Embeddings of manifolds with boundary: classification
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1 Introduction
In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In 5 we introduce an invariant of embedding of a -manifold in -space for even . In 7 which is independent from 4, 5 and 6 we state generalisations of theorems from 2 to highly-connected manifolds.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3]. In those pages mostly results for closed manifolds are stated.
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
(a) or
(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 4.1.
Part (b) in case can be found in [Edwards1968, 4, Corollary 5]. Case is clear. Both parts of this theorem are special cases of the Theorem 7.2. Case can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 7). All stated theorems of 2 and 7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Proofs of Theorem 2.1.b
The first proof uses immersions, while an alternative proof does not.
Proof. Proof of theorem 2.1.b. By strong Whitney immersion theorem there exist an immersion . Since is connected and has non-empty boundary, it follows that collapses to an -dimensional subcomplex of some triangulation of . By general position we may assume that is an embedding, because . Since is an immersion, it follows that has a sufficiently small regular neighbourhood such that is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism . The composition is an embedding of .
This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.
For the alternative proof we need some lemmas.
Lemma 3.1. [Wall1966] Assume that is a closed connected smooth -manifold. Then have handle decomposition with indices of attaching map at most .
Lemma 3.2. Assume that is a closed smooth -manifold and is an attaching map such that . If there is embedding , then extends to an embedding of .
Proof. Alternative proof of theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of with attaching maps with indices less or equal to . Denote by the manifold obtained from by the attaching first handles. Define an embedding recursively. Take any embedding . Let us define using . Denote by the -th attaching map. Since and , by lemma 3.2 it follows that there is extension of to an embedding .
Lemma 3.3. Let be a closed smooth -manifold and , , , are smooth embeddings such that on . Suppose that on there is a field of pairwise orthogonal normal vectors whose restriction to is tangent to . Then extends to a smooth embedding .
4 Example of non-isotopic embeddings
The following example is folklore.
Example 4.1. 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 .
Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let be a map of degree . (To prove part (a) it is sufficient to take as the identity map of as a map of degree one and the constant map as a map of degree zero.) Define by the formula .
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.Denote .
Example 4.2. Let . Assume . Then there exists a bijection defined by the formula .
The surjectivity of is given analogously to Proposition 4.1(b). The injectivity of follows from forgetful bijection between embeddings of and a cylinder.
This example shows that Theorem 7.4 fails for .
Example 4.3. 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 4.4. (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)).
5 Seifert linking form
In this section assume that
- is any closed orientable connected -manifold,
- is any embedding,
- if the (co)homology coefficients are omitted, then they are ,
- is even and is torsion free (these two assumptions are not required in Lemma 5.4).
By we denote the closure of the complement in to an closed -ball. Thus is the -sphere.
The following folklore result holds.
Lemma 5.1. 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 5.2. Denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 5.3 ( is well-defined). 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.
Lemma 5.4. 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 5.4.
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 5.3 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 5.5. For every the following equality holds:
This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] 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 5.4.
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.
6 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).
Definition 6.1. For each even define an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where is Whitney invariant, [Skopenkov2016e, 5].
Lemma 6.2. 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 6.3 of if is even and is torsion-free. Take a collection such that . For each such that define
where .
Note also that depends on choice of collection . The following Theorems hold for any choice of .
Theorem 6.4. Let be a closed connected orientable -manifold with torsion-free, , even. The map
is one-to-one.
Lemma 6.5. For each even and each the following equality holds: .
An equivalemt statement of Theorem 6.4:
Theorem 6.6. 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]
7 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 8 for a generalization.
Denote by a closed -manifold. By denote the complement in to an open -ball. Thus is the -sphere.
Theorem 7.1. Assume that is a closed -connected -manifold.
(a) If , then embeds into .
(b) If and , then embeds into .
Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.
Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.
Theorem 7.2. 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.
Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.
For part (b) is a corollary of Theorem 7.4 below. For part (b) coincides with Theorem 2.2b.
Conjecture 7.3. Assume that is a closed -connected -manifold. Then any two embeddings of in are isotopic.
We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.
Theorem 7.4. 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 6.6, which describes and differs from the general case.
8 Comments on non-spherical boundary
Theorem 8.1. Assume that is a compact -connected -manifold, , is -connected and . Then embeds into .
This is [Wall1965, Theorem on p.567].
Theorem 8.2. 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 7.2.
For a compact connected -manifold with boundary, the property of having an -dimensional spine is close to -connectedness. Indeed, the following theorem holds.
Theorem 8.3. Every compact connected -manifold with boundary for which is -connected, , and , has an -dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].
9 Comments on immersions
Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in is homotopy equivalent to the space of linear monomorphisms from to .
Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If is immersible in with a normal -field, then is immersible in .
Theorem 9.3. Every -manifold with non-empty boundary is immersible in .
Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every -manifold is immersible in .
Denote by is Stiefel manifold of -frames in .
Theorem 9.5. 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 . Let us construct such a linear monomorphism by skeleta of . It is clear that a linear monomorphism exists on -skeleton of .
The obstruction to extend the linear monomorphism from -skeleton to -skeleton lies 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.
Theorem 9.6. Suppose is a connected -manifold with non-empty boudary, 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 extend the homotopy from -skeleton to -skeleton lies 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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