# Embeddings of manifolds with boundary: classification

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## 1 Introduction

In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In $\S$${{Authors|Mikhail Fedorov}} {{Stub}} == Introduction == ; \label{sec::intro} In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In \S\ref{sec::linking_form} we introduce an invariant of embedding of a n-manifold in (n-1)-space for even n. In \S\ref{sec::generalisations} which is independent from \S\ref{sec::example}, \S\ref{sec::linking_form} and \S\ref{sec::classification} we state generalisations of theorems from \S\ref{sec::general_theorems} to highly-connected manifolds. For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[\S\S$4 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$6 which is independent from $\S$$\S$3, $\S$$\S$4 and $\S$$\S$5 we state generalisations of theorems from $\S$$\S$2 to highly-connected manifolds.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category.

We state the simplest results. These results 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.

We do not claim the references we give are references to original proofs.

## 2 Embedding and unknotting theorems

Theorem 2.1. Assume that $N$$N$ is a closed compact $n$$n$-manifold. Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb 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$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

This theorem is a corollary of strong Whitney immersion theorem. For the Diff case of this result see [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$$N$ into $\mathbb R^m$$\mathbb 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, which is shown by the construction of the Hopf link.

Theorem 2.4. 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$$N$ into $\mathbb R^m$$\mathbb 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. Both parts of this theorem are special cases of the Theorem 6.4.

Inequality in part (a) is sharp, see Proposition 3.1.

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

## 3 Example on non-isotopic embeddings

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

The following example is folklore.

Proposition 3.1. Let $N=S^k\times [0, 1]$$N=S^k\times [0, 1]$ be the cylinder over $S^k$$S^k$. (a) Then there exist non-isotopic embeddings of $N$$N$ into $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$.

(b) Then for each $a\in\mathbb Z$$a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$$f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$$\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$.

Proof. Let $h\colon S^k\to S^k$$h\colon S^k\to S^k$ be a map of degree $a$$a$. To prove part (a) it is sufficient to take the identity map of $S^k$$S^k$ as a map of degree one and the constant map as a map of degree zero. Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$$g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$$g(x, t) = (x, h(x)t)$, where $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Let $f=\mathrm ig$$f=\mathrm ig$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is the standard embedding.Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$$\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$

$\square$$\square$

## 4 Seifert linking form

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. If the (co)homology coefficients are omitted, then we assume them to be $\mathbb Z$$\mathbb Z$.

Denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

Example 4.1. For $N=S^k\times S^1$$N=S^k\times S^1$ and each $k\ge2$$k\ge2$ there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ given by $\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given by Proposition 3.1(b).

The following folklore result holds.

Lemma 4.2. Assume $N$$N$ is a closed orientable connected $n$$n$-manifold, $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free. Then for 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)\cong H_1(N_0, \partial N_0)\cong H_1(N)$$\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$$f(N_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$. Since $H_1(N)$$H_1(N)$ is torsion free, it follows that $\bar e=0$$\bar e=0$.

Since $N_0$$N_0$ has non-empty boundary, we have that $N_0$$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)$$f(N_0)$. Since $\bar e=0$$\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

Denote by $x, y$$x, y$ two disjoint $(n-1)$$(n-1)$-cycles in $N_0$$N_0$ with integer coefficients.

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

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

where $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the results of the shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Lemma 4.4 ($L$$L$ is well-defined). For even $n$$n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ the integer $L(f)(x, y)$$L(f)(x, y)$:

• is well-defined, i.e. does not change when $s$$s$ is replaced by $s'$$s'$,
• does not change when $x$$x$ or $y$$y$ are changed to homologous cycles and,
• does not change when $f$$f$ is changed to an isotopic embedding.

The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.

We will need the following supporting lemma.

