Embeddings of manifolds with boundary: classification
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− | Observe that analog of Theorem \ref{thm::special_Haef_Zem} (a) fails for $m = 2n - 1$, i.e. for the first non trivial case. More precisely, the following statement holds. | + | Observe that analog of Theorem \ref{thm::special_Haef_Zem} (a) fails for $m = 2n - 1$, i.e. for the first non trivial case. More precisely, the following folklore statement holds. |
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Revision as of 11:30, 30 April 2020
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
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. Them embeds into .
Theorem 2.2. Assume that is a compact PL -manifold with nonempty boundary. Then PL embeds into .
This result is a special case of [Hudson1969, Theorem 8.3]. See also [Horvatic1971, Theorem 5.2].
Theorem 2.3. Assume that is a closed compact -connected -manifold and . Them embeds into .
This result can be found in [Irwin1965, Corollary 1.3].
Theorem 2.4. Assume that is a compact PL -manifold with nonempty boundary and is -connected. Then PL embeds into for each .
This result can be found in [Hudson1969, Theorem 8.3].
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. This construction is based on the Hopf link. Let be the Hopf link. The image of the Hopf link is the union of two -spheres which can be described as follows: the spheres are and in . Denote by and restrictions of to the first and second components respectively. Then the embedding of can be obtained by the following folrmula:
Embedding is not isotopic to the standard embedding, because the components of its boundary are linked.
About the Hopf link see also [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. 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 ??%\ref{}
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.
- [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
- [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).
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. Them embeds into .
Theorem 2.2. Assume that is a compact PL -manifold with nonempty boundary. Then PL embeds into .
This result is a special case of [Hudson1969, Theorem 8.3]. See also [Horvatic1971, Theorem 5.2].
Theorem 2.3. Assume that is a closed compact -connected -manifold and . Them embeds into .
This result can be found in [Irwin1965, Corollary 1.3].
Theorem 2.4. Assume that is a compact PL -manifold with nonempty boundary and is -connected. Then PL embeds into for each .
This result can be found in [Hudson1969, Theorem 8.3].
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. This construction is based on the Hopf link. Let be the Hopf link. The image of the Hopf link is the union of two -spheres which can be described as follows: the spheres are and in . Denote by and restrictions of to the first and second components respectively. Then the embedding of can be obtained by the following folrmula:
Embedding is not isotopic to the standard embedding, because the components of its boundary are linked.
About the Hopf link see also [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. 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 ??%\ref{}
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.
- [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
- [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).
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. Them embeds into .
Theorem 2.2. Assume that is a compact PL -manifold with nonempty boundary. Then PL embeds into .
This result is a special case of [Hudson1969, Theorem 8.3]. See also [Horvatic1971, Theorem 5.2].
Theorem 2.3. Assume that is a closed compact -connected -manifold and . Them embeds into .
This result can be found in [Irwin1965, Corollary 1.3].
Theorem 2.4. Assume that is a compact PL -manifold with nonempty boundary and is -connected. Then PL embeds into for each .
This result can be found in [Hudson1969, Theorem 8.3].
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. This construction is based on the Hopf link. Let be the Hopf link. The image of the Hopf link is the union of two -spheres which can be described as follows: the spheres are and in . Denote by and restrictions of to the first and second components respectively. Then the embedding of can be obtained by the following folrmula:
Embedding is not isotopic to the standard embedding, because the components of its boundary are linked.
About the Hopf link see also [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. 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 ??%\ref{}
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.
- [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
- [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).
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. Them embeds into .
Theorem 2.2. Assume that is a compact PL -manifold with nonempty boundary. Then PL embeds into .
This result is a special case of [Hudson1969, Theorem 8.3]. See also [Horvatic1971, Theorem 5.2].
Theorem 2.3. Assume that is a closed compact -connected -manifold and . Them embeds into .
This result can be found in [Irwin1965, Corollary 1.3].
Theorem 2.4. Assume that is a compact PL -manifold with nonempty boundary and is -connected. Then PL embeds into for each .
This result can be found in [Hudson1969, Theorem 8.3].
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. This construction is based on the Hopf link. Let be the Hopf link. The image of the Hopf link is the union of two -spheres which can be described as follows: the spheres are and in . Denote by and restrictions of to the first and second components respectively. Then the embedding of can be obtained by the following folrmula:
Embedding is not isotopic to the standard embedding, because the components of its boundary are linked.
About the Hopf link see also [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. 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 ??%\ref{}
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.
- [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
- [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).