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, $\S$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\S$1, $\S$$\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$$\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$$\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 Embedding theorems

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

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

This result is a special case of [Hudson1969, Theorem 8.3]. See also [Horvatic1971, Theorem 5.2].

Theorem 2.3. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n\ge2k+2$$n\ge2k+2$. Then $N$$N$ embeds into $\R^{2n-k}$$\R^{2n-k}$.

PL case of this result can be found in [Irwin1965, Corollary 1.3].

Theorem 2.4. Assume that $N$$N$ is a compact $n$$n$-manifold with nonempty boundary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $n\ge2k+2$$n\ge2k+2$. Then $N$$N$ embeds into $\R^{2n-k-1}$$\R^{2n-k-1}$.

PL case of this result can be found in [Hudson1969, Theorem 8.3].

3 Unknotting Theorems

Theorem 3.1. Assume that $N$$N$ is a compact $n$$n$-manifold and either

(a) $m \ge 2n+2$$m \ge 2n+2$ or

(b) $N$$N$ is connected and $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$.

Then any two embeddings of $N$$N$ into $\R^m$$\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$$\S$ 2].

Theorem 3.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold with non-empty boundary and either

(a) $m \ge 2n$$m \ge 2n$ or

(b) $N$$N$ is $1$$1$-connected, $m \ge 2n - 1\ge3$$m \ge 2n - 1\ge3$.

Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Part (a) of this theorem in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, Corollary 5]. Case $n=1$$n=1$ is clear.

Theorem 3.3. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $n\ge2k + 2$$n\ge2k + 2$, $m \ge 2n - k + 1$$m \ge 2n - k + 1$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

See Theorem 2.4 of [Skopenkov2016c, $\S$$\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 3.4. Assume that $N$$N$ is a $k$$k$-connected $n$$n$-manifold with non-empty boundary. Then for each $n\ge k+3$$n\ge k+3$ and $m\ge2n-k$$m\ge2n-k$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Theorem 3.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].

4 Examples

Observe that analog of Theorem 3.2 (a) fails for $m = 2n - 1$$m = 2n - 1$, i.e. for the first non trivial case. More precisely, the following folklore statement holds.

Proposition 4.1. Let $N=S^{n-1}\times [0, 1]$$N=S^{n-1}\times [0, 1]$ be the cylinder over $S^{n-1}$$S^{n-1}$. Then there exist non isotopic embeddings of $N$$N$ to $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Proof. Let $f:S^{n-1}\sqcup S^{n-1}\to \mathbb R^{2n-1}$$f:S^{n-1}\sqcup S^{n-1}\to \mathbb R^{2n-1}$ be the Hopf link. The image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows: the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$. Denote by $f_1$$f_1$ and $f_2$$f_2$ restrictions of $f$$f$ to the first and second components respectively. Then the embedding $g$$g$ of $N$$N$ can be obtained by the following folrmula:

$\displaystyle g(x, t) = tf_1(x) + (1-t)f_2(x), \text{ where } x\in S^{n-1}, t\in [0,1].$ $$\displaystyle g(x, t) = tf_1(x) + (1-t)f_2(x), \text{ where } x\in S^{n-1}, t\in [0,1].$$

Embedding $g$$g$ is not isotopic to the standard embedding, because the components of its boundary are linked.

$\square$$\square$

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

5 Invariants

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

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

The following folklore result holds.

Lemma 5.1. For each even $n$$n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ exists a nowhere vanishing normal field to $f(N_0)$$f(N_0)$.

For even $n$$n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$$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)$$x, y\in H_{n-1}(N_0)$ are two homology classes, realized by closed connected orientation submanifolds of $N_0$$N_0$, $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the submanifolds $f(x), f(y)$$f(x), f(y)$ shifted by $s$$s$.

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

Denote by $\mathrm{Emb}^{m}N_0$$\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$$N_0$ into $\mathbb R^{m}$$\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}$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_{n-2}$ to be the class of the cycle on which two general position normal fields on $N_0$$N_0$ are linearly dependent.

Lemma 5.2.(O. Saeki)

Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Let $s,s'$$s,s'$ be two sections of $T|_{x\cup y}$$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)$$d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$$s,s'$ being homotopic as sections of $\pi$$\pi$.

