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 Embedding theorems
Theorem 2.1. Assume that is a closed compact -manifold. Then embeds into .
This is well-known 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]. This result is a special case of Theorem 2.4. See also [Horvatic1971, Theorem 5.2] for the PL case.
Theorem 2.3. Assume that is a closed compact -connected -manifold and . Then embeds into .
The Diff case of this result is proved in [Haefliger1961, Existence Theorem (a)], the PL case of this result is proved in [Irwin1965, Corollary 1.3].
Theorem 2.4. 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].
3 Unknotting Theorems
Theorem 3.1. 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].
Theorem 3.2. Assume that is a compact connected -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.
Theorem 3.3. Assume that is a closed -connected -manifold. Then for each , any two embeddings of into are isotopic.
See Theorem 2.4 of [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 3.4. Assume that is a -connected -manifold with non-empty boundary. Then for each and any two embeddings of into are isotopic.
Theorem 3.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].
4 Examples
Observe that analog of Theorem 3.2 (a) fails for , i.e. for the first non trivial case. More precisely, the following folklore statement holds.
Proposition 4.1. Let be the cylinder over . Then there exist non isotopic embeddings of to .
Proof. Recall is the standard embedding. Define , where is a fixed point.
Define . Then embeddings and are not isotopic to each other.
See also about the Hopf link in [Skopenkov2016h, 2].
5 Invariants
Denote by the linking coefficient ([Skopenkov2016h, 3, remark 3.2d]) of two cycles with disjoint support.
By we will denote a closed connected -manifold. Let be a closed -ball in . Denote .
The following folklore result holds.
Lemma 5.1. For each even and each embedding exists a nowhere vanishing normal field to .
For even and every embedding denote by
where are two homology classes, realized by closed connected orientation submanifolds of , is a nowhere vanishing normal field to and are the submanifolds shifted by .
Denote by reduction modulo .
Denote by the set embeddings of into up to isotopy.
Define the dual to Steifel-Whitney class to be the class of the cycle on which two general position normal fields on are linearly dependent.
Lemma 5.2.(O. Saeki)
Let be an embedding. Let be the boundary of a tubular neighborhood of . Given two homology classes , let be two sections of . Then
where is (Poincare dual to) the first obstruction to being homotopic as sections of .
This Lemma is proved in [Saeki1999, Lemma2.2] for , but the proof is valid in all dimensions.
Lemma 5.3. Let , then
Proof. Observe
Denote by the normal vector field opposite to . If we shift the link by , we get the link and the will not change. Hence,
By lemma 5.2
Finally, let us show that . If we generically perturb it will become linearly dependent with only on a 2--dimensional cycle in , such that by definition. On the other hand the linear homotopy of to perturbed degenerates on . Thus .
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.
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).