Embeddings of manifolds with boundary: classification
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 1 Introduction
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. In this page we present results peculiar for manifold with non-empty boundary.
If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
We state only the results that 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. Sometimes we give references to such direct proofs but we do not claim these are original proofs.
 2 Classification theorems
Theorem 2.1. Assume that is a closed compact -manifold. Then embeds into .
This is well-known strong Whitney embedding theorem.
Theorem 2.2. Assume that is a compact -manifold with nonempty boundary. Then embeds into .
Theorem 2.3. Assume that is a compact -manifold and either
(b) is connected and .
Then any two embeddings of into are isotopic.
The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see Theorems 2.1 and 2.2 respectively of [Skopenkov2016c, 2]. Note that inequality in part (a) is sharp, see the Hopf link.
Theorem 2.4. Assume that is a compact -manifold with non-empty boundary and either
(b) is -connected, .
Then any two embeddings of into are isotopic.
Part (a) of this theorem in case can be found in [Edwards1968, 4, Corollary 5]. Case is clear. Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.
These basic results can be generalized to the highly-connected manifolds (see 6).
 3 Example
The following example is folklore.
Proposition 3.1. Let be the cylinder over . Then there exist non-isotopic embeddings of to .
Proof. Define by the formula , where . Define by the formula .Recall that is the standard embedding. Then embeddings and are not isotopic. Indeed, the components of are not linked while the components of are linked [Skopenkov2016h, 3, remark 3.2d].
 4 Invariants
Let be a closed connected -manifold. By we will denote the complement in to an open -ball. Thus is the -sphere.
The following folklore result holds.
Lemma 4.1. Suppose is torsion free. For each even and 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 on . In the following paragraph we show that .
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 . This implies because is torsion free.
Since has non-empty boundary it follows that is homotopy equivalent to an -complex. The dimension of this complex equals the dimension of normal space to at any point of . Hence since the Euler class is zero there exists a nowhere vanishing normal vector field to .
Lemma 4.2. Let be an embedding. Given two homology classes , let be two normal vector fields of . Then
where is (Poincare dual to) the first obstruction to being homotopic in the class of the non-zero vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Denote by the reduction modulo .
Definition 4.3. For even and every embedding denote by
where are two closed connected oriented submanifolds of , is a nowhere vanishing normal field to and are the result of shift of by .
Note that does not change when or are changed to homologic submanifolds or when is changed to an isotopic embedding. Thus is a bilinear form on
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields on are linearly dependent.
Lemma 4.4. Let be an embedding. Then for every the following equality holds.
This Lemma was stated in [Tonkonog2010], here we give proof covering a minor gap.
Proof. We get
The first line is clear.
If we shift the link by , we get the link . The linking coefficient will not change after this shift.
The third line follows from Lemma 4.2.
Finally, let us show that . If we generically perturb it will become linearly dependent with only on a 2--dimensional cycle in . And by definition. On the other hand the linear homotopy of to perturbed degenerates on . Thus .
 5 Classification
Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.
Denote by the set embeddings of into up to isotopy.
Theorem 5.1. 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, and is the standard pairing.
 6 A generalization to highly-connected manifolds
Theorem 6.1. Assume that is a closed compact -connected -manifold and . Then embeds into .
Theorem 6.2. Assume that is a compact -manifold with nonempty boundary, is -connected and . Then embeds into .
The PL case of this result is proved in [Hudson1969, Theorem 8.3]. For the Diff case see [Haefliger1961, 1.7, remark 2], where Haefliger proposes to use the deleted pruduct criterion to obtain this result.
Theorem 6.3. Assume that is a closed -connected -manifold. Then for each , any two embeddings of into are isotopic.
Theorem 6.4. Assume that is a -connected -manifold with non-empty boundary. Then for each and any two embeddings of into are isotopic.
 7 References
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- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [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.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, submitted to Bull. Man. Atl.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).