Lemma 4.5. Let $f:N_0\to \mathbb R^{2n-1}$$f:N_0\to \mathbb R^{2n-1}$ be an embedding. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

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

where $s(y)$$s(y)$ is the result of the shift of $f(y)$$f(y)$ by $s$$s$, and $d(s,s')\in H_2(N_0)$$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 nowhere vanishing vector fields.

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

Proof of Lemma 4.4. The first bullet point follows because:
\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}

Here the second equality follows from Lemma 4.5.

For each two homologous $(n-1)$$(n-1)$-cycles $x, x'$$x, x'$ in $N_0$$N_0$, the image of the homology between $x$$x$ and $x'$$x'$ is a $n$$n$-chain $X$$X$ of $f(N_0)$$f(N_0)$ such that $\partial X = f(x) - f(x')$$\partial X = f(x) - f(x')$. Since $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$, this implies that the supports of $s(y)$$s(y)$ and $X$$X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$$\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.

Since isotopy of $f$$f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$$f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy.

$\square$$\square$

Lemma 4.4 implies that $L(f)$$L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$$H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter.

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

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

Lemma 4.6. Let $f:N_0\to \mathbb R^{2n-1}$$f:N_0\to \mathbb R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 4.6. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get
\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \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 congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$$s(X)\sqcup f(Y)$ by $-s$$-s$, we get the link $f(X)\sqcup -s(Y)$$f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 4.5.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$$s'$ a general perturbation of $s$$s$. We get:

$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$

The first equality holds because $s'$$s'$ and $s$$s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$$s'$ and $-s$$-s$ degenerates only at those points $x$$x$ where $s'(x)=s(x)$$s'(x)=s(x)$. These points $x$$x$ are exactly points where $s'(x)$$s'(x)$ and $s(x)$$s(x)$ are linearly dependent. All those point $x$$x$ form a $2$$2$-cycle modulo two in $N_0$$N_0$. The homotopy class of this $2$$2$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$$\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class.

$\square$$\square$

## 5 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. Denote $\mathrm{Emb}^mN$$\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$$f\colon N\to\mathbb R^m$ up to isotopy. For a free Abelian group $A$$A$, let $B_n^∗A$$B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$$\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$$\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$$\phi(x, x)$ is even for each $x$$x$ (the second condition automatically holds for n odd).

Definition 5.1. For each even $n$$n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$$W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$$g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$$f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$$W\Lambda([f]) = W(g)$, where $W$$W$ is Whitney invariant, [Skopenkov2016e, $\S$$\S$5].

Lemma 5.2. The invariant $W\Lambda$$W\Lambda$ is well-defined for $n\ge4$$n\ge4$.

Proof. Note that Unknotting Spheres Theorem implies that $\partial N_0$$\partial N_0$ unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus $f|_{\partial N_0}$$f|_{\partial N_0}$ can be extended to embedding of an $n$$n$-ball $B^n$$B^n$ into $\mathbb R^{2n}$$\mathbb R^{2n}$. Unknotting Spheres Theorem implies that $n$$n$-sphere unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus all extensions of $f$$f$ are isotopic in PL category. Note also that if $f$$f$ and $g$$g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$$W$ is invariant for PL embeddings.

$\square$$\square$

Definition 5.3 of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion-free. Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$$\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$$W\Lambda(f_z)=z$. For each $f$$f$ such that $W\Lambda(f)=z$$W\Lambda(f)=z$ define

$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$

where $x,y\in H_{n-1}(N_0)$$x,y\in H_{n-1}(N_0)$.

Note also that $G$$G$ depends on choice of collection $\{f_z\}$$\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$$\{f_z\}$.

Theorem 5.4. 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. The map

$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$

is one-to-one.

Lemma 5.5. For each even $n\in H_{n-1}(N)$$n\in H_{n-1}(N)$ and each $x$$x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$$W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$.

An equivalemt statement of Theorem 5.4:

Theorem 5.6. 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)= \bar w_2(N_0)\cap\rho_2(x\cap y)$$\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class.