Lemma 5.3. Let $f\in\mathrm{Emb}^{2n-1}N_0$$f\in\mathrm{Emb}^{2n-1}N_0$, then

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

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

Denote by $-s$$-s$ the normal vector field opposite to $s$$s$. If we shift the link $s(x)\sqcup f(y)$$s(x)\sqcup f(y)$ by $-s$$-s$, we get the link $f(x), -s(y)$$f(x), -s(y)$ and the $\mathrm{lk}$$\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 ??%\ref{}

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

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

$\square$$\square$

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.
• [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
• [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.
• [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. == Embedding theorems == ; {{beginthm|Theorem}} Assume that $N$ is a closed compact $n$-manifold. Then $N$ embeds into $\R^{2n}$. {{endthm}} {{beginthm|Theorem}} Assume that $N$ is a compact $n$-manifold with nonempty boundary. Then $N$ embeds into $\R^{2n-1}$. {{endthm}} This result is a special case of \cite[Theorem 8.3]{Hudson1969}. See also \cite[Theorem 5.2]{Horvatic1971}. {{beginthm|Theorem}} Assume that $N$ is a closed compact $k$-connected $n$-manifold and $n\ge2k+2$. Then $N$ embeds into $\R^{2n-k}$. {{endthm}} PL case of this result can be found in \cite[Corollary 1.3]{Irwin1965}. {{beginthm|Theorem}} 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}} PL case of this result can be found in \cite[Theorem 8.3]{Hudson1969}. == 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 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$$\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$$\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 Embedding theorems

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

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

This result is a special case of [Hudson1969, Theorem 8.3]. See also [Horvatic1971, Theorem 5.2].

Theorem 2.3. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n\ge2k+2$$n\ge2k+2$. Then $N$$N$ embeds into $\R^{2n-k}$$\R^{2n-k}$.

PL case of this result can be found in [Irwin1965, Corollary 1.3].

Theorem 2.4. Assume that $N$$N$ is a compact $n$$n$-manifold with nonempty boundary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $n\ge2k+2$$n\ge2k+2$. Then $N$$N$ embeds into $\R^{2n-k-1}$$\R^{2n-k-1}$.

PL case of this result can be found in [Hudson1969, Theorem 8.3].

3 Unknotting Theorems

Theorem 3.1. Assume that $N$$N$ is a compact $n$$n$-manifold and either

(a) $m \ge 2n+2$$m \ge 2n+2$ or

(b) $N$$N$ is connected and $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$.

Then any two embeddings of $N$$N$ into $\R^m$$\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$$\S$ 2].

Theorem 3.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold with non-empty boundary and either

(a) $m \ge 2n$$m \ge 2n$ or

(b) $N$$N$ is $1$$1$-connected, $m \ge 2n - 1\ge3$$m \ge 2n - 1\ge3$.

Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Part (a) of this theorem in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, Corollary 5]. Case $n=1$$n=1$ is clear.

Theorem 3.3. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $n\ge2k + 2$$n\ge2k + 2$, $m \ge 2n - k + 1$$m \ge 2n - k + 1$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

See Theorem 2.4 of [Skopenkov2016c, $\S$$\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 3.4. Assume that $N$$N$ is a $k$$k$-connected $n$$n$-manifold with non-empty boundary. Then for each $n\ge k+3$$n\ge k+3$ and $m\ge2n-k$$m\ge2n-k$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Theorem 3.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].

4 Examples

Observe that analog of Theorem 3.2 (a) fails for $m = 2n - 1$$m = 2n - 1$, i.e. for the first non trivial case. More precisely, the following folklore statement holds.

