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 Classification theorems

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

This is well-known strong Whitney embedding theorem.

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}$.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].

Theorem 2.3. 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]. Note that inequality in part (a) is sharp, see the Hopf link.

Theorem 2.4. Assume that $N$$N$ is a compact $n$$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. Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.

These basic results can be generalized to the highly-connected manifolds (see $\S$$\S$6).

3 Example

The following example is folklore.

Proposition 3.1. 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. Define $g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$$g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_1(x, t) = (x, t1_{n-1})$$g_1(x, t) = (x, t1_{n-1})$, where $1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$$1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$. Define $g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$$g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_2(x, t) = (x, tx)$$g_2(x, t) = (x, tx)$.

Recall that $\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$$\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$ is the standard embedding. Then embeddings $\mathrm ig_1$$\mathrm ig_1$ and $\mathrm ig_2$$\mathrm ig_2$ are not isotopic. Indeed, the components of $\mathrm ig_1(S^{n-1}\times \{0, 1\})$$\mathrm ig_1(S^{n-1}\times \{0, 1\})$ are not linked while the components of $\mathrm ig_2(S^{n-1}\times \{0, 1\})$$\mathrm ig_2(S^{n-1}\times \{0, 1\})$ are linked [Skopenkov2016h, $\S$$\S$ 3, remark 3.2d].

$\square$$\square$

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

4 Invariants

Let $N$$N$ be a closed connected $n$$n$-manifold. By $N_0$$N_0$ we will denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 4.1. Suppose $H_1(N, \mathbb Z)$$H_1(N, \mathbb Z)$ is torsion free. For each even $n$$n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0, \mathbb Z)\cong H_1(N_0, \partial N_0, \mathbb Z)\cong H_1(N, \mathbb Z)$$\bar e=\bar e(f)\in H^{n-1}(N_0, \mathbb Z)\cong H_1(N_0, \partial N_0, \mathbb Z)\cong H_1(N, \mathbb Z)$ to existence of a nowhere vanishing normal vector field on $N_0$$N_0$. In the following paragraph we show that $\bar e = 0$$\bar e = 0$.

A normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$ has dimension $n-1$$n-1$. As $n$$n$ is even thus $n-1$$n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$$\bar e=-\bar e$. This implies $\bar e=0$$\bar e=0$ because $H_1(N, \mathbb Z)$$H_1(N, \mathbb Z)$ is torsion free.

Since $N_0$$N_0$ has non-empty boundary it follows that $N_0$$N_0$ is homotopy equivalent to an $(n-1)$$(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$. Hence since the Euler class is zero there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

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

Lemma 4.2. Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Given two homology classes $[x],[y]\in H_{n-1}(N_0, \Z)$$[x],[y]\in H_{n-1}(N_0, \Z)$, let $s,s'$$s,s'$ be two normal vector fields of $f(N_0)$$f(N_0)$. 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 in the class of the non-zero vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$$n=3$, but the proof is valid in all dimensions.

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

Definition 4.3. 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)$$x, y\in H_{n-1}(N_0)$ are two closed connected oriented submanifolds of $N_0$$N_0$, $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the result of shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Note that $L$$L$ does not change when $x$$x$ or $y$$y$ are changed to homologic submanifolds or when $f$$f$ is changed to an isotopic embedding. Thus $L(f)$$L(f)$ is a bilinear form on $H_{n-1}(N_0, \mathbb Z)$$H_{n-1}(N_0, \mathbb Z)$

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0, \mathbb Z_2)$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0, \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields on $N_0$$N_0$ are linearly dependent.

Lemma 4.4. Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0, \mathbb Z)$$X, Y \in H_{n-1}(N_0, \mathbb Z)$ the following equality holds.

$\displaystyle \rho_2(L(f)(X, Y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$ $$\displaystyle \rho_2(L(f)(X, Y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in [Tonkonog2010], here we give proof covering a minor gap.

Proof. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\equiv_2 \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\equiv_2 \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y \end{aligned}$$

The first line is clear.

If we shift the link $s(c_x)\sqcup f(c_y)$$s(c_x)\sqcup f(c_y)$ by $-s$$-s$, we get the link $f(c_x)\sqcup -s(c_y)$$f(c_x)\sqcup -s(c_y)$. The linking coefficient will not change after this shift.