This is the main Theorem of [Tonkonog2010]

## 6 A generalization to highly-connected manifolds

Theorem 6.1. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n\geq2k+2$$n\geq2k+2$. Then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [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 $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

For the Diff case see [Haefliger1961, $\S$$\S$ 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result). For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].

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 $\mathbb R^m$$\mathbb R^m$ are isotopic.

See Theorem 2.4 of the 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 $\mathbb R^m$$\mathbb R^m$ are isotopic.

For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. Denote by $\mathrm{Emb}^{m}N_0$$\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$$N_0$ into $\mathbb R^{m}$$\mathbb R^{m}$ up to isotopy.

Theorem 6.5. Assume $N$$N$ is a closed orientable $k$$k$-connected manifold embeddable into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Then for each $k\ge1$$k\ge1$ there exists a bijection

$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$

where $\mathbb Z_{(s)}$$\mathbb Z_{(s)}$ denote $\mathbb Z$$\mathbb Z$ for $s$$s$ even and $\mathbb Z_2$$\mathbb Z_2$ for $s$$s$ odd.

For definition of $W_0'$$W_0'$ and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2($W_0'$$W_0'$)]. Latter Theorem is essetialy known result. Compare to the Theorem 5.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

, $\S]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated. If the category is omitted, then we assume the smooth (DIFF) category. We state the simplest results. These results can be deduced from [[Some_calculations_involving_configuration_spaces_of_distinct_points|the Haefliger-Weber deleted product criterion]] \cite[$\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$]{Skopenkov2002} for the PL case. Usually there exist easier direct proofs than deduction from this criterion. We do not claim the references we give are references to original proofs. == Embedding and unknotting theorems == ; \label{sec::general_theorems} {{beginthm|Theorem}} Assume that$N$is a closed compact$n$-manifold. Then$N$embeds into$\mathbb 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$embeds into$\mathbb R^{2n-1}$. {{endthm}} This theorem is a corollary of [[Wikipedia:Whitney_immersion_theorem|strong Whitney immersion theorem]]. For the Diff case of this result see \cite[Theorem 4.6]{Hirsch1961a}. For the PL case see references for Theorem \ref{thm::k_connect_boundary} below and \cite[Theorem 5.2]{Horvatic1971}. {{beginthm|Theorem}}\label{thm::closed_unknotting} Assume that$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$into$\mathbb 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$2]{Skopenkov2016c}. Note that inequality in part (a) is sharp, which is shown by the construction of [[High_codimension_links#Examples|the Hopf link]]. {{beginthm|Theorem}} \label{thm::special_Haef_Zem} Assume that$N$is a compact connected$n$-manifold with non-empty boundary and either (a)$m \ge 2n$or (b)$Nis \S4 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$6 which is independent from $\S$$\S$3, $\S$$\S$4 and $\S$$\S$5 we state generalisations of theorems from $\S$$\S$2 to highly-connected manifolds. For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. In those pages mostly results for closed manifolds are stated. If the category is omitted, then we assume the smooth (DIFF) category. We state the simplest results. These results 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$$\alpha\partial$] for the PL case. Usually there exist easier direct proofs than deduction from this criterion. We do not claim the references we give are references to original proofs. ## 2 Embedding and unknotting theorems Theorem 2.1. Assume that $N$$N$ is a closed compact $n$$n$-manifold. Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$. This is well-known strong Whitney embedding theorem. Theorem 2.2. Assume that $N$$N$ is a compact $n$$n$-manifold with nonempty boundary. Then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$. This theorem is a corollary of strong Whitney immersion theorem. For the Diff case of this result see [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2]. Theorem 2.3. Assume that $N$$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$$N$ into $\mathbb R^m$$\mathbb 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$$\S$ 2]. Note that inequality in part (a) is sharp, which is shown by the construction of the Hopf link. Theorem 2.4. Assume that $N$$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 $\mathbb R^m$$\mathbb R^m$ are isotopic. Part (a) of this theorem in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, Corollary 5]. Case $n=1$$n=1$ is clear. Both parts of this theorem are special cases of the Theorem 6.4. Inequality in part (a) is sharp, see Proposition 3.1. These basic results can be generalized to the highly-connected