Embeddings of manifolds with boundary: classification

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1 Introduction

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $ {{Stub}} == Introduction == <wikitex>; For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\S$ 1, $\S$ 3]. 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

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

This is well-known strong Whitney embedding theorem.

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].

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].

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 $\S$ 5).

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.

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

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.

$\square$ $\square$

See also about the Hopf link in [Skopenkov2016h, $\S$ 2].

4 Invariants

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

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

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

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

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

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.

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

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

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,

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

By lemma 4.2

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

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 $\rho_2(d(s, -s)) = w_{n-2}(N_0)$ .

$\square$ $\square$

5 A generalization to highly-connected manifolds

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].

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.

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].

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

[Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
[Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
[Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
[Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
[Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
[Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
[Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
[Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)

[Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
[Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
[Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$ , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
[Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
[Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
[Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).

, $\S]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated. In this page we present results peculiar for manifold with non-empty boundary. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category. We state only the results that can be deduced from [[Some_calculations_involving_configuration_spaces_of_distinct_points|the Haefliger-Weber deleted product criterion]] \cite[$\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 == ; First we state some well-known results. They can be generalized to the highly-connected cases (see $\S$\ref{prt::highly_connected}). === Embedding 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}. This result is a special case of Theorem \ref{thm::k_connect_boundary}. See also \cite[Theorem 5.2]{Horvatic1971} for the PL case. === Unknotting Theorems === ; {{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, $\S$ $\S$ 3]. 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

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

This is well-known strong Whitney embedding theorem.

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].

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].

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 $\S$ 5).

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.

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

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.

$\square$ $\square$

See also about the Hopf link in [Skopenkov2016h, $\S$ 2].

4 Invariants

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

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

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

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

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

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.

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

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

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,

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

By lemma 4.2

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

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 $\rho_2(d(s, -s)) = w_{n-2}(N_0)$ .

$\square$ $\square$

5 A generalization to highly-connected manifolds

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].

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.

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].

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

[Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
[Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
[Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
[Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
[Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
[Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
[Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
[Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)

[Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
[Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
[Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$ , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
[Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
[Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
[Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).

$-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 part (a) holds both for connected and not connected manifolds, it differs from the case for closed manifolds, compare to Theorem \ref{thm::closed_unknotting}. == A generalization to highly-connected manifolds == ; \label{prt::highly_connected} === Embedding theorems === ; {{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 proved in \cite[Existence Theorem (a)]{Haefliger1961}, the PL case of this result is proved 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}. === Unknotting Theorems === ; {{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 2.4]] of \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}. == 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 $i_{2n-1,n}\colon D^{n-1}\times S^n \to \R^{2n-1}$ is the standard embedding. Define $g_1\colon S^n\times [0, 1] \to D^{n-1}\times S^n, g_1(x, t) = (x, ta)$, where $a\in \partial D^{n-1}$ is an arbitrary point. Define $g_2\colon S^n\times [0, 1] \to D^{n-1}\times S^n, g_2(x, t) = (x, tx)$. Then embeddings $i_{2n-1,n}g_1$ and $i_{2n-1,n}g_2$ are not isotopic to each other. {{endproof}} See also about [[High_codimension_links#Examples|the Hopf link]] in \cite[$\S$ 2]{Skopenkov2016h}. == Invariants == ; Here we state all other results concerning embeddion of manifolds with boundary except some results when the classification of embeddiongs coinsides with one of immersions, see \cite[Theorem 2.3]{Skopenkov2010}. Result below can be found in parpers by O. Saeki and D Tonkonog. 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}} == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S1, $\S$ $\S$ 3]. 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

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

This is well-known strong Whitney embedding theorem.

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].

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].

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 $\S$ 5).

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.

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

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.

$\square$ $\square$

See also about the Hopf link in [Skopenkov2016h, $\S$ 2].

4 Invariants

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

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

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

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

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

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.

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

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

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,

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

By lemma 4.2

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

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 $\rho_2(d(s, -s)) = w_{n-2}(N_0)$ .

$\square$ $\square$

5 A generalization to highly-connected manifolds

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].

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.

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].

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

[Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
[Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
[Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
[Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
[Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
[Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
[Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
[Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)

[Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
[Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
[Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$ , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
[Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
[Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
[Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).