Proposition 4.1. Let $N=S^{n-1}\times [0, 1]$$N=S^{n-1}\times [0, 1]$ be the cylinder over $S^{n-1}$$S^{n-1}$. Then there exist non isotopic embeddings of $N$$N$ to $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Proof. Let $f:S^{n-1}\sqcup S^{n-1}\to \mathbb R^{2n-1}$$f:S^{n-1}\sqcup S^{n-1}\to \mathbb R^{2n-1}$ be the Hopf link. The image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows: the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$. Denote by $f_1$$f_1$ and $f_2$$f_2$ restrictions of $f$$f$ to the first and second components respectively. Then the embedding $g$$g$ of $N$$N$ can be obtained by the following folrmula:

$\displaystyle g(x, t) = tf_1(x) + (1-t)f_2(x), \text{ where } x\in S^{n-1}, t\in [0,1].$ $$\displaystyle g(x, t) = tf_1(x) + (1-t)f_2(x), \text{ where } x\in S^{n-1}, t\in [0,1].$$

Embedding $g$$g$ is not isotopic to the standard embedding, because the components of its boundary are linked.

$\square$$\square$

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

5 Invariants

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

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

The following folklore result holds.

Lemma 5.1. For each even $n$$n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ exists a nowhere vanishing normal field to $f(N_0)$$f(N_0)$.

For even $n$$n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$$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)$$x, y\in H_{n-1}(N_0)$ are two homology classes, realized by closed connected orientation submanifolds of $N_0$$N_0$, $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the submanifolds $f(x), f(y)$$f(x), f(y)$ shifted by $s$$s$.

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

Denote by $\mathrm{Emb}^{m}N_0$$\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$$N_0$ into $\mathbb R^{m}$$\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}$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_{n-2}$ to be the class of the cycle on which two general position normal fields on $N_0$$N_0$ are linearly dependent.

Lemma 5.2.(O. Saeki)

Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Let $s,s'$$s,s'$ be two sections of $T|_{x\cup y}$$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)$$d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$$s,s'$ being homotopic as sections of $\pi$$\pi$.

Lemma 5.3. Let $f\in\mathrm{Emb}^{2n-1}N_0$$f\in\mathrm{Emb}^{2n-1}N_0$, then

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

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

Denote by $-s$$-s$ the normal vector field opposite to $s$$s$. If we shift the link $s(x)\sqcup f(y)$$s(x)\sqcup f(y)$ by $-s$$-s$, we get the link $f(x), -s(y)$$f(x), -s(y)$ and the $\mathrm{lk}$$\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 ??%\ref{}

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

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

$\square$$\square$

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.
• [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
• [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.
• [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. {{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}} 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 $\S$1, $\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$$\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$$\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 Embedding theorems

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

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

This result is a special case of [Hudson1969, Theorem 8.3]. See also [Horvatic1971, Theorem 5.2].

Theorem 2.3. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n\ge2k+2$$n\ge2k+2$. Then $N$$N$ embeds into $\R^{2n-k}$$\R^{2n-k}$.

PL case of this result can be found in [Irwin1965, Corollary 1.3].

Theorem 2.4. Assume that $N$$N$ is a compact $n$$n$-manifold with nonempty boundary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $n\ge2k+2$$n\ge2k+2$. Then $N$$N$ embeds into $\R^{2n-k-1}$$\R^{2n-k-1}$.

PL case of this result can be found in [Hudson1969, Theorem 8.3].

3 Unknotting Theorems

Theorem 3.1. Assume that $N$$N$ is a compact $n$$n$-manifold and either

(a) $m \ge 2n+2$$m \ge 2n+2$ or

(b) $N$$N$ is connected and $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$.

Then any two embeddings of $N$$N$ into $\R^m$$\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$$\S$ 2].

Theorem 3.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold with non-empty boundary and either

(a) $m \ge 2n$$m \ge 2n$ or

(b) $N$$N$ is $1$$1$-connected, $m \ge 2n - 1\ge3$$m \ge 2n - 1\ge3$.

Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Part (a) of this theorem in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, Corollary 5]. Case $n=1$$n=1$ is clear.

Theorem 3.3. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $n\ge2k + 2$$n\ge2k + 2$, $m \ge 2n - k + 1$$m \ge 2n - k + 1$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

See Theorem 2.4 of [Skopenkov2016c, $\S$$\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 3.4. Assume that $N$$N$ is a $k$$k$-connected $n$$n$-manifold with non-empty boundary. Then for each $n\ge k+3$$n\ge k+3$ and $m\ge2n-k$$m\ge2n-k$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Theorem 3.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].

