Embeddings of manifolds with boundary: classification
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<!--We state the results below in piecewise-linear (PL) category.--> | <!--We state the results below in piecewise-linear (PL) category.--> | ||
− | We do not provide references to the original proofs of stated results. | + | We do not claim to provide references to the original proofs of stated results. |
+ | We state only the results that can be deduced from Haefliger-Weber theorem, see \cite[$\S$ 5]{Skopenkov2006}. | ||
{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
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{{beginthm|Theorem}}\label{th::unknotting} | {{beginthm|Theorem}}\label{th::unknotting} | ||
− | Assume $N$ is a compact connected $n$-manifold and either | + | Assume that $N$ is a compact connected $n$-manifold and either |
<!--(a) \label{item_GP} (General position theorem)--> | <!--(a) \label{item_GP} (General position theorem)--> | ||
− | $m \ge 2n+2$ or | + | (a) $m \ge 2n+2$ or |
<!--(b) \label{item_WW} (Whitney-Wu unknotting theorem)--> | <!--(b) \label{item_WW} (Whitney-Wu unknotting theorem)--> | ||
− | $m\ | + | (b) $m \ge 2n+1 \ge 5$. |
Then any two embeddings of $N$ into $\R^m$ are isotopic. | Then any two embeddings of $N$ into $\R^m$ are isotopic. | ||
{{endthm}} | {{endthm}} | ||
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{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
[The Haefliger-Zeeman unknotting Theorem] | [The Haefliger-Zeeman unknotting Theorem] | ||
− | + | Assume that $N$ is a closed $k$-connected $n$-manifold. Then for each $n\ge2k + 2$, $m \ge 2n - k + 1$ any two embeddings of $N$ into $\R^m$ are isotopic. | |
{{endthm}} | {{endthm}} | ||
{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
− | + | Assume that $N$ is a $k$-connected $n$-manifold with non-empty boundary. | |
+ | Then for every $m-k\ge3$ and $m\ge2n-k$ any two embeddings of $N$ into $\R^m$ are isotopic. | ||
<!--Assume that $N$ is a compact $n$-manifold, $\partial N\neq\emptyset$. If $N$ is $k$-connected, $m\ge2n-k$ and $m-k\ge3$ then any two embeddings of $N$ into $\R^m$ are isotopic. --> | <!--Assume that $N$ is a compact $n$-manifold, $\partial N\neq\emptyset$. If $N$ is $k$-connected, $m\ge2n-k$ and $m-k\ge3$ then any two embeddings of $N$ into $\R^m$ are isotopic. --> | ||
<!--Let $f, g\colon N \to \R^q$ be PL-embeddings, $q-n > 3$. Suppose $(N, \partial N)$ is $(2n-q)$-connected. Then $f$ and $g$ are isotopic.--> | <!--Let $f, g\colon N \to \R^q$ be PL-embeddings, $q-n > 3$. Suppose $(N, \partial N)$ is $(2n-q)$-connected. Then $f$ and $g$ are isotopic.--> |
Revision as of 12:54, 29 March 2020
This page has not been refereed. The information given here might be incomplete or provisional. |
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.
We do not claim to provide references to the original proofs of stated results. We state only the results that can be deduced from Haefliger-Weber theorem, see [Skopenkov2006, 5].
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 that is a compact connected -manifold and either (a) or (b) . Then any two embeddings of into are isotopic.
The condition stands for General Position Theorem and the condition 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 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 easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006, 5]. Both condition are special cases of the Haefliger-Zeeman unknotting Theorem stated below, see [Penrose&Whitehead&Zeeman1961, Theorem 1.2b].
Theorem 2.3. [The Haefliger-Zeeman unknotting Theorem] Assume that is a closed -connected -manifold. Then for each , any two embeddings of into are isotopic.
Theorem 2.4. Assume that is a -connected -manifold with non-empty boundary. Then for every and 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
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [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.
We do not claim to provide references to the original proofs of stated results. We state only the results that can be deduced from Haefliger-Weber theorem, see [Skopenkov2006, 5].
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 that is a compact connected -manifold and either (a) or (b) . Then any two embeddings of into are isotopic.
The condition stands for General Position Theorem and the condition 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 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 easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006, 5]. Both condition are special cases of the Haefliger-Zeeman unknotting Theorem stated below, see [Penrose&Whitehead&Zeeman1961, Theorem 1.2b].
Theorem 2.3. [The Haefliger-Zeeman unknotting Theorem] Assume that is a closed -connected -manifold. Then for each , any two embeddings of into are isotopic.
Theorem 2.4. Assume that is a -connected -manifold with non-empty boundary. Then for every and 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
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [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.
We do not claim to provide references to the original proofs of stated results. We state only the results that can be deduced from Haefliger-Weber theorem, see [Skopenkov2006, 5].
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 that is a compact connected -manifold and either (a) or (b) . Then any two embeddings of into are isotopic.
The condition stands for General Position Theorem and the condition 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 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 easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006, 5]. Both condition are special cases of the Haefliger-Zeeman unknotting Theorem stated below, see [Penrose&Whitehead&Zeeman1961, Theorem 1.2b].
Theorem 2.3. [The Haefliger-Zeeman unknotting Theorem] Assume that is a closed -connected -manifold. Then for each , any two embeddings of into are isotopic.
Theorem 2.4. Assume that is a -connected -manifold with non-empty boundary. Then for every and 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
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [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.