Embeddings of manifolds with boundary: classification
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Then any two embeddings of $N$ into $\R^m$ are isotopic. | Then any two embeddings of $N$ into $\R^m$ are isotopic. | ||
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− | Part (a) of this theorem in case $n>2$ can be found in \cite[$\S$ 4, Corollary 5]{Edwards1968}. Case $n=1$ is clear. Case $n=2$ has a short direct proof or can be deduced from Haefliger-Weber deleted product criterion \cite[Theorem 5.5]{Skopenkov2006}. | + | Part (a) of this theorem in case $n>2$ can be found in \cite[$\S$ 4, Corollary 5]{Edwards1968}. Case $n=1$ is clear. |
+ | <!--Case $n=2$ has a short direct proof or can be deduced from Haefliger-Weber deleted product criterion \cite[Theorem 5.5]{Skopenkov2006}.--> | ||
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Revision as of 13:13, 9 April 2020
This page has not been refereed. The information given here might be incomplete or provisional. |
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.
Theorem 1.1.
Every compact PLTex syntax error-manifold with nonempty boundary PL embeds into .
This result can be found in [Horvatic1971, Theorem 5.2].
Theorem 1.2.
Every compactTex syntax error-connected PL
Tex syntax error-manifold with nonempty boundary PL embeds into for each .
This result can be found in [Hudson1969, Theorem 8.3].
2 Unknotting Theorems
Theorem 2.1.
Assume that is a compactTex syntax error-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 2.2.
Assume that is a compact connectedTex syntax error-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 2.3.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold. Then for each , any two embeddings of into are isotopic.
See Theorem 2.4 of [Skopenkov2016c, 2], [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 2.4.
Assume that is aTex syntax error-connected
Tex syntax error-manifold with non-empty boundary.
Then for each and any two embeddings of into are isotopic.
Theorem 2.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].
3 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
- [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.
- [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.
Theorem 1.1.
Every compact PLTex syntax error-manifold with nonempty boundary PL embeds into .
This result can be found in [Horvatic1971, Theorem 5.2].
Theorem 1.2.
Every compactTex syntax error-connected PL
Tex syntax error-manifold with nonempty boundary PL embeds into for each .
This result can be found in [Hudson1969, Theorem 8.3].
2 Unknotting Theorems
Theorem 2.1.
Assume that is a compactTex syntax error-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 2.2.
Assume that is a compact connectedTex syntax error-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 2.3.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold. Then for each , any two embeddings of into are isotopic.
See Theorem 2.4 of [Skopenkov2016c, 2], [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 2.4.
Assume that is aTex syntax error-connected
Tex syntax error-manifold with non-empty boundary.
Then for each and any two embeddings of into are isotopic.
Theorem 2.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].
3 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
- [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.
- [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.
Theorem 1.1.
Every compact PLTex syntax error-manifold with nonempty boundary PL embeds into .
This result can be found in [Horvatic1971, Theorem 5.2].
Theorem 1.2.
Every compactTex syntax error-connected PL
Tex syntax error-manifold with nonempty boundary PL embeds into for each .
This result can be found in [Hudson1969, Theorem 8.3].
2 Unknotting Theorems
Theorem 2.1.
Assume that is a compactTex syntax error-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 2.2.
Assume that is a compact connectedTex syntax error-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 2.3.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold. Then for each , any two embeddings of into are isotopic.
See Theorem 2.4 of [Skopenkov2016c, 2], [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 2.4.
Assume that is aTex syntax error-connected
Tex syntax error-manifold with non-empty boundary.
Then for each and any two embeddings of into are isotopic.
Theorem 2.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].
3 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
- [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.
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).