Embeddings of manifolds with boundary: classification
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Contents |
1 Introduction
Recall that some Unknotting Theorems hold for manifolds with boundary [Skopenkov2016c, 3], [Skopenkov2006, 2]. In this page we present results peculiar for manifold with non-empty boundary.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
Theorem 1.1. Every -manifold with nonempty boundary PL-embeds into .
This result can be found in [Horvatic1971, theorem 5.2]
2 Unknotting Theorems
Theorem 2.1. Assume is a compact connected -manifold and either or . Then any two embeddings of into are isotopic.
Condition stands for General Position Theorem and condition stands for Whitney-Wu Unknotting Theorem, see Unknotting Theorems, theorems 2.1 and 2.2 respectivly.
Theorem 2.2. Assume is a compact connected -manifold with non-empty boundary and one of the following conditions holds:
(a)
(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. Case can be derived from deleted square criterion, [Skopenkov2006].
Theorem 2.3. [The Haefliger-Zeeman unknotting theorem] For every , and closed -connected -manifold , any two embeddings of into are isotopic.
Theorem 2.4. Assume is a compact -manifold, . If is -connected, and then any two embeddings of into are isotopic.
This result can be found in [Hudson1969, Theorem 10.3]
3 Construction and examples
...
4 Invariants
...
5 Classification
Theorem 5.1.[Becker-Glover] Let be a closed homologically -connected -manifold and . The cone map is one-to-one for and is surjective for .
6 Further discussion
...
7 References
- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [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
- [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.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
Theorem 1.1. Every -manifold with nonempty boundary PL-embeds into .
This result can be found in [Horvatic1971, theorem 5.2]
2 Unknotting Theorems
Theorem 2.1. Assume is a compact connected -manifold and either or . Then any two embeddings of into are isotopic.
Condition stands for General Position Theorem and condition stands for Whitney-Wu Unknotting Theorem, see Unknotting Theorems, theorems 2.1 and 2.2 respectivly.
Theorem 2.2. Assume is a compact connected -manifold with non-empty boundary and one of the following conditions holds:
(a)
(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. Case can be derived from deleted square criterion, [Skopenkov2006].
Theorem 2.3. [The Haefliger-Zeeman unknotting theorem] For every , and closed -connected -manifold , any two embeddings of into are isotopic.
Theorem 2.4. Assume is a compact -manifold, . If is -connected, and then any two embeddings of into are isotopic.
This result can be found in [Hudson1969, Theorem 10.3]
3 Construction and examples
...
4 Invariants
...
5 Classification
Theorem 5.1.[Becker-Glover] Let be a closed homologically -connected -manifold and . The cone map is one-to-one for and is surjective for .
6 Further discussion
...
7 References
- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [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
- [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.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
Theorem 1.1. Every -manifold with nonempty boundary PL-embeds into .
This result can be found in [Horvatic1971, theorem 5.2]
2 Unknotting Theorems
Theorem 2.1. Assume is a compact connected -manifold and either or . Then any two embeddings of into are isotopic.
Condition stands for General Position Theorem and condition stands for Whitney-Wu Unknotting Theorem, see Unknotting Theorems, theorems 2.1 and 2.2 respectivly.
Theorem 2.2. Assume is a compact connected -manifold with non-empty boundary and one of the following conditions holds:
(a)
(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. Case can be derived from deleted square criterion, [Skopenkov2006].
Theorem 2.3. [The Haefliger-Zeeman unknotting theorem] For every , and closed -connected -manifold , any two embeddings of into are isotopic.
Theorem 2.4. Assume is a compact -manifold, . If is -connected, and then any two embeddings of into are isotopic.
This result can be found in [Hudson1969, Theorem 10.3]
3 Construction and examples
...
4 Invariants
...
5 Classification
Theorem 5.1.[Becker-Glover] Let be a closed homologically -connected -manifold and . The cone map is one-to-one for and is surjective for .
6 Further discussion
...
7 References
- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [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
- [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.