Embeddings of manifolds with boundary: classification

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== Unknotting Theorems ==
== Unknotting Theorems ==
<wikitex>;
<wikitex>;
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If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
{{beginthm|Theorem}}\label{th::unknotting}
{{beginthm|Theorem}}\label{th::unknotting}
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Assume $N$ is a compact connected $n$-manifold and one of the following conditions holds:
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Assume $N$ is a compact connected $n$-manifold and either
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<!--(a) \label{item_GP} (General position theorem)-->
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(a) \label{item_GP} (General position theorem)
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$m \ge 2n+2$ or
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$m \ge 2n+2$;
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<!--(b) \label{item_WW} (Whitney-Wu unknotting theorem)-->
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$m\ge2n+1\ge5$.
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(b) \label{item_WW} (Whitney-Wu unknotting theorem)
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$m\ge2n+1\ge5$;
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−
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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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Condition $m \ge 2n+2$ stands for General Position Theorem and condition $m\ge2n+1\ge5$ stands for Whitney-Wu Unknotting Theorem, see [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Unknotting Theorems]], theprems 2.1 and 2.2 respectivly.
{{beginthm|Theorem}}
{{beginthm|Theorem}}
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Assume $N$ is a compact connected $n$-manifold with non-empty boundary and one of the following conditions holds:
Assume $N$ is a compact connected $n$-manifold with non-empty boundary and one of the following conditions holds:
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(a) \label{item_HZ0}
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(a) $m \ge 2n\neq4$
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$m \ge 2n\neq4$
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(b) \label{item_HZ1}
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(b) $N$ is $1$-connected, $m \ge 2n - 1\ge3$
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$N$ is $1$-connected, $m \ge 2n - 1\ge3 (?)$
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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 can be found in \cite[$\S$ 4, corollary 5]{Edwards1968}
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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 derived from deleted square criterion.
{{beginthm|Theorem}}
{{beginthm|Theorem}}
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Assume $N$ is a compact $n$-manifold, $\partial N\neq\emptyset$. 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.
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[The Haefliger-Zeeman unknotting theorem]
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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}}
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−
This result can be found in \cite[Theorem 10.3]{Hudson1969}
{{beginthm|Theorem}}
{{beginthm|Theorem}}
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[The Haefliger-Zeeman unknotting theorem]
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Assume $N$ is a compact $n$-manifold, $\partial N\neq\emptyset$. If $N$ is $k$-connected, $m\ge2n-k$ and $m-k>3$ then any two embeddings of $N$ into $\R^m$ are isotopic.
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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.
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<!--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.-->
{{endthm}}
{{endthm}}
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This result can be found in \cite[Theorem 10.3]{Hudson1969}
</wikitex>
</wikitex>

Revision as of 16:41, 22 March 2020


This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

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 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].

2 Unknotting Theorems

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

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.

Condition m \ge 2n+2 stands for General Position Theorem and condition m\ge2n+1\ge5 stands for Whitney-Wu Unknotting Theorem, see Unknotting Theorems, theprems 2.1 and 2.2 respectivly.

Theorem 2.2. Every n-manifold N with nonempty boundary PL-embeds into \R^{2n-1}.

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

Theorem 2.3. Assume N is a compact connected n-manifold with non-empty boundary and one of the following conditions holds:

(a) m \ge 2n\neq4

(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 derived from deleted square criterion.

Theorem 2.4. [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 2.5. Assume N is a compact n-manifold, \partial N\neq\emptyset. If N is k-connected, m\ge2n-k and m-k>3 then 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}. == Unknotting Theorems == ; {{beginthm|Theorem}}\label{th::unknotting} Assume $N$ is a compact connected $n$-manifold and one of the following conditions holds: (a) \label{item_GP} (General position theorem) $m \ge 2n+2$; (b) \label{item_WW} (Whitney-Wu unknotting theorem) $m\ge2n+1\ge5$; Then any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} {{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} {{beginthm|Theorem}} Assume $N$ is a compact connected $n$-manifold with non-empty boundary and one of the following conditions holds: (a) \label{item_HZ0} $m \ge 2n\neq4$ (b) \label{item_HZ1} $N$ 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].

2 Unknotting Theorems

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

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.

Condition m \ge 2n+2 stands for General Position Theorem and condition m\ge2n+1\ge5 stands for Whitney-Wu Unknotting Theorem, see Unknotting Theorems, theprems 2.1 and 2.2 respectivly.

Theorem 2.2. Every n-manifold N with nonempty boundary PL-embeds into \R^{2n-1}.

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

Theorem 2.3. Assume N is a compact connected n-manifold with non-empty boundary and one of the following conditions holds:

(a) m \ge 2n\neq4

(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 derived from deleted square criterion.

Theorem 2.4. [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 2.5. Assume N is a compact n-manifold, \partial N\neq\emptyset. If N is k-connected, m\ge2n-k and m-k>3 then 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 can be found in \cite[$\S$ 4, corollary 5]{Edwards1968} {{beginthm|Theorem}} Assume $N$ is a compact $n$-manifold, $\partial N\neq\emptyset$. 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. {{endthm}} This result can be found in \cite[Theorem 10.3]{Hudson1969} {{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}} == 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].

2 Unknotting Theorems

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

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.

Condition m \ge 2n+2 stands for General Position Theorem and condition m\ge2n+1\ge5 stands for Whitney-Wu Unknotting Theorem, see Unknotting Theorems, theprems 2.1 and 2.2 respectivly.

Theorem 2.2. Every n-manifold N with nonempty boundary PL-embeds into \R^{2n-1}.

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

Theorem 2.3. Assume N is a compact connected n-manifold with non-empty boundary and one of the following conditions holds:

(a) m \ge 2n\neq4

(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 derived from deleted square criterion.

Theorem 2.4. [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 2.5. Assume N is a compact n-manifold, \partial N\neq\emptyset. If N is k-connected, m\ge2n-k and m-k>3 then 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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