The third line follows from Lemma 4.2.

Finally, let us show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. If we generically perturb $-s$$-s$ it will become linearly dependent with $s$$s$ only on a 2--dimensional cycle $C$$C$ in $N_0$$N_0$. And $\rho_2([C]) = w_{n-2}(N_0)$$\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 I = d(s, -s)$$C\times I = d(s, -s)$. Thus $\rho_2d(s, -s) = w_{n-2}(N_0)$$\rho_2d(s, -s) = w_{n-2}(N_0)$.

$\square$$\square$

5 Classification

Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.

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.

Theorem 5.1. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$$L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$$\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$$\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class, and $\langle\cdot,\cdot\rangle$$\langle\cdot,\cdot\rangle$ is the standard pairing.

6 A generalization to highly-connected manifolds

Theorem 6.1. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n>2k+2$$n>2k+2$. Then $N$$N$ embeds into $\R^{2n-k}$$\R^{2n-k}$.

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

Theorem 6.2. 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}$.

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

Theorem 6.3. 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 survey [Skopenkov2016c, $\S$$\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 6.4. 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 2.4 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

7 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.
• [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.
• [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
• [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 == ; {{beginthm|Theorem}} Assume that $N$ is a closed compact $n$-manifold. Then $N$ embeds into $\R^{2n}$. {{endthm}} This is well-known [[Wikipedia:Whitney_embedding_theorem|strong Whitney embedding theorem]]. {{beginthm|Theorem}} Assume that $N$ is a compact $n$-manifold with nonempty boundary. Then $N$ embeds into $\R^{2n-1}$. {{endthm}} The Diff case of this result is proved in \cite[Theorem 4.6]{Hirsch1961a}. For the PL case see references for Theorem \ref{thm::k_connect_boundary} below and \cite[Theorem 5.2]{Horvatic1971}. {{beginthm|Theorem}}\label{thm::closed_unknotting} Assume that $N$ is a compact $n$-manifold and either (a) $m \ge 2n+2$ or (b) $N$ is connected and $m \ge 2n+1 \ge 5$. Then any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorems 2.1 and 2.2]] respectively of \cite[$\S$ 2]{Skopenkov2016c}. Note that inequality in part (a) is sharp, see [[High_codimension_links#Examples|the Hopf link]]. {{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$$\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\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 Classification theorems

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

This is well-known strong Whitney embedding theorem.

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}$.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].

Theorem 2.3. 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]. Note that inequality in part (a) is sharp, see the Hopf link.

Theorem 2.4. Assume that $N$$N$ is a compact $n$$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. Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.

These basic results can be generalized to the highly-connected manifolds (see $\S$$\S$6).

3 Example

The following example is folklore.

Proposition 3.1. 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. Define $g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$$g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_1(x, t) = (x, t1_{n-1})$$g_1(x, t) = (x, t1_{n-1})$, where $1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$$1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$. Define $g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$$g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_2(x, t) = (x, tx)$$g_2(x, t) = (x, tx)$.

Recall that $\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$$\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$ is the standard embedding. Then embeddings $\mathrm ig_1$$\mathrm ig_1$ and $\mathrm ig_2$$\mathrm ig_2$ are not isotopic. Indeed, the components of $\mathrm ig_1(S^{n-1}\times \{0, 1\})$$\mathrm ig_1(S^{n-1}\times \{0, 1\})$ are not linked while the components of $\mathrm ig_2(S^{n-1}\times \{0, 1\})$$\mathrm ig_2(S^{n-1}\times \{0, 1\})$ are linked [Skopenkov2016h, $\S$$\S$ 3, remark 3.2d].

$\square$$\square$

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

4 Invariants

Let $N$$N$ be a closed connected $n$$n$-manifold. By $N_0$$N_0$ we will denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 4.1. Suppose $H_1(N, \mathbb Z)$$H_1(N, \mathbb Z)$ is torsion free. For each even $n$$n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0, \mathbb Z)\cong H_1(N_0, \partial N_0, \mathbb Z)\cong H_1(N, \mathbb Z)$$\bar e=\bar e(f)\in H^{n-1}(N_0, \mathbb Z)\cong H_1(N_0, \partial N_0, \mathbb Z)\cong H_1(N, \mathbb Z)$ to existence of a nowhere vanishing normal vector field on $N_0$$N_0$. In the following paragraph we show that $\bar e = 0$$\bar e = 0$.

A normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$ has dimension $n-1$$n-1$. As $n$$n$ is even thus $n-1$$n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$$\bar e=-\bar e$. This implies $\bar e=0$$\bar e=0$ because $H_1(N, \mathbb Z)$$H_1(N, \mathbb Z)$ is torsion free.

Since $N_0$$N_0$ has non-empty boundary it follows that $N_0$$N_0$ is homotopy equivalent to an $(n-1)$$(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$. Hence since the Euler class is zero there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

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

Lemma 4.2. Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Given two homology classes $[x],[y]\in H_{n-1}(N_0, \Z)$$[x],[y]\in H_{n-1}(N_0, \Z)$, let $s,s'$$s,s'$ be two normal vector fields of $f(N_0)$$f(N_0)$. 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 in the class of the non-zero vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$$n=3$, but the proof is valid in all dimensions.

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

Definition 4.3. 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)$$x, y\in H_{n-1}(N_0)$ are two closed connected oriented submanifolds of $N_0$$N_0$, $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the result of shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Note that $L$$L$ does not change when $x$$x$ or $y$$y$ are changed to homologic submanifolds or when $f$$f$ is changed to an isotopic embedding. Thus $L(f)$$L(f)$ is a bilinear form on $H_{n-1}(N_0, \mathbb Z)$$H_{n-1}(N_0, \mathbb Z)$

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0, \mathbb Z_2)$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0, \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields on $N_0$$N_0$ are linearly dependent.

Lemma 4.4. Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0, \mathbb Z)$$X, Y \in H_{n-1}(N_0, \mathbb Z)$ the following equality holds.

$\displaystyle \rho_2(L(f)(X, Y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$ $$\displaystyle \rho_2(L(f)(X, Y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in [Tonkonog2010], here we give proof covering a minor gap.

Proof. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\equiv_2 \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\equiv_2 \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y \end{aligned}$$

The first line is clear.

If we shift the link $s(c_x)\sqcup f(c_y)$$s(c_x)\sqcup f(c_y)$ by $-s$$-s$, we get the link $f(c_x)\sqcup -s(c_y)$$f(c_x)\sqcup -s(c_y)$. The linking coefficient will not change after this shift.

The third line follows from Lemma 4.2.

Finally, let us show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. If we generically perturb $-s$$-s$ it will become linearly dependent with $s$$s$ only on a 2--dimensional cycle $C$$C$ in $N_0$$N_0$. And $\rho_2([C]) = w_{n-2}(N_0)$$\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 I = d(s, -s)$$C\times I = d(s, -s)$. Thus $\rho_2d(s, -s) = w_{n-2}(N_0)$$\rho_2d(s, -s) = w_{n-2}(N_0)$.

$\square$$\square$

5 Classification

Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.

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.

Theorem 5.1. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$$L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$$\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$$\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class, and $\langle\cdot,\cdot\rangle$$\langle\cdot,\cdot\rangle$ is the standard pairing.

6 A generalization to highly-connected manifolds

Theorem 6.1. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n>2k+2$$n>2k+2$. Then $N$$N$ embeds into $\R^{2n-k}$$\R^{2n-k}$.

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

Theorem 6.2. 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}$.

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

Theorem 6.3. 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 survey [Skopenkov2016c, $\S$$\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 6.4. 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 2.4 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

7 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.
• [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.
• [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
• [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. Inequality in part (a) is sharp, see Proposition \ref{exm::linked_boundary}. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem \ref{thm::closed_unknotting}. These basic results can be generalized to the highly-connected manifolds (see $\S$\ref{prt::highly_connected}). == Example == ; The following example is folklore. {{beginthm|Proposition}} \label{exm::linked_boundary} 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}} Define $g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_1(x, t) = (x, t1_{n-1})$, where \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 Classification theorems

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

This is well-known strong Whitney embedding theorem.

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}$.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].

Theorem 2.3. 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]. Note that inequality in part (a) is sharp, see the Hopf link.

Theorem 2.4. Assume that $N$$N$ is a compact $n$$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. Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.

