Embeddings of manifolds with boundary: classification

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1 Introduction

Recall that some Unknotting Theorems hold for manifolds with boundary [Skopenkov2016c, $\S$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; <!--Most of this page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.--> Recall that some [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Unknotting Theorems]] hold for manifolds with boundary \cite[$\S]{Skopenkov2016c}, \cite[$\S]{Skopenkov2006}. In this page we present results peculiar for manifold with non-empty boundary. For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\S$3], [Skopenkov2006, $\S$$\S$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, $\S$$\S$1, $\S$$\S$3].

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 Haefliger-Weber theorem, see [Skopenkov2006, $\S$$\S$ 5]. Note that the deleted product approach do not give the most short proofs possible. Sometimes we give references to direct proofs but we do not claim to provide references to the original proofs of stated results.

Theorem 1.1. Every $n$$n$-manifold $N$$N$ with nonempty boundary PL embeds into $\R^{2n-1}$$\R^{2n-1}$.

This result can be found in [Horvatic1971, theorem 5.2]

2 Unknotting Theorems

Theorem 2.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either (a) $m \ge 2n+2$$m \ge 2n+2$ or (b) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$. Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ 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, $\S$$\S$ 2].

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold with non-empty boundary and one of the following conditions holds:

(a) $m \ge 2n$$m \ge 2n$

(b) $N$$N$ is $1$$1$-connected, $m \ge 2n - 1\ge3$$m \ge 2n - 1\ge3$

Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Part (a) of this theorem in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, corollary 5]. Case $n=1$$n=1$ is clear. Case $n=2$$n=2$ has a short direct proof or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006, $\S$$\S$ 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 $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $n\ge2k + 2$$n\ge2k + 2$, $m \ge 2n - k + 1$$m \ge 2n - k + 1$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Theorem 2.4. Assume that $N$$N$ is a $k$$k$-connected $n$$n$-manifold with non-empty boundary. Then for every $m-k\ge3$$m-k\ge3$ and $m\ge2n-k$$m\ge2n-k$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ 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 $N$$N$ be a closed homologically $k$$k$-connected $n$$n$-manifold and $m\ge 3n/2+2$$m\ge 3n/2+2$. The cone map $\Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N)$$\Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N)$ is one-to-one for $m\ge 2n-2k$$m\ge 2n-2k$ and is surjective for $m=2n-2k-1$$m=2n-2k-1$.

[Becker&Glover1971]

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.

, $\S]{Skopenkov2016c}. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category. We state only the results that can be deduced from Haefliger-Weber theorem, see \cite[$\S$ 5]{Skopenkov2006}. Note that the deleted product approach do not give the most short proofs possible. Sometimes we give references to direct proofs but we do not claim to provide references to the original proofs of stated results. {{beginthm|Theorem}} Every $n$-manifold $N$ with nonempty boundary PL embeds into $\R^{2n-1}$. {{endthm}} This result can be found in \cite[theorem 5.2]{Horvatic1971} == Unknotting Theorems == ; {{beginthm|Theorem}}\label{th::unknotting} Assume that $N$ is a compact connected $n$-manifold and either (a) $m \ge 2n+2$ or (b) $m \ge 2n+1 \ge 5$. Then any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|theorems 2.1 and 2.2]] respectively of \cite[$\S$ 2]{Skopenkov2016c}. {{beginthm|Theorem}} Assume that $N$ is a compact connected $n$-manifold with non-empty boundary and one of the following conditions holds: (a) $m \ge 2n$ (b) $N$ is \S3], [Skopenkov2006, $\S$$\S$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, $\S$$\S$1, $\S$$\S$3].

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 Haefliger-Weber theorem, see [Skopenkov2006, $\S$$\S$ 5]. Note that the deleted product approach do not give the most short proofs possible. Sometimes we give references to direct proofs but we do not claim to provide references to the original proofs of stated results.

Theorem 1.1. Every $n$$n$-manifold $N$$N$ with nonempty boundary PL embeds into $\R^{2n-1}$$\R^{2n-1}$.

This result can be found in [Horvatic1971, theorem 5.2]

2 Unknotting Theorems

Theorem 2.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either (a) $m \ge 2n+2$$m \ge 2n+2$ or (b) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$. Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ 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, $\S$$\S$ 2].

