Embeddings of manifolds with boundary: classification

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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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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 $n=2$ can be easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion \cite{Skopenkov2006}. Both condition are special cases of the Haefliger-Zeeman unknotting theorem stated below, see \cite[Theorem 1.2b]{Penrose&Whitehead&Zeeman1961}
+
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 $n=2$ can be easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion \cite{Skopenkov2006}. Both condition are special cases of the Haefliger-Zeeman unknotting Theorem stated below, see \cite[Theorem 1.2b]{Penrose&Whitehead&Zeeman1961}.
{{beginthm|Theorem}}
{{beginthm|Theorem}}
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[The Haefliger-Zeeman unknotting theorem]
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[The Haefliger-Zeeman unknotting Theorem]
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.
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 12:33, 26 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, \S3], [Skopenkov2006, \S2]. 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, \S1, \S3].

If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

We do not provide references to the original proofs of stated results.

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 N is a compact connected n-manifold and either m \ge 2n+2 or m\ge2n+1\ge5. Then any two embeddings of N into \R^m are isotopic.

The condition m \ge 2n+2 stands for General Position Theorem and the condition m\ge2n+1\ge5 stands for Whitney-Wu Unknotting Theorem, see theorems 2.1 and 2.2 respectivly of [Skopenkov2016c, \S 2].

Theorem 2.2. 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 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 can be easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006]. 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] 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.

Theorem 2.4. For every m\ge2n-k and m-k\ge3 and k-connected n-manifold N with non-empty boundary, any two embeddings of N into \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 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.

[Becker&Glover1971]

6 Further discussion

...

7 References

, $\S]{Skopenkov2016c}. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category. We do not 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 $N$ is a compact connected $n$-manifold and either $m \ge 2n+2$ or $m\ge2n+1\ge5$. Then any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} The condition $m \ge 2n+2$ stands for General Position Theorem and the condition $m\ge2n+1\ge5$ stands for Whitney-Wu Unknotting Theorem, see [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|theorems 2.1 and 2.2]] respectivly 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, \S2]. 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, \S1, \S3].

If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

We do not provide references to the original proofs of stated results.

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 N is a compact connected n-manifold and either m \ge 2n+2 or m\ge2n+1\ge5. Then any two embeddings of N into \R^m are isotopic.

The condition m \ge 2n+2 stands for General Position Theorem and the condition m\ge2n+1\ge5 stands for Whitney-Wu Unknotting Theorem, see theorems 2.1 and 2.2 respectivly of [Skopenkov2016c, \S 2].

Theorem 2.2. 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 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 can be easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006]. 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] 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.

Theorem 2.4. For every m\ge2n-k and m-k\ge3 and k-connected n-manifold N with non-empty boundary, any two embeddings of N into \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 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.

[Becker&Glover1971]

6 Further discussion

...

7 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 $n=2$ can be easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion \cite{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] 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}} {{beginthm|Theorem}} For every $m\ge2n-k$ and $m-k\ge3$ and $k$-connected $n$-manifold $N$ with non-empty boundary, 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, \S2]. 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, \S1, \S3].

If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

We do not provide references to the original proofs of stated results.

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 N is a compact connected n-manifold and either m \ge 2n+2 or m\ge2n+1\ge5. Then any two embeddings of N into \R^m are isotopic.

The condition m \ge 2n+2 stands for General Position Theorem and the condition m\ge2n+1\ge5 stands for Whitney-Wu Unknotting Theorem, see theorems 2.1 and 2.2 respectivly of [Skopenkov2016c, \S 2].

Theorem 2.2. 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 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 can be easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006]. 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] 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.

Theorem 2.4. For every m\ge2n-k and m-k\ge3 and k-connected n-manifold N with non-empty boundary, any two embeddings of N into \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 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.

[Becker&Glover1971]

6 Further discussion

...

7 References

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