Embeddings of manifolds with boundary: classification

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Here we state all other results concerning embeddings of manifolds with boundary except some results when the classification of embeddiongs coinsides with the classification of immersions, see \cite[Theorem 2.3]{Skopenkov2010}. Result below can be found in parpers by O. Saeki and D Tonkonog, see \cite{Saeki1999}, \cite{Tonkonog2010}. Here we give proofs with some minor improvments.
Here we state all other results concerning embeddings of manifolds with boundary except some results when the classification of embeddiongs coinsides with the classification of immersions, see \cite[Theorem 2.3]{Skopenkov2010}. Result below can be found in parpers by O. Saeki and D Tonkonog, see \cite{Saeki1999}, \cite{Tonkonog2010}. Here we give proofs with some minor improvments.
−
Denote by $\mathrm{lk}$ [[High_codimension_links#The_linking_coefficient|the linking coefficient]] (\cite[$\S$ 3, remark 3.2d]{Skopenkov2016h}) of two cycles with disjoint support.
+
Denote by $\mathrm{lk}$ [[High_codimension_links#The_linking_coefficient|the linking coefficient]] \cite[$\S$ 3, remark 3.2d]{Skopenkov2016h} of two cycles with disjoint support.
By $N$ we will denote a closed connected $n$-manifold. Let $B^n$ be a closed $n$-ball in $N$. Denote $N_0:=Cl(N-B^n)$.
By $N$ we will denote a closed connected $n$-manifold. Let $B^n$ be a closed $n$-ball in $N$. Denote $N_0:=Cl(N-B^n)$.

Revision as of 11:46, 14 May 2020


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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]. 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.

2 Classification theorems

Theorem 2.1. Assume that N is a closed compact n-manifold. Then N embeds into \R^{2n}.

This is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that N is a compact n-manifold with nonempty boundary. Then N embeds into \R^{2n-1}.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 5.2 below and [Horvatic1971, Theorem 5.2].

Theorem 2.3. 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 2.4. Assume that N is a compact 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. Observe that inequality in part (a) is sharp both for connected and not connected manifolds. This differs from the case for closed manifolds, compare to Theorem 2.3.

These basic results can be generalized to the highly-connected cases (see \S5).

3 Examples

Observe that analog of Theorem 2.4 (a) fails for m = 2n - 1, i.e. for the first non trivial case. More precisely, the following folklore statement holds.

Proposition 3.1. Let N=S^{n-1}\times [0, 1] be the cylinder over S^{n-1}. Then there exist non-isotopic embeddings of N to \mathbb R^{2n-1}.

Proof. Recall \mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1} is the standard embedding. Denote 1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}. Define g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1} by formula g_1(x, t) = (x, t1_{n-1}).

Define g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}, g_2(x, t) = (x, tx).

Then embeddings \mathrm i_{2n-1,n-1}g_1 and \mathrm i_{2n-1,n-1}g_2 are not isotopic to each other, because the components of boundary of \mathrm i_{2n-1,n-1}g_1(S^{n-1}\times [0, 1]) are not linked and conponents of boundary of \mathrm i_{2n-1,n-1}g_2(S^{n-1}\times [0, 1]) are linked.
\square

This construction is analogous to the Hopf link, see [Skopenkov2016h, \S 2].

4 Invariants

Here we state all other results concerning embeddings of manifolds with boundary except some results when the classification of embeddiongs coinsides with the classification of immersions, see [Skopenkov2010, Theorem 2.3]. Result below can be found in parpers by O. Saeki and D Tonkonog, see [Saeki1999], [Tonkonog2010]. Here we give proofs with some minor improvments.

Denote by \mathrm{lk} the linking coefficient [Skopenkov2016h, \S 3, remark 3.2d] of two cycles with disjoint support.

By N we will denote a closed connected n-manifold. Let B^n be a closed n-ball in N. Denote N_0:=Cl(N-B^n).

The following folklore result holds.

