defined by the formula .
The surjectivity of is given analogously to Proposition 3.1(b). The injectivity of follows from forgetful bijection between embeddings of and a cylinder.
This example shows that Theorem 6.4 fails for .
Example 3.3. Let be the connected sum of two tori. Then there exists a surjection defined by the formula .
To prove the surjectivity of it is sufficient to take linked -spheres in and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 3.4. (a) Let be the punctured 2-torus containing the meridian and the parallel of the torus. For each embedding denote by the normal field of -length vectors to defined by orientation on (see figure (b)). Then there exists a surjection defined by the formula .
(b) Let be two embeddings shown on figure (a). Figure (c) shows that and which proves the intuitive fact that and are not isotopic. (Notice that the restrictions of and on are isotopic!) If we use the opposite normal vector field , the values of and will change but will still be different (see figure (d)).
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. If the (co)homology coefficients are omitted, then we assume them to be .
The following folklore result holds.
Lemma 4.1. Assume is a closed orientable connected -manifold, is even and is torsion free. Then for each embedding there exists a nowhere vanishing normal vector field to .
Proof. There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to .
A normal space to at any point of has dimension . As is even thus is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore . Since is torsion free, it follows that .
Since has non-empty boundary, we have that is homotopy equivalent to an -complex. The dimension of this complex equals the dimension of normal space to at any point of . Since , it follows that there exists a nowhere vanishing normal vector field to .
Denote by two disjoint -cycles in with integer coefficients.
Definition 4.2. For even and every embedding denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 4.3 ( is well-defined). For even and every embedding the integer :
- is well-defined, i.e. does not change when is replaced by ,
- does not change when or are changed to homologous cycles and,
- does not change when is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.4. Let be an embedding. Let be two nowhere vanishing normal vector fields to . Then
where is the result of the shift of by , and is (Poincare dual to) the first obstruction to being homotopic in the class of the nowhere vanishing vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Here the second equality follows from Lemma 4.4.
For each two homologous -cycles in , the image of the homology between and is a -chain of such that . Since is a nowhere vanishing normal field to , this implies that the supports of and are disjoint. Hence .
Since isotopy of is a map from to , it follows that this isotopy gives an isotopy of the link . Now the third bullet point follows because the linking coefficient is preserved under isotopy.
Lemma 4.3 implies that generates a bilinear form denoted by the same letter.
Denote by the reduction modulo .
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 4.5. Let be an embedding. Then for every the following equality holds:
This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
The first congruence is clear.
The second equality holds because if we shift the link by , we get the link and the linking coefficient will not change after this shift.
The third equality follows from Lemma 4.4.
Thus it is sufficient to show that . Denote by a general perturbation of . We get:
The first equality holds because and are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between and degenerates only at those points where . These points are exactly points where and are linearly dependent. All those point form a -cycle modulo two in . The homotopy class of this -cycle is by the definition of Stiefel-Whitney class.
5 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. For a free Abelian group , let be the group of bilinear forms such that and is even for each (the second condition automatically holds for n odd).
Lemma 5.2. The invariant is well-defined for .
Proof. Note that Unknotting Spheres Theorem implies that unknots in . Thus can be extended to embedding of an -ball into . Unknotting Spheres Theorem implies that -sphere unknots in . Thus all extensions of are isotopic in PL category. Note also that if and are isotopic then their extensions are isotopic as well. And Whitney invariant is invariant for PL embeddings.
Definition 5.3 of if is even and is torsion-free. Take a collection such that . For each such that define
Note also that depends on choice of collection . The following Theorems hold for any choice of .
Theorem 5.4. Let be a closed connected orientable -manifold with torsion-free, , even. The map
Lemma 5.5. For each even and each the following equality holds: .
An equivalemt statement of Theorem 5.4:
Theorem 5.6. Let be a closed connected orientable -manifold with torsion-free, , even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms such that . Here is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 7 for a generalization.
Denote by a closed -manifold. By denote the complement in to an open -ball. Thus is the -sphere.
Theorem 6.1. Assume that is a closed -connected -manifold.
(a) If , then embeds into .
(b) If and , then embeds into .
Theorem 6.2. Assume that is a closed -connected -manifold.
(a) If and , then any two embeddings of into are isotopic.
(b) If and and then any two embeddings of into are isotopic.
Conjecture 6.3. Assume that is a closed -connected -manifold. Then any two embeddings of in are isotopic.
Theorem 6.4. Assume is a closed -connected -manifold. Then for each there exists a bijection
where denote for even and for odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2()]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 5.6, which describes and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1. Assume that is a compact -connected -manifold, , is -connected and . Then embeds into .
This is [Wall1965, Theorem on p.567].
Theorem 7.2. Assume that is a -manifold. If has -dimensional spine, , , then any two embeddings of into are isotopic.
Proof is similar to the proof of theorem 6.2.
For a compact connected -manifold with boundary, the property of having an -dimensional spine is close to -connectedness. Indeed, the following theorem holds.
Theorem 7.3. Every compact connected -manifold with boundary for which is -connected, , and , has an -dimensional spine.
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch] The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from to .
Theorem 8.2. If is immersible in with a transversal -field then it is immersible in .
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3. Every -manifold with non-empty boundary is immersible in .
Theorem 8.4.[Whitney] Every -manifold is immersible in .
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5. Suppose is a -manifold with non-empty boudary, is -connected. Then is immersible in for each .
Proof. It suffices to show that exists an immersion of in . It suffices to show that exists a linear monomorphism from to . Lets cunstruct such linear monomorphist on each -skeleton of . It is clear that linear monomorphism exists on -skeleton of .
The obstruction to continue the linear monomorphism from -skeleton to -skeleton lies in , where is Stiefel manifold of -frames in .
For we know .
For we have since is -connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into with tranversal linearly independent fields. It is true because is -connected.
Theorem 8.6. Suppose is a -manifold with non-empty boudary, (N, \partial N) is -connected and . Then every two immersions of in are regulary homotopic.
Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from to . Lets cunstruct such homotopy on each -skeleton of . It is clear that homotopy exists on -skeleton of .
The obstruction to continue the homotopy from -skeleton to -skeleton lies in , where is Stiefel manifold of -frames in .
For we know .
For we have since is -connected and has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
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