Embeddings of manifolds with boundary: classification

Revision as of 11:21, 14 May 2020

Contents

1 Introduction

2 Classification theorems

3 Examples

4 Invariants

5 A generalization to highly-connected manifolds

6 References

2 Classification theorems

3 Examples

4 Invariants

5 A generalization to highly-connected manifolds

6 References

2 Classification theorems

3 Examples

4 Invariants

5 A generalization to highly-connected manifolds

6 References

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@@ Line 27: / Line 27: @@
 == Classification theorems ==
-<wikitex>;
-First we state some well-known results. They can be generalized to the highly-connected cases (see $\S$\ref{prt::highly_connected}).
-</wikitex>
-=== Embedding theorems ===
 <wikitex>;
 {{beginthm|Theorem}}
@@ Line 41: / Line 37: @@
 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}. This result is a special case of Theorem \ref{thm::k_connect_boundary}. See also \cite[Theorem 5.2]{Horvatic1971} for the PL case.
+The Diff case of this result is proved in \cite[Theorem 4.6]{Hirsch1961a}.<!--This result is a special case of Theorem \ref{thm::k_connect_boundary}.--> For the PL case see references for Theorem \ref{thm::k_connect_boundary} below and \cite[Theorem 5.2]{Horvatic1971}.
-</wikitex>
-=== Unknotting Theorems ===
-<wikitex>;
 {{beginthm|Theorem}}\label{thm::closed_unknotting}
 Assume that $N$ is a compact $n$-manifold and either
@@ Line 68: / Line 60: @@
 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}.-->
-Observe that part (a) holds both for connected and not connected manifolds, it differs from the case for closed manifolds, compare to Theorem \ref{thm::closed_unknotting}.
+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}.
-</wikitex>
-== A generalization to highly-connected manifolds ==
+These basic results can be generalized to the highly-connected cases (see $\S$\ref{prt::highly_connected}).
-<wikitex>; \label{prt::highly_connected}
-</wikitex>
-=== Embedding theorems ===
-<wikitex>;
-{{beginthm|Theorem}}
-Assume that $N$ is a closed compact $k$-connected $n$-manifold and $n>2k+2$<!--$k\le n-3$-->. Then $N$ embeds into $\R^{2n-k}$.
-{{endthm}}
-The Diff case of this result is proved in \cite[Existence Theorem (a)]{Haefliger1961}, the PL case of this result is proved 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 $\R^{2n-k-1}$.<!--$n\ge k+3$-->
-{{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}.
-</wikitex>
-=== Unknotting Theorems ===
-<wikitex>;
-{{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}}
-See [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.4]] of \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.
-<!--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.-->
-{{endthm}}
-Theorem \ref{thm::special_Haef_Zem} is a special cases of the latter result<!--, see \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963}, \cite[Existence Theorem (b) in p. 47]{Haefliger1961}-->.
-See also \cite[Theorem 10.3]{Hudson1969}.
 </wikitex>
@@ Line 115: / Line 74: @@
 {{endthm}}
 {{beginproof}}
-Recall $i_{2n-1,n}\colon D^{n-1}\times S^n \to \R^{2n-1}$ is the standard embedding.
+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 $g_1\colon S^n\times [0, 1] \to D^{n-1}\times S^n, g_1(x, t) = (x, ta)$, where $a\in \partial D^{n-1}$ is an arbitrary point.
+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 $g_2\colon S^n\times [0, 1] \to D^{n-1}\times S^n, g_2(x, t) = (x, tx)$.
+Define $g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}, g_2(x, t) = (x, tx)$.
-<!--This construction is based on [[High_codimension_links#Examples|the Hopf link]].Let $f:S^{n-1}\sqcup S^{n-1}\to \mathbb R^{2n-1}$ be the Hopf link. The image of the Hopf link is the union of two $q$-spheres which can be described as follows: the spheres are $\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$. Denote by $f_1$ and $f_2$ restrictions of $f$ to the first and second components respectively.
+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.
-Then the embedding $g$ of $N$ can be obtained by the following folrmula:
-$$g(x, t) = tf_1(x) + (1-t)f_2(x), \text{ where } x\in S^{n-1}, t\in [0,1].$$-->Then embeddings $i_{2n-1,n}g_1$ and $i_{2n-1,n}g_2$ are not isotopic to each other.
 {{endproof}}
 See also about [[High_codimension_links#Examples|the Hopf link]] in \cite[$\S$ 2]{Skopenkov2016h}.<!--Let $S^q\subset\R^m$ be the standard sphere in $m$-space. Then on $S^q$ exist a normal field $s$ non-homotopic to the standard normal field. Denote by $s(S^q)$ the shift of $S^q$ along field $s$. Then the bounded connected sum of $S^q\times D^k$ and $s(S^q)\times D^k$ is not isotopic to trivial embedding.-->
@@ Line 180: / Line 137: @@
 {{endproof}}
 </wikitex>
+== A generalization to highly-connected manifolds ==
+<wikitex>; \label{prt::highly_connected}
+{{beginthm|Theorem}}
+Assume that $N$ is a closed compact $k$-connected $n$-manifold and $n>2k+2$<!--$k\le n-3$-->. 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 $\R^{2n-k-1}$.<!--$n\ge k+3$-->
+{{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}
+<!--[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}}
+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.
+<!--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.-->
+{{endthm}}
+Theorem \ref{thm::special_Haef_Zem} is a special cases of the latter result<!--, see \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963}, \cite[Existence Theorem (b) in p. 47]{Haefliger1961}-->.
+See also \cite[Theorem 10.3]{Hudson1969}, where the Theorem is proved modulo [[Isotopy#Concordance|concordance implies isotopy theorem]].
+</wikitex>
 <!--== Classification ==
 <wikitex>;