Lemma 4.1. For each even n and each embedding f\colon N_0 \to \mathbb R^{2n-1} exists a nowhere vanishing normal field to f(N_0).

For even n and every embedding f\colon N_0\to\mathbb R^{2n-1} denote by

\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),

where x, y\in H_{n-1}(N_0) are two homology classes, realized by closed connected orientation submanifolds of N_0, s is a nowhere vanishing normal field to f(N_0) and s(x), s(y) are the submanifolds f(x), f(y) shifted by s.

Denote by \rho_2 \colon H_*(N; \mathbb Z)\to H_*(N;\mathbb Z_2) reduction modulo 2.

Denote by \mathrm{Emb}^{m}N_0 the set embeddings of N_0 into \mathbb R^{m} up to isotopy.

Define the dual to Steifel-Whitney class \mathrm{PD}\bar w_{n-2}(N_0)\in H_{n-2} to be the class of the cycle on which two general position normal fields on N_0 are linearly dependent.

Lemma 4.2.

Let f:N_0\to \R^{2n-1} be an embedding. Let T be the boundary of a tubular neighborhood of f(N_0). Given two homology classes [x],[y]\in H_{n-1}(N_0, \Z), let s,s' be two sections of T|_{x\cup y}. Then

\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y

where d(s,s')\in H_2(N_0) is (Poincare dual to) the first obstruction to s,s' being homotopic as sections of \pi.

This Lemma is proved in [Saeki1999, Lemma2.2] for n=3, but the proof is valid in all dimensions.

Lemma 4.3. Let f\in\mathrm{Emb}^{2n-1}N_0, then

\displaystyle \rho_2(L(f)(x, y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2x\cap\rho_2y.

Proof. Observe \rho_2(L(f)(x, y)) = \rho_2(\mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y))).

Denote by -s the normal vector field opposite to s. If we shift the link s(x)\sqcup f(y) by -s, we get the link f(x), -s(y) and the \mathrm{lk} will not change. Hence,

\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y)) = \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)).

By lemma 4.2

\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)) = d(s, -s)\cap x\cap y.

Finally, let us show that \rho_2(d(s, -s)) = \mathrm{PD}\bar w_{n-2}(N_0). If we generically perturb -s it will become linearly dependent with s only on a 2--dimensional cycle C in N_0, such that \rho_2([C]) = w_{n-2}(N_0) by definition. On the other hand the linear homotopy of s to perturbed -s degenerates on C\times \mathrm I = d(s, -s). Thus \rho_2(d(s, -s)) = w_{n-2}(N_0).

\square

5 A generalization to highly-connected manifolds

Theorem 5.1. Assume that N is a closed compact k-connected n-manifold and n>2k+2. Then N embeds into \R^{2n-k}.

The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].

Theorem 5.2. Assume that N is a compact n-manifold with nonempty boundary, (N, \partial N) is k-connected and n\ge2k+2. Then N embeds into \R^{2n-k-1}.

The PL case of this result is proved in [Hudson1969, Theorem 8.3]. For the Diff case see [Haefliger1961, \S 1.7, remark 2], where Haefliger proposes to use the.

Theorem 5.3. 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 survey [Skopenkov2016c, \S 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 5.4. 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 2.4 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3], where the Theorem is proved modulo concordance implies isotopy theorem.

6 References

, $\S]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated. In this page we present results peculiar for manifold with non-empty boundary. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category. We state only the results that can be deduced from [[Some_calculations_involving_configuration_spaces_of_distinct_points|the Haefliger-Weber deleted product criterion]] \cite[$\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. == Classification theorems == ; {{beginthm|Theorem}} Assume that $N$ is a closed compact $n$-manifold. Then $N$ embeds into $\R^{2n}$. {{endthm}} This is well-known [[Wikipedia:Whitney_embedding_theorem|strong Whitney embedding theorem]]. {{beginthm|Theorem}} Assume that $N$ is a compact $n$-manifold with nonempty boundary. Then $N$ embeds into $\R^{2n-1}$. {{endthm}} The Diff case of this result is proved in \cite[Theorem 4.6]{Hirsch1961a}. For the PL case see references for Theorem \ref{thm::k_connect_boundary} below and \cite[Theorem 5.2]{Horvatic1971}. {{beginthm|Theorem}}\label{thm::closed_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 $n$-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.

