Embeddings of manifolds with boundary: classification

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See [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.2]] of \cite[$\S$ 2]{Skopenkov2016c}, \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}.
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See [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.4]] of \cite[$\S$ 2]{Skopenkov2016c}, \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}.
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{{beginthm|Theorem}}

Revision as of 14:10, 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 compact PL n-manifold N with nonempty boundary PL embeds into \R^{2n-1}.

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

Theorem 1.2. Every compact k-connected PL n-manifold N with nonempty boundary PL embeds into \R^{2n-k-1} for each n\ge k+3.

This result can be found in [Hudson1969, Theorem 8.3]

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 2.3. 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 Theorem 2.4 of [Skopenkov2016c, \S 2], [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

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.

Theorem 2.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].


3 References

, $\S]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated. In this page we present results peculiar for manifold with non-empty boundary. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category. We state only the results that can be deduced from [[Some_calculations_involving_configuration_space_of_distinct_pairs|the Haefliger-Weber deleted product criterion]] \cite[$\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 compact PL $n$-manifold $N$ with nonempty boundary PL embeds into $\R^{2n-1}$. {{endthm}} This result can be found in \cite[Theorem 5.2]{Horvatic1971} {{beginthm|Theorem}} Every compact $k$-connected PL $n$-manifold $N$ with nonempty boundary PL embeds into $\R^{2n-k-1}$ for each $n\ge k+3$. {{endthm}} This result can be found in \cite[Theorem 8.3]{Hudson1969} == 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 compact PL n-manifold N with nonempty boundary PL embeds into \R^{2n-1}.

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

Theorem 1.2. Every compact k-connected PL n-manifold N with nonempty boundary PL embeds into \R^{2n-k-1} for each n\ge k+3.

This result can be found in [Hudson1969, Theorem 8.3]

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 2.3. 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 Theorem 2.4 of [Skopenkov2016c, \S 2], [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

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.

Theorem 2.2 is a special cases of the latter result. 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}. {{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#Unknotting theorems|Theorem 2.2]] of \cite[$\S$ 2]{Skopenkov2016c}, \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}. {{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}} Theorem \ref{thm::special_Haef_Zem} is a special cases of the latter result. 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 compact PL n-manifold N with nonempty boundary PL embeds into \R^{2n-1}.

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

Theorem 1.2. Every compact k-connected PL n-manifold N with nonempty boundary PL embeds into \R^{2n-k-1} for each n\ge k+3.

This result can be found in [Hudson1969, Theorem 8.3]

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 2.3. 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 Theorem 2.4 of [Skopenkov2016c, \S 2], [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

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.

Theorem 2.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].


3 References

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