4 Examples

Observe that analog of Theorem 3.2 (a) fails for $m = 2n - 1$$m = 2n - 1$, i.e. for the first non trivial case. More precisely, the following folklore statement holds.

Proposition 4.1. Let $N=S^{n-1}\times [0, 1]$$N=S^{n-1}\times [0, 1]$ be the cylinder over $S^{n-1}$$S^{n-1}$. Then there exist non isotopic embeddings of $N$$N$ to $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Proof. Let $f:S^{n-1}\sqcup S^{n-1}\to \mathbb R^{2n-1}$$f:S^{n-1}\sqcup S^{n-1}\to \mathbb R^{2n-1}$ be the Hopf link. The image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows: the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$. Denote by $f_1$$f_1$ and $f_2$$f_2$ restrictions of $f$$f$ to the first and second components respectively. Then the embedding $g$$g$ of $N$$N$ can be obtained by the following folrmula:

$\displaystyle g(x, t) = tf_1(x) + (1-t)f_2(x), \text{ where } x\in S^{n-1}, t\in [0,1].$ $$\displaystyle g(x, t) = tf_1(x) + (1-t)f_2(x), \text{ where } x\in S^{n-1}, t\in [0,1].$$

Embedding $g$$g$ is not isotopic to the standard embedding, because the components of its boundary are linked.

$\square$$\square$

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

5 Invariants

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

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

The following folklore result holds.

Lemma 5.1. For each even $n$$n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ exists a nowhere vanishing normal field to $f(N_0)$$f(N_0)$.

For even $n$$n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$$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)$$x, y\in H_{n-1}(N_0)$ are two homology classes, realized by closed connected orientation submanifolds of $N_0$$N_0$, $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the submanifolds $f(x), f(y)$$f(x), f(y)$ shifted by $s$$s$.

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

Denote by $\mathrm{Emb}^{m}N_0$$\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$$N_0$ into $\mathbb R^{m}$$\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}$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_{n-2}$ to be the class of the cycle on which two general position normal fields on $N_0$$N_0$ are linearly dependent.

Lemma 5.2.(O. Saeki)

Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Let $s,s'$$s,s'$ be two sections of $T|_{x\cup y}$$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)$$d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$$s,s'$ being homotopic as sections of $\pi$$\pi$.

Lemma 5.3. Let $f\in\mathrm{Emb}^{2n-1}N_0$$f\in\mathrm{Emb}^{2n-1}N_0$, then

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

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

Denote by $-s$$-s$ the normal vector field opposite to $s$$s$. If we shift the link $s(x)\sqcup f(y)$$s(x)\sqcup f(y)$ by $-s$$-s$, we get the link $f(x), -s(y)$$f(x), -s(y)$ and the $\mathrm{lk}$$\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 ??%\ref{}

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

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

$\square$$\square$

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.
• [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
• [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.
• [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).

\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 the embedding $g$ of $N$ can be obtained by the following folrmula: $$g(x, t) = tf_1(x) + (1-t)f_2(x), \text{ where } x\in S^{n-1}, t\in [0,1].$$ Embedding $g$ is not isotopic to the standard embedding, because the components of its boundary are linked. {{endproof}} See also about [[High_codimension_links#Examples|the Hopf link]] in \cite[$\S$ 2]{Skopenkov2016h}. == Invariants == ; 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}}(O. Saeki) \label{Lsaeki} Let $f:N_0\to \R^{2n-1}$ be an embedding. 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}} {{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{} $$\mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)) = d(s, -s)\cap x\cap y.$$ 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$$\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$$\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 Embedding theorems

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

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

This result is a special case of [Hudson1969, Theorem 8.3]. See also [Horvatic1971, Theorem 5.2].

Theorem 2.3. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n\ge2k+2$$n\ge2k+2$. Then $N$$N$ embeds into $\R^{2n-k}$$\R^{2n-k}$.

PL case of this result can be found in [Irwin1965, Corollary 1.3].

Theorem 2.4. Assume that $N$$N$ is a compact $n$$n$-manifold with nonempty boundary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $n\ge2k+2$$n\ge2k+2$. Then $N$$N$ embeds into $\R^{2n-k-1}$$\R^{2n-k-1}$.