2 Classification theorems

Theorem 2.1. Assume that N is a closed compact n-manifold. Then N embeds into \R^{2n}.

This is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that N is a compact n-manifold with nonempty boundary. Then N embeds into \R^{2n-1}.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 5.2 below and [Horvatic1971, Theorem 5.2].

Theorem 2.3. 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 2.4. Assume that N is a compact 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. Observe that inequality in part (a) is sharp both for connected and not connected manifolds. This differs from the case for closed manifolds, compare to Theorem 2.3.

These basic results can be generalized to the highly-connected cases (see \S5).

3 Examples

Observe that analog of Theorem 2.4 (a) fails for m = 2n - 1, i.e. for the first non trivial case. More precisely, the following folklore statement holds.

Proposition 3.1. Let N=S^{n-1}\times [0, 1] be the cylinder over S^{n-1}. Then there exist non-isotopic embeddings of N to \mathbb R^{2n-1}.

Proof. Recall \mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1} is the standard embedding. Denote 1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}. Define g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1} by formula g_1(x, t) = (x, t1_{n-1}).

Define g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}, g_2(x, t) = (x, tx).

Then embeddings \mathrm i_{2n-1,n-1}g_1 and \mathrm i_{2n-1,n-1}g_2 are not isotopic to each other, because the components of boundary of \mathrm i_{2n-1,n-1}g_1(S^{n-1}\times [0, 1]) are not linked and conponents of boundary of \mathrm i_{2n-1,n-1}g_2(S^{n-1}\times [0, 1]) are linked.
\square

This construction is analogous to the Hopf link, see [Skopenkov2016h, \S 2].

4 Invariants

Here we state all other results concerning embeddings of manifolds with boundary except some results when the classification of embeddiongs coinsides with the classification of immersions, see [Skopenkov2010, Theorem 2.3]. Result below can be found in parpers by O. Saeki and D Tonkonog, see [Saeki1999], [Tonkonog2010]. Here we give proofs with some minor improvments.

Denote by \mathrm{lk} the linking coefficient [Skopenkov2016h, \S 3, remark 3.2d] of two cycles with disjoint support.

By N we will denote a closed connected n-manifold. Let B^n be a closed n-ball in N. Denote N_0:=Cl(N-B^n).

The following folklore result holds.

Lemma 4.1. For each even n and each embedding f\colon N_0 \to \mathbb R^{2n-1} exists a nowhere vanishing normal field to f(N_0).

For even n and every embedding f\colon N_0\to\mathbb R^{2n-1} denote by

\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),

where x, y\in H_{n-1}(N_0) are two homology classes, realized by closed connected orientation submanifolds of N_0, s is a nowhere vanishing normal field to f(N_0) and s(x), s(y) are the submanifolds f(x), f(y) shifted by s.

Denote by \rho_2 \colon H_*(N; \mathbb Z)\to H_*(N;\mathbb Z_2) reduction modulo 2.

Denote by \mathrm{Emb}^{m}N_0 the set embeddings of N_0 into \mathbb R^{m} up to isotopy.

Define the dual to Steifel-Whitney class \mathrm{PD}\bar w_{n-2}(N_0)\in H_{n-2} to be the class of the cycle on which two general position normal fields on N_0 are linearly dependent.

Lemma 4.2.

Let f:N_0\to \R^{2n-1} be an embedding. Let T be the boundary of a tubular neighborhood of f(N_0). Given two homology classes [x],[y]\in H_{n-1}(N_0, \Z), let s,s' be two sections of T|_{x\cup y}. Then

\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y

where d(s,s')\in H_2(N_0) is (Poincare dual to) the first obstruction to s,s' being homotopic as sections of \pi.