These basic results can be generalized to the highly-connected manifolds (see $\S$$\S$6).

3 Example

The following example is folklore.

Proposition 3.1. 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. Define $g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$$g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_1(x, t) = (x, t1_{n-1})$$g_1(x, t) = (x, t1_{n-1})$, where $1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$$1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$. Define $g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$$g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_2(x, t) = (x, tx)$$g_2(x, t) = (x, tx)$.

Recall that $\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$$\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$ is the standard embedding. Then embeddings $\mathrm ig_1$$\mathrm ig_1$ and $\mathrm ig_2$$\mathrm ig_2$ are not isotopic. Indeed, the components of $\mathrm ig_1(S^{n-1}\times \{0, 1\})$$\mathrm ig_1(S^{n-1}\times \{0, 1\})$ are not linked while the components of $\mathrm ig_2(S^{n-1}\times \{0, 1\})$$\mathrm ig_2(S^{n-1}\times \{0, 1\})$ are linked [Skopenkov2016h, $\S$$\S$ 3, remark 3.2d].

$\square$$\square$

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

4 Invariants

Let $N$$N$ be a closed connected $n$$n$-manifold. By $N_0$$N_0$ we will denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 4.1. Suppose $H_1(N, \mathbb Z)$$H_1(N, \mathbb Z)$ is torsion free. For each even $n$$n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0, \mathbb Z)\cong H_1(N_0, \partial N_0, \mathbb Z)\cong H_1(N, \mathbb Z)$$\bar e=\bar e(f)\in H^{n-1}(N_0, \mathbb Z)\cong H_1(N_0, \partial N_0, \mathbb Z)\cong H_1(N, \mathbb Z)$ to existence of a nowhere vanishing normal vector field on $N_0$$N_0$. In the following paragraph we show that $\bar e = 0$$\bar e = 0$.

A normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$ has dimension $n-1$$n-1$. As $n$$n$ is even thus $n-1$$n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$$\bar e=-\bar e$. This implies $\bar e=0$$\bar e=0$ because $H_1(N, \mathbb Z)$$H_1(N, \mathbb Z)$ is torsion free.

Since $N_0$$N_0$ has non-empty boundary it follows that $N_0$$N_0$ is homotopy equivalent to an $(n-1)$$(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$. Hence since the Euler class is zero there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

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

Lemma 4.2. Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Given two homology classes $[x],[y]\in H_{n-1}(N_0, \Z)$$[x],[y]\in H_{n-1}(N_0, \Z)$, let $s,s'$$s,s'$ be two normal vector fields of $f(N_0)$$f(N_0)$. 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 in the class of the non-zero vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$$n=3$, but the proof is valid in all dimensions.

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

Definition 4.3. 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)$$x, y\in H_{n-1}(N_0)$ are two closed connected oriented submanifolds of $N_0$$N_0$, $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the result of shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Note that $L$$L$ does not change when $x$$x$ or $y$$y$ are changed to homologic submanifolds or when $f$$f$ is changed to an isotopic embedding. Thus $L(f)$$L(f)$ is a bilinear form on $H_{n-1}(N_0, \mathbb Z)$$H_{n-1}(N_0, \mathbb Z)$

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0, \mathbb Z_2)$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0, \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields on $N_0$$N_0$ are linearly dependent.

Lemma 4.4. Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0, \mathbb Z)$$X, Y \in H_{n-1}(N_0, \mathbb Z)$ the following equality holds.

$\displaystyle \rho_2(L(f)(X, Y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$ $$\displaystyle \rho_2(L(f)(X, Y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in [Tonkonog2010], here we give proof covering a minor gap.

Proof. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\equiv_2 \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\equiv_2 \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y \end{aligned}$$

The first line is clear.

If we shift the link $s(c_x)\sqcup f(c_y)$$s(c_x)\sqcup f(c_y)$ by $-s$$-s$, we get the link $f(c_x)\sqcup -s(c_y)$$f(c_x)\sqcup -s(c_y)$. The linking coefficient will not change after this shift.

The third line follows from Lemma 4.2.

