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.

Theorem 1.1. Every compact PL $n$$n$-manifold $N$$N$ with nonempty boundary PL embeds into $\R^{2n-1}$$\R^{2n-1}$.

This result can be found in [Horvatic1971, Theorem 5.2].

Theorem 1.2. Every compact $k$$k$-connected PL $n$$n$-manifold $N$$N$ with nonempty boundary PL embeds into $\R^{2n-k-1}$$\R^{2n-k-1}$ for each $n\ge k+3$$n\ge k+3$.

This result can be found in [Hudson1969, Theorem 8.3].

2 Unknotting Theorems

Theorem 2.1. 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 2.2. 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 2.3. 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], [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 2.4. 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 2.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].

3 Construction and examples

Observe that the claim of the Theorem 2.2 fails for $m = 2n - 1$$m = 2n - 1$, i.e. the first non trivial case. More precisely, the following statement holds.

Preposition 3.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold with non-empty boundary. Then there exist non isotopic embeddings of $N$$N$ to $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

4 Invariants

Denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient of two cycles.

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 $H^k(N)$$H^k(N)$ homology classes of $N$$N$ with coefficients in $\mathbb Z_2$$\mathbb Z_2$.

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

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

Lemma 4.1. Let $f\in\mathrm{Emb}^{2n-1}N_0$$f\in\mathrm{Emb}^{2n-1}N_0$, then

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

Proof. Note that $\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$

5 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
• [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.
• [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. {{beginthm|Theorem}} Every compact PL $n$-manifold $N$ with nonempty boundary PL embeds into $\R^{2n-1}$. {{endthm}} This result can be found in \cite[Theorem 5.2]{Horvatic1971}. {{beginthm|Theorem}} Every compact $k$-connected PL $n$-manifold $N$ with nonempty boundary PL embeds into $\R^{2n-k-1}$ for each $n\ge k+3$. {{endthm}} 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.

Theorem 1.1. Every compact PL $n$$n$-manifold $N$$N$ with nonempty boundary PL embeds into $\R^{2n-1}$$\R^{2n-1}$.

This result can be found in [Horvatic1971, Theorem 5.2].

Theorem 1.2. Every compact $k$$k$-connected PL $n$$n$-manifold $N$$N$ with nonempty boundary PL embeds into $\R^{2n-k-1}$$\R^{2n-k-1}$ for each $n\ge k+3$$n\ge k+3$.

This result can be found in [Hudson1969, Theorem 8.3].

2 Unknotting Theorems

Theorem 2.1. 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 2.2. 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 2.3. 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], [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 2.4. 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 2.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].

3 Construction and examples

Observe that the claim of the Theorem 2.2 fails for $m = 2n - 1$$m = 2n - 1$, i.e. the first non trivial case. More precisely, the following statement holds.

Preposition 3.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold with non-empty boundary. Then there exist non isotopic embeddings of $N$$N$ to $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

4 Invariants

Denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient of two cycles.

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 $H^k(N)$$H^k(N)$ homology classes of $N$$N$ with coefficients in $\mathbb Z_2$$\mathbb Z_2$.

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

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

Lemma 4.1. Let $f\in\mathrm{Emb}^{2n-1}N_0$$f\in\mathrm{Emb}^{2n-1}N_0$, then

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

Proof. Note that $\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$

5 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
• [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.
• [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}, \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}. == Construction and examples == ; Observe that the claim of the Theorem \ref{thm::special_Haef_Zem} fails for $m = 2n - 1$, i.e. the first non trivial case. More precisely, the following statement holds. {{beginthm|Preposition}} Assume that $N$ is a compact connected $n$-manifold with non-empty boundary. Then there exist non isotopic embeddings of $N$ to $\mathbb R^{2n-1}$. {{endthm}} == Invariants == ; Denote by $\mathrm{lk}$ the linking coefficient of two cycles. 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 $H^k(N)$ homology classes of $N$ with coefficients in $\mathbb Z_2$. Denote by $\rho_2 \colon H_*(M; \mathbb Z)\to H_*(M;\mathbb Z_2)$ reduction modulo $. Define the dual to Steifel-Whitney class $\bar w_{n-2}(N)\in H_{n-2}$ to be the class of the submanifold on which two general position normal fields on $N$ are linearly dependent. {{beginthm|Lemma}} Let $f\in\mathrm{Emb}^{2n-1}N_0$, then $$\rho_2(L(f)(x, y))\equiv \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2x\cap\rho_2y.$$ {{endthm}} {{beginproof}} Note that $\rho_2(L(f)(x, y)) = \rho_2(\mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y))).$ 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.

Theorem 1.1. Every compact PL $n$$n$-manifold $N$$N$ with nonempty boundary PL embeds into $\R^{2n-1}$$\R^{2n-1}$.

This result can be found in [Horvatic1971, Theorem 5.2].

Theorem 1.2. Every compact $k$$k$-connected PL $n$$n$-manifold $N$$N$ with nonempty boundary PL embeds into $\R^{2n-k-1}$$\R^{2n-k-1}$ for each $n\ge k+3$$n\ge k+3$.

This result can be found in [Hudson1969, Theorem 8.3].

2 Unknotting Theorems

Theorem 2.1. 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 2.2. 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 2.3. 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], [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 2.4. 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 2.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].

3 Construction and examples

Observe that the claim of the Theorem 2.2 fails for $m = 2n - 1$$m = 2n - 1$, i.e. the first non trivial case. More precisely, the following statement holds.

Preposition 3.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold with non-empty boundary. Then there exist non isotopic embeddings of $N$$N$ to $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

4 Invariants

Denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient of two cycles.

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 $H^k(N)$$H^k(N)$ homology classes of $N$$N$ with coefficients in $\mathbb Z_2$$\mathbb Z_2$.

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

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

Lemma 4.1. Let $f\in\mathrm{Emb}^{2n-1}N_0$$f\in\mathrm{Emb}^{2n-1}N_0$, then

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

Proof. Note that $\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$

5 References

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