This Lemma is proved in [Saeki1999, Lemma2.2] for n=3, but the proof is valid in all dimensions.

Lemma 4.3. Let f\in\mathrm{Emb}^{2n-1}N_0, then

\displaystyle \rho_2(L(f)(x, y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2x\cap\rho_2y.

Proof. Observe \rho_2(L(f)(x, y)) = \rho_2(\mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y))).

Denote by -s the normal vector field opposite to s. If we shift the link s(x)\sqcup f(y) by -s, we get the link f(x), -s(y) and the \mathrm{lk} will not change. Hence,

\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y)) = \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)).

By lemma 4.2

\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)) = d(s, -s)\cap x\cap y.

Finally, let us show that \rho_2(d(s, -s)) = \mathrm{PD}\bar w_{n-2}(N_0). If we generically perturb -s it will become linearly dependent with s only on a 2--dimensional cycle C in N_0, such that \rho_2([C]) = w_{n-2}(N_0) by definition. On the other hand the linear homotopy of s to perturbed -s degenerates on C\times \mathrm I = d(s, -s). Thus \rho_2(d(s, -s)) = w_{n-2}(N_0).

\square

5 A generalization to highly-connected manifolds

Theorem 5.1. Assume that N is a closed compact k-connected n-manifold and n>2k+2. Then N embeds into \R^{2n-k}.

The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].

Theorem 5.2. Assume that N is a compact n-manifold with nonempty boundary, (N, \partial N) is k-connected and n\ge2k+2. Then N embeds into \R^{2n-k-1}.

The PL case of this result is proved in [Hudson1969, Theorem 8.3]. For the Diff case see [Haefliger1961, \S 1.7, remark 2], where Haefliger proposes to use the.

Theorem 5.3. 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 survey [Skopenkov2016c, \S 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 5.4. 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 2.4 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3], where the Theorem is proved modulo concordance implies isotopy theorem.

6 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. Observe that inequality in part (a) is sharp both for connected and not connected manifolds. This differs from the case for closed manifolds, compare to Theorem \ref{thm::closed_unknotting}. These basic results can be generalized to the highly-connected cases (see $\S$\ref{prt::highly_connected}). == Examples == ; Observe that analog of Theorem \ref{thm::special_Haef_Zem} (a) fails for $m = 2n - 1$, i.e. for the first non trivial case. More precisely, the following folklore statement holds. {{beginthm|Proposition}} Let $N=S^{n-1}\times [0, 1]$ be the cylinder over $S^{n-1}$. Then there exist non-isotopic embeddings of $N$ to $\mathbb R^{2n-1}$. {{endthm}} {{beginproof}} Recall $\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$ is the standard embedding. Denote \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.

2 Classification theorems

Theorem 2.1. Assume that N is a closed compact n-manifold. Then N embeds into \R^{2n}.

This is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that N is a compact n-manifold with nonempty boundary. Then N embeds into \R^{2n-1}.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 5.2 below and [Horvatic1971, Theorem 5.2].

Theorem 2.3. 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 2.4. Assume that N is a compact 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. Observe that inequality in part (a) is sharp both for connected and not connected manifolds. This differs from the case for closed manifolds, compare to Theorem 2.3.

These basic results can be generalized to the highly-connected cases (see \S5).

3 Examples

Observe that analog of Theorem 2.4 (a) fails for m = 2n - 1, i.e. for the first non trivial case. More precisely, the following folklore statement holds.

Proposition 3.1. Let N=S^{n-1}\times [0, 1] be the cylinder over S^{n-1}. Then there exist non-isotopic embeddings of N to \mathbb R^{2n-1}.

Proof. Recall \mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1} is the standard embedding. Denote 1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}. Define g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1} by formula g_1(x, t) = (x, t1_{n-1}).