Finally, let us show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. If we generically perturb $-s$$-s$ it will become linearly dependent with $s$$s$ only on a 2--dimensional cycle $C$$C$ in $N_0$$N_0$. And $\rho_2([C]) = w_{n-2}(N_0)$$\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 I = d(s, -s)$$C\times I = d(s, -s)$. Thus $\rho_2d(s, -s) = w_{n-2}(N_0)$$\rho_2d(s, -s) = w_{n-2}(N_0)$.

$\square$$\square$

5 Classification

Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.

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.

Theorem 5.1. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$$L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$$\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$$\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class, and $\langle\cdot,\cdot\rangle$$\langle\cdot,\cdot\rangle$ is the standard pairing.

6 A generalization to highly-connected manifolds

Theorem 6.1. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n>2k+2$$n>2k+2$. Then $N$$N$ embeds into $\R^{2n-k}$$\R^{2n-k}$.

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

Theorem 6.2. 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}$.

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

Theorem 6.3. 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 survey [Skopenkov2016c, $\S$$\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 6.4. 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 2.4 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

7 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.
• [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.
• [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
• [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).

_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$. Define $g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_2(x, t) = (x, tx)$. Recall that $\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$ is the standard embedding. Then embeddings $\mathrm ig_1$ and $\mathrm ig_2$ are not isotopic. Indeed, the components of $\mathrm ig_1(S^{n-1}\times \{0, 1\})$ are [[High_codimension_links#The_linking_coefficient|not linked]] while the components of $\mathrm ig_2(S^{n-1}\times \{0, 1\})$ are [[High_codimension_links#The_linking_coefficient|linked]] \cite[$\S$ 3, remark 3.2d]{Skopenkov2016h}.{{endproof}}This construction is analogous to [[High_codimension_links#Examples|the Hopf link]], see \cite[$\S$ 2]{Skopenkov2016h}. == Invariants == ; Let $N$ be a closed connected $n$-manifold. By $N_0$ we will denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. The following folklore result holds. {{beginthm|Lemma}} Suppose $H_1(N, \mathbb Z)$ is torsion free. For each even $n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$. {{endthm}} {{beginproof}} There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0, \mathbb Z)\cong H_1(N_0, \partial N_0, \mathbb Z)\cong H_1(N, \mathbb Z)$ to existence of a nowhere vanishing normal vector field on $N_0$. In the following paragraph we show that $\bar e = 0$. A normal space to $f(N_0)$ at any point of $f(N_0)$ has dimension $n-1$. As $n$ is even thus $n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$. This implies $\bar e=0$ because $H_1(N, \mathbb Z)$ is torsion free. Since $N_0$ has non-empty boundary it follows that $N_0$ is homotopy equivalent to an $(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$ at any point of $f(N_0)$. Hence since the Euler class is zero there exists a nowhere vanishing normal vector field to $f(N_0)$. {{endproof}} 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. {{beginthm|Lemma}}\label{lmm::saeki} Let $f:N_0\to \R^{2n-1}$ be an embedding. Given two homology classes $[x],[y]\in H_{n-1}(N_0, \Z)$, let $s,s'$ be two normal vector fields of $f(N_0)$. 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 in the class of the non-zero vector fields. {{endthm}} This Lemma is proved in \cite[Lemma 2.2]{Saeki1999} for $n=3$, but the proof is valid in all dimensions. Denote by $\rho_2 \colon H_*(N; \mathbb Z)\to H_*(N;\mathbb Z_2)$ the reduction modulo $. {{beginthm|Definition}} 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 closed connected oriented submanifolds of $N_0$, $s$ is a nowhere vanishing normal field to $f(N_0)$ and $s(x), s(y)$ are the result of shift of $f(x), f(y)$ by $s$. {{endthm}} Note that $L$ does not change when $x$ or $y$ are changed to homologic submanifolds or when $f$ is changed to an isotopic embedding. Thus $L(f)$ is a bilinear form on $H_{n-1}(N_0, \mathbb Z)$ Define the dual to [[Stiefel-Whitney_characteristic_classes|Stiefel-Whitney class]] $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0, \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields on $N_0$ are linearly dependent. {{beginthm|Lemma}} Let $f:N_0\to \R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0, \mathbb Z)$ the following equality holds. $$\rho_2(L(f)(X, Y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$ {{endthm}} This Lemma was stated in \cite{Tonkonog2010}, here we give proof covering a minor gap. {{beginproof}} We get $$ \begin{aligned} L(f)(X, Y) &\equiv_2 \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \ &= d(s, -s)\cap X\cap Y \end{aligned} $$ The first line is clear. If we shift the link $s(c_x)\sqcup f(c_y)$ by $-s$, we get the link $f(c_x)\sqcup -s(c_y)$. The linking coefficient will not change after this shift. The third line follows from Lemma \ref{lmm::saeki}. Finally, let us show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. If we generically perturb $-s$ it will become linearly dependent with $s$ only on a 2--dimensional cycle $C$ in $N_0$. And $\rho_2([C]) = w_{n-2}(N_0)$ by definition. On the other hand the linear homotopy of $s$ to perturbed $-s$ degenerates on $C\times I = d(s, -s)$. Thus $\rho_2d(s, -s) = w_{n-2}(N_0)$. {{endproof}} == Classification == Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions. Denote by $\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$ into $\mathbb R^{m}$ up to isotopy. {{beginthm|Theorem}} Let $N$ be a closed connected orientable $n$-manifold with $H_1(N)$ torsion-free, $n\ge 4$, $n$ even. Then (a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection. (b) The image of $L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$. Here $\bar w_2(N_0)$ is the normal Stiefel-Whitney class, and $\langle\cdot,\cdot\rangle$ is the standard pairing. {{endthm}} == A generalization to highly-connected manifolds == ; \label{prt::highly_connected} {{beginthm|Theorem}} Assume that $N$ is a closed compact $k$-connected $n$-manifold and $n>2k+2$. Then $N$ embeds into $\R^{2n-k}$. {{endthm}} The Diff case of this result is in \cite[Existence Theorem (a)]{Haefliger1961}, the PL case of this result is in \cite[Corollary 1.3]{Irwin1965}. {{beginthm|Theorem}}\label{thm::k_connect_boundary} Assume that $N$ is a compact $n$-manifold with nonempty boundary, $(N, \partial N)$ is $k$-connected and $n\ge2k+2$. Then $N$ embeds into $\R^{2n-k-1}$. {{endthm}} The PL case of this result is proved in \cite[Theorem 8.3]{Hudson1969}. For the Diff case see \cite[$\S$ 1.7, remark 2]{Haefliger1961}, where Haefliger proposes to use the deleted pruduct criterion to obtain this result. {{beginthm|Theorem}}\label{thm::k_connect_closed_unknot} Assume that $N$ is a closed $k$-connected $n$-manifold. Then for each $n\ge2k + 2$, $m \ge 2n - k + 1$ any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} See [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.4]] of survey \cite[$\S$ 2]{Skopenkov2016c}, or \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}. {{beginthm|Theorem}} Assume that $N$ is a $k$-connected $n$-manifold with non-empty boundary. Then for each $n\ge k+3$ and $m\ge2n-k$ any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} Theorem \ref{thm::special_Haef_Zem} is a special cases of the latter result. See also \cite[Theorem 10.3]{Hudson1969}, which is proved using [[Isotopy#Concordance|concordance implies isotopy theorem]]. == 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 Classification theorems

