Embeddings of manifolds with boundary: classification

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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$1, $\S$3]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated.
For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S$1, $\S$3]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated.
<!--Most of this page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.-->
<!--Most of this page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.-->
−
<!--Recall that some [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Unknotting Theorems]] hold for manifolds with boundary \cite[$\S$3]{Skopenkov2016c}, \cite[$\S$2]{Skopenkov2006}.-->In this page we present results peculiar for manifold with non-empty boundary.
<!--Recall that some [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Unknotting Theorems]] hold for manifolds with boundary \cite[$\S$3]{Skopenkov2016c}, \cite[$\S$2]{Skopenkov2006}.-->In this page we present results peculiar for manifold with non-empty boundary.

Revision as of 11:02, 5 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, \S1, \S3]. 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, \S 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] 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 n-manifold N with nonempty boundary PL embeds into \R^{2n-1}.

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

2 Unknotting Theorems

Theorem 2.1. Assume that N is a compact n-manifold and either

(a) m \ge 2n+2 or

(b) N is connected and m \ge 2n+1 \ge 5.

Then any two embeddings of N into \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 2].

Theorem 2.2. Assume that N is a compact connected n-manifold with non-empty boundary and either

(a) m \ge 2n or

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

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

Part (a) of this theorem in case n>2 can be found in [Edwards1968, \S 4, Corollary 5]. Case n=1 is clear. Case n=2 has a short direct proof or can be deduced from Haefliger-Weber deleted product criterion [Skopenkov2006, Theorem 5.5].

Theorem 2.2 is a special cases of the following result, see [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 2.3. 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.

See Remark 1.2 (d)(ii) of [Skopenkov2016c, \S 1].

Theorem 2.4. Assume that N is a k-connected n-manifold with non-empty boundary. Then for each n\ge k+3 and m\ge2n-k any two embeddings of N into \R^m are isotopic.

See also [Hudson1969, Theorem 10.3]


3 References

, $\S]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated. In this page we present results peculiar for manifold with non-empty boundary. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category. We state only the results that can be deduced from the Haefliger-Weber deleted product criterion \cite[$\S$ 5]{Skopenkov2006}, see \cite[6.4]{Haefliger1963}, \cite[Theorem 1.1$\alpha\partial$]{Skopenkov2002} for the DIFF case and \cite[Theorem 1.3$\alpha\partial$]{Skopenkov2002} 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. {{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 $n$-manifold and either (a) $m \ge 2n+2$ or (b) $N$ is connected and $m \ge 2n+1 \ge 5$. 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}} \label{thm::special_Haef_Zem} Assume that $N$ is a compact connected $n$-manifold with non-empty boundary and either (a) $m \ge 2n$ or (b) $N$ is \S1, \S3]. 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, \S 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] 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 n-manifold N with nonempty boundary PL embeds into \R^{2n-1}.

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

2 Unknotting Theorems

Theorem 2.1. Assume that N is a compact n-manifold and either

(a) m \ge 2n+2 or

(b) N is connected and m \ge 2n+1 \ge 5.

Then any two embeddings of N into \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 2].

Theorem 2.2. Assume that N is a compact connected n-manifold with non-empty boundary and either

(a) m \ge 2n or

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

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

Part (a) of this theorem in case n>2 can be found in [Edwards1968, \S 4, Corollary 5]. Case n=1 is clear. Case n=2 has a short direct proof or can be deduced from Haefliger-Weber deleted product criterion [Skopenkov2006, Theorem 5.5].

Theorem 2.2 is a special cases of the following result, see [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 2.3. 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.

See Remark 1.2 (d)(ii) of [Skopenkov2016c, \S 1].

Theorem 2.4. Assume that N is a k-connected n-manifold with non-empty boundary. Then for each n\ge k+3 and m\ge2n-k any two embeddings of N into \R^m are isotopic.

See also [Hudson1969, Theorem 10.3]


3 References

$-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 product criterion \cite[Theorem 5.5]{Skopenkov2006}. Theorem \ref{thm::special_Haef_Zem} is a special cases of the following result, see \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963}, \cite[Existence Theorem (b) in p. 47]{Haefliger1961}. {{beginthm|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}} See [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|Remark 1.2 (d)(ii)]] of \cite[$\S$ 1]{Skopenkov2016c}. {{beginthm|Theorem}} Assume that $N$ is a $k$-connected $n$-manifold with non-empty boundary. Then for each $n\ge k+3$ and $m\ge2n-k$ any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} See also \cite[Theorem 10.3]{Hudson1969} == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S1, \S3]. 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, \S 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] 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 n-manifold N with nonempty boundary PL embeds into \R^{2n-1}.

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

2 Unknotting Theorems

Theorem 2.1. Assume that N is a compact n-manifold and either

(a) m \ge 2n+2 or

(b) N is connected and m \ge 2n+1 \ge 5.

Then any two embeddings of N into \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 2].

Theorem 2.2. Assume that N is a compact connected n-manifold with non-empty boundary and either

(a) m \ge 2n or

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

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

Part (a) of this theorem in case n>2 can be found in [Edwards1968, \S 4, Corollary 5]. Case n=1 is clear. Case n=2 has a short direct proof or can be deduced from Haefliger-Weber deleted product criterion [Skopenkov2006, Theorem 5.5].

Theorem 2.2 is a special cases of the following result, see [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 2.3. 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.

See Remark 1.2 (d)(ii) of [Skopenkov2016c, \S 1].

Theorem 2.4. Assume that N is a k-connected n-manifold with non-empty boundary. Then for each n\ge k+3 and m\ge2n-k any two embeddings of N into \R^m are isotopic.

See also [Hudson1969, Theorem 10.3]


3 References

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