Define g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}, g_2(x, t) = (x, tx).

Then embeddings \mathrm i_{2n-1,n-1}g_1 and \mathrm i_{2n-1,n-1}g_2 are not isotopic to each other, because the components of boundary of \mathrm i_{2n-1,n-1}g_1(S^{n-1}\times [0, 1]) are not linked and conponents of boundary of \mathrm i_{2n-1,n-1}g_2(S^{n-1}\times [0, 1]) are linked.
\square

This construction is analogous to the Hopf link, see [Skopenkov2016h, \S 2].

4 Invariants

Here we state all other results concerning embeddings of manifolds with boundary except some results when the classification of embeddiongs coinsides with the classification of immersions, see [Skopenkov2010, Theorem 2.3]. Result below can be found in parpers by O. Saeki and D Tonkonog, see [Saeki1999], [Tonkonog2010]. Here we give proofs with some minor improvments.

Denote by \mathrm{lk} the linking coefficient [Skopenkov2016h, \S 3, remark 3.2d] of two cycles with disjoint support.

By N we will denote a closed connected n-manifold. Let B^n be a closed n-ball in N. Denote N_0:=Cl(N-B^n).

The following folklore result holds.

Lemma 4.1. For each even n and each embedding f\colon N_0 \to \mathbb R^{2n-1} exists a nowhere vanishing normal field to f(N_0).

For even n and every embedding f\colon N_0\to\mathbb R^{2n-1} denote by

\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),

where x, y\in H_{n-1}(N_0) are two homology classes, realized by closed connected orientation submanifolds of N_0, s is a nowhere vanishing normal field to f(N_0) and s(x), s(y) are the submanifolds f(x), f(y) shifted by s.

Denote by \rho_2 \colon H_*(N; \mathbb Z)\to H_*(N;\mathbb Z_2) reduction modulo 2.

Denote by \mathrm{Emb}^{m}N_0 the set embeddings of N_0 into \mathbb R^{m} up to isotopy.

Define the dual to Steifel-Whitney class \mathrm{PD}\bar w_{n-2}(N_0)\in H_{n-2} to be the class of the cycle on which two general position normal fields on N_0 are linearly dependent.

Lemma 4.2.

Let f:N_0\to \R^{2n-1} be an embedding. Let T be the boundary of a tubular neighborhood of f(N_0). Given two homology classes [x],[y]\in H_{n-1}(N_0, \Z), let s,s' be two sections of T|_{x\cup y}. Then

\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y

where d(s,s')\in H_2(N_0) is (Poincare dual to) the first obstruction to s,s' being homotopic as sections of \pi.

This Lemma is proved in [Saeki1999, Lemma2.2] for n=3, but the proof is valid in all dimensions.

Lemma 4.3. Let f\in\mathrm{Emb}^{2n-1}N_0, then

\displaystyle \rho_2(L(f)(x, y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2x\cap\rho_2y.

Proof. Observe \rho_2(L(f)(x, y)) = \rho_2(\mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y))).

Denote by -s the normal vector field opposite to s. If we shift the link s(x)\sqcup f(y) by -s, we get the link f(x), -s(y) and the \mathrm{lk} will not change. Hence,

\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y)) = \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)).

By lemma 4.2

\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)) = d(s, -s)\cap x\cap y.

Finally, let us show that \rho_2(d(s, -s)) = \mathrm{PD}\bar w_{n-2}(N_0). If we generically perturb -s it will become linearly dependent with s only on a 2--dimensional cycle C in N_0, such that \rho_2([C]) = w_{n-2}(N_0) by definition. On the other hand the linear homotopy of s to perturbed -s degenerates on C\times \mathrm I = d(s, -s). Thus \rho_2(d(s, -s)) = w_{n-2}(N_0).

\square

5 A generalization to highly-connected manifolds

Theorem 5.1. Assume that N is a closed compact k-connected n-manifold and n>2k+2. Then N embeds into \R^{2n-k}.