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

This is well-known strong Whitney embedding theorem.

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}$.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].

Theorem 2.3. 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]. Note that inequality in part (a) is sharp, see the Hopf link.

Theorem 2.4. Assume that $N$$N$ is a compact $n$$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. Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.

These basic results can be generalized to the highly-connected manifolds (see $\S$$\S$6).

3 Example

The following example is folklore.

Proposition 3.1. 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. Define $g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$$g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_1(x, t) = (x, t1_{n-1})$$g_1(x, t) = (x, t1_{n-1})$, where $1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$$1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$. Define $g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$$g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_2(x, t) = (x, tx)$$g_2(x, t) = (x, tx)$.

Recall that $\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$$\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$ is the standard embedding. Then embeddings $\mathrm ig_1$$\mathrm ig_1$ and $\mathrm ig_2$$\mathrm ig_2$ are not isotopic. Indeed, the components of $\mathrm ig_1(S^{n-1}\times \{0, 1\})$$\mathrm ig_1(S^{n-1}\times \{0, 1\})$ are not linked while the components of $\mathrm ig_2(S^{n-1}\times \{0, 1\})$$\mathrm ig_2(S^{n-1}\times \{0, 1\})$ are linked [Skopenkov2016h, $\S$$\S$ 3, remark 3.2d].

