Embeddings of manifolds with boundary: classification
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− | Every | + | Every $n$-manifold $N$ with nonempty boundary PL-embeds into $\R^{2n-1}$. |
{{endthm}} | {{endthm}} | ||
− | This result can be found in \cite[theorem 5.2]{ | + | This result can be found in \cite[theorem 5.2]{Horvatic1971} |
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{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
Say $n$-manifold $M$ is compact, $\partial M\neq\emptyset$. Let $f, g\colon M \to S^q$ be PL-embeddings, $f \simeq g, q-n > 3$. Suppose $(M, \partial M)$ is $(2n-q)$-connected. Then $f$ and $g$ are ambient isotopic. | Say $n$-manifold $M$ is compact, $\partial M\neq\emptyset$. Let $f, g\colon M \to S^q$ be PL-embeddings, $f \simeq g, q-n > 3$. Suppose $(M, \partial M)$ is $(2n-q)$-connected. Then $f$ and $g$ are ambient isotopic. | ||
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{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
[The Haefliger-Zeeman unknotting theorem] | [The Haefliger-Zeeman unknotting theorem] | ||
− | For every $n\ge2k + 2$, $m \ge 2n - k + 1$ and closed $k$- | + | For every $n\ge2k + 2$, $m \ge 2n - k + 1$ and closed $k$-connected $n$-manifold $N$, any two embeddings of $N$ into $\R^m$ are isotopic. |
{{endthm}} | {{endthm}} |
Revision as of 16:01, 19 March 2020
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
Most of this page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.
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].
2 Unknotting Theorems
Theorem 2.1. Assume is a compact connected -manifold and one of the following conditions holds:
(a) (General position theorem) ;
(b) (Whitney-Wu unknotting theorem) ;
Then any two embeddings of into are isotopic.
Theorem 2.2. Every -manifold with nonempty boundary PL-embeds into .
This result can be found in [Horvatic1971, theorem 5.2]
Theorem 2.3. 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.
Theorem 2.4. Say -manifold is compact, . Let be PL-embeddings, . Suppose is -connected. Then and are ambient isotopic.
This result can be found in [keylist, Theorem 10.3] % Hudson Piecewise linear topology
Theorem 2.5. [The Haefliger-Zeeman unknotting theorem] For every , and closed -connected -manifold , any two embeddings of into are isotopic.
3 Construction and examples
...
4 Invariants
...
5 Classification
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
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [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.
- [keylist] Template:Keylist
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
2 Unknotting Theorems
Theorem 2.1. Assume is a compact connected -manifold and one of the following conditions holds:
(a) (General position theorem) ;
(b) (Whitney-Wu unknotting theorem) ;
Then any two embeddings of into are isotopic.
Theorem 2.2. Every -manifold with nonempty boundary PL-embeds into .
This result can be found in [Horvatic1971, theorem 5.2]
Theorem 2.3. 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.
Theorem 2.4. Say -manifold is compact, . Let be PL-embeddings, . Suppose is -connected. Then and are ambient isotopic.
This result can be found in [keylist, Theorem 10.3] % Hudson Piecewise linear topology
Theorem 2.5. [The Haefliger-Zeeman unknotting theorem] For every , and closed -connected -manifold , any two embeddings of into are isotopic.
3 Construction and examples
...
4 Invariants
...
5 Classification
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
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [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.
- [keylist] Template:Keylist
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
2 Unknotting Theorems
Theorem 2.1. Assume is a compact connected -manifold and one of the following conditions holds:
(a) (General position theorem) ;
(b) (Whitney-Wu unknotting theorem) ;
Then any two embeddings of into are isotopic.
Theorem 2.2. Every -manifold with nonempty boundary PL-embeds into .
This result can be found in [Horvatic1971, theorem 5.2]
Theorem 2.3. 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.
Theorem 2.4. Say -manifold is compact, . Let be PL-embeddings, . Suppose is -connected. Then and are ambient isotopic.
This result can be found in [keylist, Theorem 10.3] % Hudson Piecewise linear topology
Theorem 2.5. [The Haefliger-Zeeman unknotting theorem] For every , and closed -connected -manifold , any two embeddings of into are isotopic.
3 Construction and examples
...
4 Invariants
...
5 Classification
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
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [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.
- [keylist] Template:Keylist