The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].

Theorem 5.2. Assume that N is a compact n-manifold with nonempty boundary, (N, \partial N) is k-connected and n\ge2k+2. Then N embeds into \R^{2n-k-1}.

The PL case of this result is proved in [Hudson1969, Theorem 8.3]. For the Diff case see [Haefliger1961, \S 1.7, remark 2], where Haefliger proposes to use the.

Theorem 5.3. 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 survey [Skopenkov2016c, \S 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 5.4. 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 2.4 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3], where the Theorem is proved modulo concordance implies isotopy theorem.

6 References

_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$. Define $g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by formula $g_1(x, t) = (x, t1_{n-1})$. Define $g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}, g_2(x, t) = (x, tx)$. Then embeddings $\mathrm i_{2n-1,n-1}g_1$ and $\mathrm i_{2n-1,n-1}g_2$ are not isotopic to each other, because the components of boundary of $\mathrm i_{2n-1,n-1}g_1(S^{n-1}\times [0, 1])$ are not linked and conponents of boundary of $\mathrm i_{2n-1,n-1}g_2(S^{n-1}\times [0, 1])$ are linked.{{endproof}} This construction is analogous to [[High_codimension_links#Examples|the Hopf link]], see \cite[$\S$ 2]{Skopenkov2016h}. == Invariants == ; Here we state all other results concerning embeddings of manifolds with boundary except some results when the classification of embeddiongs coinsides with the classification of immersions, see \cite[Theorem 2.3]{Skopenkov2010}. Result below can be found in parpers by O. Saeki and D Tonkonog, see \cite{Saeki1999}, \cite{Tonkonog2010}. Here we give proofs with some minor improvments. Denote by $\mathrm{lk}$ [[High_codimension_links#The_linking_coefficient|the linking coefficient]] (\cite[$\S$ 3, remark 3.2d]{Skopenkov2016h}) of two cycles with disjoint support. By $N$ we will denote a closed connected $n$-manifold. Let $B^n$ be a closed $n$-ball in $N$. Denote $N_0:=Cl(N-B^n)$. The following folklore result holds. {{beginthm|Lemma}} For each even $n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$ exists a nowhere vanishing normal field to $f(N_0)$. {{endthm}} For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ denote by $$L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$ where $x, y\in H_{n-1}(N_0)$ are two homology classes, realized by closed connected orientation submanifolds of $N_0$, $s$ is a nowhere vanishing normal field to $f(N_0)$ and $s(x), s(y)$ are the submanifolds $f(x), f(y)$ shifted by $s$. Denote by $\rho_2 \colon H_*(N; \mathbb Z)\to H_*(N;\mathbb Z_2)$ reduction modulo $. Denote by $\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$ into $\mathbb R^{m}$ up to isotopy. Define the dual to [[Stiefel-Whitney_characteristic_classes|Steifel-Whitney class]] $\mathrm{PD}\bar w_{n-2}(N_0)\in H_{n-2}$ to be the class of the cycle on which two general position normal fields on $N_0$ are linearly dependent. {{beginthm|Lemma}} \label{Lsaeki} Let $f:N_0\to \R^{2n-1}$ be an embedding. Let $T$ be the boundary of a tubular neighborhood of $f(N_0)$. Given two homology classes $[x],[y]\in H_{n-1}(N_0, \Z)$, let $s,s'$ be two sections of $T|_{x\cup y}$. Then $$\mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$ where $d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$ being homotopic as sections of $\pi$. {{endthm}} This Lemma is proved in \cite[Lemma2.2]{Saeki1999} for $n=3$, but the proof is valid in all dimensions. {{beginthm|Lemma}} Let $f\in\mathrm{Emb}^{2n-1}N_0$, then $$\rho_2(L(f)(x, y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2x\cap\rho_2y.$$ {{endthm}} {{beginproof}} Observe $\rho_2(L(f)(x, y)) = \rho_2(\mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y))).$ Denote by $-s$ the normal vector field opposite to $s$. If we shift the link $s(x)\sqcup f(y)$ by $-s$, we get the link $f(x), -s(y)$ and the $\mathrm{lk}$ will not change. Hence, $$\mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y)) = \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)).$$ By lemma \ref{Lsaeki} $$\mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)) = d(s, -s)\cap x\cap y.$$ Finally, let us show that $\rho_2(d(s, -s)) = \mathrm{PD}\bar w_{n-2}(N_0)$. If we generically perturb $-s$ it will become linearly dependent with $s$ only on a 2--dimensional cycle $C$ in $N_0$, such that $\rho_2([C]) = w_{n-2}(N_0)$ by definition. On the other hand the linear homotopy of $s$ to perturbed $-s$ degenerates on $C\times \mathrm I = d(s, -s)$. Thus $\rho_2(d(s, -s)) = w_{n-2}(N_0)$. {{endproof}} == A generalization to highly-connected manifolds == ; \label{prt::highly_connected} {{beginthm|Theorem}} Assume that $N$ is a closed compact $k$-connected $n$-manifold and $n>2k+2$. Then $N$ embeds into $\R^{2n-k}$. {{endthm}} The Diff case of this result is in \cite[Existence Theorem (a)]{Haefliger1961}, the PL case of this result is in \cite[Corollary 1.3]{Irwin1965}. {{beginthm|Theorem}}\label{thm::k_connect_boundary} Assume that $N$ is a compact $n$-manifold with nonempty boundary, $(N, \partial N)$ is $k$-connected and $n\ge2k+2$. Then $N$ embeds into $\R^{2n-k-1}$. {{endthm}} The PL case of this result is proved in \cite[Theorem 8.3]{Hudson1969}. For the Diff case see \cite[$\S$ 1.7, remark 2]{Haefliger1961}, where Haefliger proposes to use the. {{beginthm|Theorem}}\label{thm::k_connect_closed_unknot} 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 2.4]] of survey \cite[$\S$ 2]{Skopenkov2016c}, or \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}, where the Theorem is proved modulo [[Isotopy#Concordance|concordance implies isotopy theorem]]. == 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.