PL case of this result can be found in [Hudson1969, Theorem 8.3].

3 Unknotting Theorems

Theorem 3.1. Assume that $N$$N$ is a compact $n$$n$-manifold and either

(a) $m \ge 2n+2$$m \ge 2n+2$ or

(b) $N$$N$ is connected and $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$.

Then any two embeddings of $N$$N$ into $\R^m$$\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$$\S$ 2].

Theorem 3.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold with non-empty boundary and either

(a) $m \ge 2n$$m \ge 2n$ or

(b) $N$$N$ is $1$$1$-connected, $m \ge 2n - 1\ge3$$m \ge 2n - 1\ge3$.

Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Part (a) of this theorem in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, Corollary 5]. Case $n=1$$n=1$ is clear.

Theorem 3.3. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $n\ge2k + 2$$n\ge2k + 2$, $m \ge 2n - k + 1$$m \ge 2n - k + 1$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

See Theorem 2.4 of [Skopenkov2016c, $\S$$\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 3.4. Assume that $N$$N$ is a $k$$k$-connected $n$$n$-manifold with non-empty boundary. Then for each $n\ge k+3$$n\ge k+3$ and $m\ge2n-k$$m\ge2n-k$ any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Theorem 3.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].

4 Examples

Observe that analog of Theorem 3.2 (a) fails for $m = 2n - 1$$m = 2n - 1$, i.e. for the first non trivial case. More precisely, the following folklore statement holds.

Proposition 4.1. Let $N=S^{n-1}\times [0, 1]$$N=S^{n-1}\times [0, 1]$ be the cylinder over $S^{n-1}$$S^{n-1}$. Then there exist non isotopic embeddings of $N$$N$ to $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Proof. Let $f:S^{n-1}\sqcup S^{n-1}\to \mathbb R^{2n-1}$$f:S^{n-1}\sqcup S^{n-1}\to \mathbb R^{2n-1}$ be the Hopf link. The image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows: the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$. Denote by $f_1$$f_1$ and $f_2$$f_2$ restrictions of $f$$f$ to the first and second components respectively. Then the embedding $g$$g$ of $N$$N$ can be obtained by the following folrmula:

$\displaystyle g(x, t) = tf_1(x) + (1-t)f_2(x), \text{ where } x\in S^{n-1}, t\in [0,1].$ $$\displaystyle g(x, t) = tf_1(x) + (1-t)f_2(x), \text{ where } x\in S^{n-1}, t\in [0,1].$$

Embedding $g$$g$ is not isotopic to the standard embedding, because the components of its boundary are linked.

$\square$$\square$

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

5 Invariants

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

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

The following folklore result holds.

Lemma 5.1. For each even $n$$n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ exists a nowhere vanishing normal field to $f(N_0)$$f(N_0)$.

For even $n$$n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$$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)$$x, y\in H_{n-1}(N_0)$ are two homology classes, realized by closed connected orientation submanifolds of $N_0$$N_0$, $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the submanifolds $f(x), f(y)$$f(x), f(y)$ shifted by $s$$s$.

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

Denote by $\mathrm{Emb}^{m}N_0$$\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$$N_0$ into $\mathbb R^{m}$$\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}$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_{n-2}$ to be the class of the cycle on which two general position normal fields on $N_0$$N_0$ are linearly dependent.

Lemma 5.2.(O. Saeki)

Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Let $s,s'$$s,s'$ be two sections of $T|_{x\cup y}$$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)$$d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$$s,s'$ being homotopic as sections of $\pi$$\pi$.

Lemma 5.3. Let $f\in\mathrm{Emb}^{2n-1}N_0$$f\in\mathrm{Emb}^{2n-1}N_0$, then

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

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

Denote by $-s$$-s$ the normal vector field opposite to $s$$s$. If we shift the link $s(x)\sqcup f(y)$$s(x)\sqcup f(y)$ by $-s$$-s$, we get the link $f(x), -s(y)$$f(x), -s(y)$ and the $\mathrm{lk}$$\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 ??%\ref{}

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

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

$\square$$\square$

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