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold with non-empty boundary and one of the following conditions holds:

(a) $m \ge 2n$$m \ge 2n$

(b) $N$$N$ is $1$$1$-connected, $m \ge 2n - 1\ge3$$m \ge 2n - 1\ge3$

Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Part (a) of this theorem in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, corollary 5]. Case $n=1$$n=1$ is clear. Case $n=2$$n=2$ has a short direct proof or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006, $\S$$\S$ 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 $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $n\ge2k + 2$$n\ge2k + 2$, $m \ge 2n - k + 1$$m \ge 2n - k + 1$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Theorem 2.4. Assume that $N$$N$ is a $k$$k$-connected $n$$n$-manifold with non-empty boundary. Then for every $m-k\ge3$$m-k\ge3$ and $m\ge2n-k$$m\ge2n-k$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ 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 $N$$N$ be a closed homologically $k$$k$-connected $n$$n$-manifold and $m\ge 3n/2+2$$m\ge 3n/2+2$. The cone map $\Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N)$$\Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N)$ is one-to-one for $m\ge 2n-2k$$m\ge 2n-2k$ and is surjective for $m=2n-2k-1$$m=2n-2k-1$.

[Becker&Glover1971]

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.

$-connected, $m \ge 2n - 1\ge3$ Then any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} 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 square criterion \cite[$\S$ 5]{Skopenkov2006}. Both condition are special cases of the Haefliger-Zeeman unknotting Theorem stated below, see \cite[Theorem 1.2b]{Penrose&Whitehead&Zeeman1961}. {{beginthm|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}} {{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. {{endthm}} This result can be found in \cite[Theorem 10.3]{Hudson1969} == Construction and examples == ; ... == Invariants == ; ... == Classification == ; {{beginthm|Theorem}}[Becker-Glover] Let $N$ be a closed homologically $k$-connected $n$-manifold and $m\ge 3n/2+2$. The cone map $\Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N)$ is one-to-one for $m\ge 2n-2k$ and is surjective for $m=2n-2k-1$. {{endthm}} \cite{Becker&Glover1971} == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S3], [Skopenkov2006, $\S$$\S$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, $\S$$\S$1, $\S$$\S$3].

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 Haefliger-Weber theorem, see [Skopenkov2006, $\S$$\S$ 5]. Note that the deleted product approach do not give the most short proofs possible. Sometimes we give references to direct proofs but we do not claim to provide references to the original proofs of stated results.

Theorem 1.1. Every $n$$n$-manifold $N$$N$ with nonempty boundary PL embeds into $\R^{2n-1}$$\R^{2n-1}$.

This result can be found in [Horvatic1971, theorem 5.2]

2 Unknotting Theorems

Theorem 2.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either (a) $m \ge 2n+2$$m \ge 2n+2$ or (b) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$. Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ 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, $\S$$\S$ 2].

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold with non-empty boundary and one of the following conditions holds:

(a) $m \ge 2n$$m \ge 2n$

(b) $N$$N$ is $1$$1$-connected, $m \ge 2n - 1\ge3$$m \ge 2n - 1\ge3$

Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Part (a) of this theorem in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, corollary 5]. Case $n=1$$n=1$ is clear. Case $n=2$$n=2$ has a short direct proof or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006, $\S$$\S$ 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 $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $n\ge2k + 2$$n\ge2k + 2$, $m \ge 2n - k + 1$$m \ge 2n - k + 1$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Theorem 2.4. Assume that $N$$N$ is a $k$$k$-connected $n$$n$-manifold with non-empty boundary. Then for every $m-k\ge3$$m-k\ge3$ and $m\ge2n-k$$m\ge2n-k$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ 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 $N$$N$ be a closed homologically $k$$k$-connected $n$$n$-manifold and $m\ge 3n/2+2$$m\ge 3n/2+2$. The cone map $\Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N)$$\Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N)$ is one-to-one for $m\ge 2n-2k$$m\ge 2n-2k$ and is surjective for $m=2n-2k-1$$m=2n-2k-1$.

[Becker&Glover1971]

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.