2 Classification theorems

Theorem 2.1. Assume that N is a closed compact n-manifold. Then N embeds into \R^{2n}.

This is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that N is a compact n-manifold with nonempty boundary. Then N embeds into \R^{2n-1}.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 5.2 below and [Horvatic1971, Theorem 5.2].

Theorem 2.3. 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 2.4. Assume that N is a compact 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. Observe that inequality in part (a) is sharp both for connected and not connected manifolds. This differs from the case for closed manifolds, compare to Theorem 2.3.

These basic results can be generalized to the highly-connected cases (see \S5).

3 Examples

Observe that analog of Theorem 2.4 (a) fails for m = 2n - 1, i.e. for the first non trivial case. More precisely, the following folklore statement holds.

Proposition 3.1. Let N=S^{n-1}\times [0, 1] be the cylinder over S^{n-1}. Then there exist non-isotopic embeddings of N to \mathbb R^{2n-1}.

Proof. Recall \mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1} is the standard embedding. Denote 1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}. Define g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1} by formula g_1(x, t) = (x, t1_{n-1}).

Define g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}, g_2(x, t) = (x, tx).

Then embeddings \mathrm i_{2n-1,n-1}g_1 and \mathrm i_{2n-1,n-1}g_2 are not isotopic to each other, because the components of boundary of \mathrm i_{2n-1,n-1}g_1(S^{n-1}\times [0, 1]) are not linked and conponents of boundary of \mathrm i_{2n-1,n-1}g_2(S^{n-1}\times [0, 1]) are linked.
\square

This construction is analogous to the Hopf link, see [Skopenkov2016h, \S 2].

