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 .
The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 5.2 below and [Horvatic1971, Theorem 5.2].
Theorem 2.3. Assume that is a compact -manifold and either
(a) or
(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, which is shown by the construction of the Hopf link.
Theorem 2.4. Assume that is a compact -manifold with non-empty boundary and either
(a) or
(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.
This theorem is a special case of the Theorem 5.4 .
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 5).
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 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 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 the linking coefficient [Skopenkov2016h, 3, remark 3.2d] of two disjoint cycles.
Denote by two disjoint -cycles in with integer coefficients.
Lemma 4.2. 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.
Definition 4.3. For even and every embedding denote by
where is a nowhere vanishing normal field to and are the results of the shift of by .
Note that does not change when or are changed to homological cycles or when is changed to an isotopic embedding. Thus 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 4.4. Let be an embedding. Then for every the following equality holds:
This Lemma was stated in a unpublished update of [Tonkonog2010], the following complite proof is inspired by Tonkonog.
Proof of Lemma 4.4 Let be the normal field to opposite to . We get
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 4.2.
Thus it is sufficient to show that . Denote by a general perturbation of . We get:
The first equality holds because the linear homotopy of and degenerates on a -cycle in homotopic to the cycle where and are liniarly dependent. The second equality holds because is a general perturbation of . Thus .
.
Denote by $\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$ into $\mathbb R^{m}$ up to isotopy.
Theorem 4.5. Let $N$ be a closed connected orientable $n$-manifold with $H_1(N)$ torsion-free, $n\ge 4$, $n$ even. Then
(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.
(b) The image of $L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$. Here $\bar w_2(N_0)$ is the normal Stiefel-Whitney class, and $\langle\cdot,\cdot\rangle$ is the standard pairing.
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5 A generalization to highly-connected manifolds
Theorem 5.1. Assume that is a closed compact -connected -manifold and . Then embeds into .
The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].
Theorem 5.2. 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] (there Haefliger proposes to use the deleted product criterion to obtain this result).
Theorem 5.3. Assume that is a closed -connected -manifold. Then for each , any two embeddings of into are isotopic.
See Theorem 2.4 of survey [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 5.4. Assume that is a -connected -manifold with non-empty boundary. Then for each and any two embeddings of into are isotopic.
See also [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
6 References
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [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, to appear in 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).