Embeddings of manifolds with boundary: classification

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<!--We state the results below in piecewise-linear (PL) category.-->
<!--We state the results below in piecewise-linear (PL) category.-->
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We state only the results that can be deduced from Haefliger-Weber deleted product criterion, see \cite[$\S$ 5]{Skopenkov2006}. Note that the deleted product approach do not give the most short proofs possible.
+
We state only the results that can be deduced from Haefliger-Weber deleted product criterion, see \cite[$\S$ 5]{Skopenkov2006}, \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 proofs easier or shorter then deduction from this criterion.
Sometimes we give references to direct proofs but we do not claim to provide references to the original proofs of stated results.
Sometimes we give references to direct proofs but we do not claim to provide references to the original proofs of stated results.
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Every $n$-manifold $N$ with nonempty boundary PL embeds into $\R^{2n-1}$.
Every $n$-manifold $N$ with nonempty boundary PL embeds into $\R^{2n-1}$.
{{endthm}}
{{endthm}}
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This result can be found in \cite[theorem 5.2]{Horvatic1971}
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This result can be found in \cite[Theorem 5.2]{Horvatic1971}
</wikitex>
</wikitex>
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{{beginthm|Theorem}}\label{th::unknotting}
{{beginthm|Theorem}}\label{th::unknotting}
Assume that $N$ is a compact $n$-manifold and either
Assume that $N$ is a compact $n$-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)-->
(a) $m \ge 2n+2$ or
(a) $m \ge 2n+2$ or
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<!--(b) \label{item_WW} (Whitney-Wu unknotting theorem)-->
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<!--(b) \label{item_WW} (Whitney-Wu unknotting Theorem)-->
(b) $N$ is connected and $m \ge 2n+1 \ge 5$.
(b) $N$ is connected and $m \ge 2n+1 \ge 5$.
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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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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}.
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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}
{{beginthm|Theorem}} \label{thm::special_Haef_Zem}
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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 $n=2$ has a short direct proof or can be deduced from Haefliger-Weber deleted square criterion \cite[$\S$ 5]{Skopenkov2006}.
+
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}.
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Theorem \ref{thm::special_Haef_Zem} is a special cases of the following result, see \cite[Theorem 1.2b]{Penrose&Whitehead&Zeeman1961} and \cite{Skopenkov2006}.
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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}}
{{beginthm|Theorem}}
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{{beginthm|Theorem}}
{{beginthm|Theorem}}
Assume that $N$ is a $k$-connected $n$-manifold with non-empty boundary.
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.
+
Then for every $n\ge k+3$ and $m\ge2n-k$ any two embeddings of $N$ into $\R^m$ are isotopic.
<!--Assume that $N$ is a compact $n$-manifold, $\partial N\neq\emptyset$. If $N$ is $k$-connected, $m\ge2n-k$ and $m-k\ge3$ then any two embeddings of $N$ into $\R^m$ are isotopic. -->
<!--Assume that $N$ is a compact $n$-manifold, $\partial N\neq\emptyset$. If $N$ is $k$-connected, $m\ge2n-k$ and $m-k\ge3$ then any two embeddings of $N$ into $\R^m$ are isotopic. -->
<!--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.-->
<!--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.-->

Revision as of 14:46, 29 March 2020


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

Contents

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].

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.

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 deleted product criterion, see [Skopenkov2006, \S 5], [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 proofs easier or shorter then deduction from this criterion. 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-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 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 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. [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.

Theorem 2.4. Assume that N is a k-connected n-manifold with non-empty boundary. Then for every n\ge k+3 and m\ge2n-k 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

, $\S]{Skopenkov2016c}. Recall that some [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Unknotting Theorems]] hold for manifolds with boundary \cite[$\S]{Skopenkov2016c}, \cite[$\S]{Skopenkov2006}. In this page we present results peculiar for manifold with non-empty boundary. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category. We state only the results that can be deduced from Haefliger-Weber deleted product criterion, see \cite[$\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 $n$-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 one of the following conditions holds: (a) $m \ge 2n$ (b) $N$ is \S1, \S3].

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.

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 deleted product criterion, see [Skopenkov2006, \S 5], [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 proofs easier or shorter then deduction from this criterion. 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-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 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 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. [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.

Theorem 2.4. Assume that N is a k-connected n-manifold with non-empty boundary. Then for every n\ge k+3 and m\ge2n-k 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 $n=2$ has a short direct proof or can be deduced from Haefliger-Weber deleted square criterion \cite[$\S$ 5]{Skopenkov2006}. Theorem \ref{thm::special_Haef_Zem} is a special cases of the following result, see \cite[Theorem 1.2b]{Penrose&Whitehead&Zeeman1961} and \cite{Skopenkov2006}. {{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]]\S1, \S3].

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.

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 deleted product criterion, see [Skopenkov2006, \S 5], [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 proofs easier or shorter then deduction from this criterion. 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-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 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 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. [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.

Theorem 2.4. Assume that N is a k-connected n-manifold with non-empty boundary. Then for every n\ge k+3 and m\ge2n-k 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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