$\square$$\square$

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

4 Invariants

Let $N$$N$ be a closed connected $n$$n$-manifold. By $N_0$$N_0$ we will denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 4.1. Suppose $H_1(N, \mathbb Z)$$H_1(N, \mathbb Z)$ is torsion free. For each even $n$$n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0, \mathbb Z)\cong H_1(N_0, \partial N_0, \mathbb Z)\cong H_1(N, \mathbb Z)$$\bar e=\bar e(f)\in H^{n-1}(N_0, \mathbb Z)\cong H_1(N_0, \partial N_0, \mathbb Z)\cong H_1(N, \mathbb Z)$ to existence of a nowhere vanishing normal vector field on $N_0$$N_0$. In the following paragraph we show that $\bar e = 0$$\bar e = 0$.

A normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$ has dimension $n-1$$n-1$. As $n$$n$ is even thus $n-1$$n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$$\bar e=-\bar e$. This implies $\bar e=0$$\bar e=0$ because $H_1(N, \mathbb Z)$$H_1(N, \mathbb Z)$ is torsion free.

Since $N_0$$N_0$ has non-empty boundary it follows that $N_0$$N_0$ is homotopy equivalent to an $(n-1)$$(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$. Hence since the Euler class is zero there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

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

Lemma 4.2. Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Given two homology classes $[x],[y]\in H_{n-1}(N_0, \Z)$$[x],[y]\in H_{n-1}(N_0, \Z)$, let $s,s'$$s,s'$ be two normal vector fields of $f(N_0)$$f(N_0)$. 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 in the class of the non-zero vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$$n=3$, but the proof is valid in all dimensions.

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

Definition 4.3. 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)$$x, y\in H_{n-1}(N_0)$ are two closed connected oriented submanifolds of $N_0$$N_0$, $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the result of shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Note that $L$$L$ does not change when $x$$x$ or $y$$y$ are changed to homologic submanifolds or when $f$$f$ is changed to an isotopic embedding. Thus $L(f)$$L(f)$ is a bilinear form on $H_{n-1}(N_0, \mathbb Z)$$H_{n-1}(N_0, \mathbb Z)$

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0, \mathbb Z_2)$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0, \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields on $N_0$$N_0$ are linearly dependent.

Lemma 4.4. Let $f:N_0\to \R^{2n-1}$$f:N_0\to \R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0, \mathbb Z)$$X, Y \in H_{n-1}(N_0, \mathbb Z)$ the following equality holds.

$\displaystyle \rho_2(L(f)(X, Y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$ $$\displaystyle \rho_2(L(f)(X, Y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in [Tonkonog2010], here we give proof covering a minor gap.

Proof. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\equiv_2 \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\equiv_2 \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y \end{aligned}$$

The first line is clear.

If we shift the link $s(c_x)\sqcup f(c_y)$$s(c_x)\sqcup f(c_y)$ by $-s$$-s$, we get the link $f(c_x)\sqcup -s(c_y)$$f(c_x)\sqcup -s(c_y)$. The linking coefficient will not change after this shift.

The third line follows from Lemma 4.2.

Finally, let us show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. If we generically perturb $-s$$-s$ it will become linearly dependent with $s$$s$ only on a 2--dimensional cycle $C$$C$ in $N_0$$N_0$. And $\rho_2([C]) = w_{n-2}(N_0)$$\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 I = d(s, -s)$$C\times I = d(s, -s)$. Thus $\rho_2d(s, -s) = w_{n-2}(N_0)$$\rho_2d(s, -s) = w_{n-2}(N_0)$.

$\square$$\square$

5 Classification

Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.

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.

Theorem 5.1. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$$L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$$\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$$\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class, and $\langle\cdot,\cdot\rangle$$\langle\cdot,\cdot\rangle$ is the standard pairing.

6 A generalization to highly-connected manifolds

Theorem 6.1. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n>2k+2$$n>2k+2$. Then $N$$N$ embeds into $\R^{2n-k}$$\R^{2n-k}$.

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

Theorem 6.2. 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}$.

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

Theorem 6.3. 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 survey [Skopenkov2016c, $\S$$\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 6.4. 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 2.4 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

7 References

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