4 Invariants

Here we state all other results concerning embeddings of manifolds with boundary except some results when the classification of embeddiongs coinsides with the classification of immersions, see [Skopenkov2010, Theorem 2.3]. Result below can be found in parpers by O. Saeki and D Tonkonog, see [Saeki1999], [Tonkonog2010]. Here we give proofs with some minor improvments.

Denote by \mathrm{lk} the linking coefficient [Skopenkov2016h, \S 3, remark 3.2d] of two cycles with disjoint support.

By N we will denote a closed connected n-manifold. Let B^n be a closed n-ball in N. Denote N_0:=Cl(N-B^n).

The following folklore result holds.

Lemma 4.1. For each even n and each embedding f\colon N_0 \to \mathbb R^{2n-1} exists a nowhere vanishing normal field to f(N_0).

For even n and every embedding f\colon N_0\to\mathbb R^{2n-1} denote by

\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),

where x, y\in H_{n-1}(N_0) are two homology classes, realized by closed connected orientation submanifolds of N_0, s is a nowhere vanishing normal field to f(N_0) and s(x), s(y) are the submanifolds f(x), f(y) shifted by s.

Denote by \rho_2 \colon H_*(N; \mathbb Z)\to H_*(N;\mathbb Z_2) reduction modulo 2.

Denote by \mathrm{Emb}^{m}N_0 the set embeddings of N_0 into \mathbb R^{m} up to isotopy.

Define the dual to Steifel-Whitney class \mathrm{PD}\bar w_{n-2}(N_0)\in H_{n-2} to be the class of the cycle on which two general position normal fields on N_0 are linearly dependent.

Lemma 4.2.

Let f:N_0\to \R^{2n-1} be an embedding. Let T be the boundary of a tubular neighborhood of f(N_0). Given two homology classes [x],[y]\in H_{n-1}(N_0, \Z), let s,s' be two sections of T|_{x\cup y}. Then

\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y

where d(s,s')\in H_2(N_0) is (Poincare dual to) the first obstruction to s,s' being homotopic as sections of \pi.

This Lemma is proved in [Saeki1999, Lemma2.2] for n=3, but the proof is valid in all dimensions.

Lemma 4.3. Let f\in\mathrm{Emb}^{2n-1}N_0, then

\displaystyle \rho_2(L(f)(x, y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2x\cap\rho_2y.

Proof. Observe \rho_2(L(f)(x, y)) = \rho_2(\mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y))).

Denote by -s the normal vector field opposite to s. If we shift the link s(x)\sqcup f(y) by -s, we get the link f(x), -s(y) and the \mathrm{lk} will not change. Hence,

\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y)) = \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)).

By lemma 4.2

\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)) = d(s, -s)\cap x\cap y.

Finally, let us show that \rho_2(d(s, -s)) = \mathrm{PD}\bar w_{n-2}(N_0). If we generically perturb -s it will become linearly dependent with s only on a 2--dimensional cycle C in N_0, such that \rho_2([C]) = w_{n-2}(N_0) by definition. On the other hand the linear homotopy of s to perturbed -s degenerates on C\times \mathrm I = d(s, -s). Thus \rho_2(d(s, -s)) = w_{n-2}(N_0).

\square

5 A generalization to highly-connected manifolds

Theorem 5.1. Assume that N is a closed compact k-connected n-manifold and n>2k+2. Then N embeds into \R^{2n-k}.

The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].

Theorem 5.2. Assume that N is a compact n-manifold with nonempty boundary, (N, \partial N) is k-connected and n\ge2k+2. Then N embeds into \R^{2n-k-1}.

The PL case of this result is proved in [Hudson1969, Theorem 8.3]. For the Diff case see [Haefliger1961, \S 1.7, remark 2], where Haefliger proposes to use the.

Theorem 5.3. 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 survey [Skopenkov2016c, \S 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 5.4. 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 2.4 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3], where the Theorem is proved modulo concordance implies isotopy theorem.

6 References

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