manifolds (see $\S$$\S$6). ## 3 Example on non-isotopic embeddings Denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient of two disjoint cycles with integer coefficient. The following example is folklore. Proposition 3.1. Let $N=S^k\times [0, 1]$$N=S^k\times [0, 1]$ be the cylinder over $S^k$$S^k$. (a) Then there exist non-isotopic embeddings of $N$$N$ into $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$. (b) Then for each $a\in\mathbb Z$$a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$$f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$$\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$. Proof. Let $h\colon S^k\to S^k$$h\colon S^k\to S^k$ be a map of degree $a$$a$. To prove part (a) it is sufficient to take the identity map of $S^k$$S^k$ as a map of degree one and the constant map as a map of degree zero. Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$$g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$$g(x, t) = (x, h(x)t)$, where $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$. Let $f=\mathrm ig$$f=\mathrm ig$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is the standard embedding.Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$$\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$ $\square$$\square$ ## 4 Seifert linking form Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we 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. If the (co)homology coefficients are omitted, then we assume them to be $\mathbb Z$$\mathbb Z$. Denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles. Example 4.1. For $N=S^k\times S^1$$N=S^k\times S^1$ and each $k\ge2$$k\ge2$ there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ given by $\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$. The surjectivity of $l$$l$ is given by Proposition 3.1(b). The following folklore result holds. Lemma 4.2. Assume $N$$N$ is a closed orientable connected $n$$n$-manifold, $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free. Then for 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)\cong H_1(N_0, \partial N_0)\cong H_1(N)$$\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$$f(N_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$. Since $H_1(N)$$H_1(N)$ is torsion free, it follows that $\bar e=0$$\bar e=0$. Since $N_0$$N_0$ has non-empty boundary, we have that $N_0$$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)$$f(N_0)$. Since $\bar e=0$$\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$. $\square$$\square$ Denote by $x, y$$x, y$ two disjoint $(n-1)$$(n-1)$-cycles in $N_0$$N_0$ with integer coefficients. Definition 4.3. For even $n$$n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ denote $\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$ where $s$$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 results of the shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$. Lemma 4.4 ($L$$L$ is well-defined). For even $n$$n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ the integer $L(f)(x, y)$$L(f)(x, y)$: • is well-defined, i.e. does not change when $s$$s$ is replaced by $s'$$s'$, • does not change when $x$$x$ or $y$$y$ are changed to homologous cycles and, • does not change when $f$$f$ is changed to an isotopic embedding. The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple. We will need the following supporting lemma. Lemma 4.5. Let $f:N_0\to \mathbb R^{2n-1}$$f:N_0\to \mathbb R^{2n-1}$ be an embedding. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then $\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$ where $s(y)$$s(y)$ is the result of the shift of $f(y)$$f(y)$ by $s$$s$, and $d(s,s')\in H_2(N_0)$$d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$$s,s'$ being homotopic in the class of the nowhere vanishing vector fields. This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$$n=3$, but the proof is valid in all dimensions. Proof of Lemma 4.4. The first bullet point follows because: \displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned} Here the second equality follows from Lemma 4.5. For each two homologous $(n-1)$$(n-1)$-cycles $x, x'$$x, x'$ in $N_0$$N_0$, the image of the homology between $x$$x$ and $x'$$x'$ is a $n$$n$-chain $X$$X$ of $f(N_0)$$f(N_0)$ such that $\partial X = f(x) - f(x')$$\partial X = f(x) - f(x')$. Since $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$, this implies that the supports of $s(y)$$s(y)$ and $X$$X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$$\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$. Since isotopy of $f$$f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$$f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy. $\square$$\square$ Lemma 4.4 implies that $L(f)$$L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$$H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter. Denote by $\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$$\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$$2$. Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$$\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 to $f(N_0)$$f(N_0)$ are linearly dependent. Lemma 4.6. Let $f:N_0\to \mathbb R^{2n-1}$$f:N_0\to \mathbb R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds: $\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$ This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2]. Proof of Lemma 4.6. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get \displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \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 congruence is clear. The second equality holds because if we shift the link $s(X)\sqcup f(Y)$$s(X)\sqcup f(Y)$ by $-s$$-s$, we get the link $f(X)\sqcup -s(Y)$$f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift. The third equality follows from Lemma 4.5. Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$$s'$ a general perturbation of $s$$s$. We get: $\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$ The first equality holds because $s'$$s'$ and $s$$s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$$s'$ and $-s$$-s$ degenerates only at those points $x$$x$ where $s'(x)=s(x)$$s'(x)=s(x)$. These points $x$$x$ are exactly points where $s'(x)$$s'(x)$ and $s(x)$$s(x)$ are linearly dependent. All those point $x$$x$ form a $2$$2$-cycle modulo two in $N_0$$N_0$. The homotopy class of this $2$$2$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$$\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class. $\square$$\square$ ## 5 Classification theorems Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary. Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we 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. Denote $\mathrm{Emb}^mN$$\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$$f\colon N\to\mathbb R^m$ up to isotopy. For a free Abelian group $A$$A$, let $B_n^∗A$$B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$$\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$$\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$$\phi(x, x)$ is even for each $x$$x$ (the second condition automatically holds for n odd). Definition 5.1. For each even $n$$n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$$W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$$g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$$f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$$W\Lambda([f]) = W(g)$, where $W$$W$ is Whitney invariant, [Skopenkov2016e, $\S$$\S$5]. Lemma 5.2. The invariant $W\Lambda$$W\Lambda$ is well-defined for $n\ge4$$n\ge4$. Proof. Note that Unknotting Spheres Theorem implies that $\partial N_0$$\partial N_0$ unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus $f|_{\partial N_0}$$f|_{\partial N_0}$ can be extended to embedding of an $n$$n$-ball $B^n$$B^n$ into $\mathbb R^{2n}$$\mathbb R^{2n}$. Unknotting Spheres Theorem implies that $n$$n$-sphere unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus all extensions of $f$$f$ are isotopic in PL category. Note also that if $f$$f$ and $g$$g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$$W$ is invariant for PL embeddings. $\square$$\square$ Definition 5.3 of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion-free. Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$$\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$$W\Lambda(f_z)=z$. For each $f$$f$ such that $W\Lambda(f)=z$$W\Lambda(f)=z$ define $\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$ where $x,y\in H_{n-1}(N_0)$$x,y\in H_{n-1}(N_0)$. Note also that $G$$G$ depends on choice of collection $\{f_z\}$$\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$$\{f_z\}$. Theorem 5.4. 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$$n$ even. The map $\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$ is one-to-one. Lemma 5.5. For each even $n\in H_{n-1}(N)$$n\in H_{n-1}(N)$ and each $x$$x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$$W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$. An equivalemt statement of Theorem 5.4: Theorem 5.6. 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)= \bar w_2(N_0)\cap\rho_2(x\cap y)$$\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class. This is the main Theorem of [Tonkonog2010] ## 6 A generalization to highly-connected manifolds Theorem 6.1. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n\geq2k+2$$n\geq2k+2$. Then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [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 $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. For the Diff case see [Haefliger1961, $\S$$\S$ 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result). For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2]. 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 $\mathbb R^m$$\mathbb R^m$ are isotopic. See Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\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 $\mathbb R^m$$\mathbb R^m$ are isotopic. For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem. By $N_0$$N_0$ we 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. Denote by $\mathrm{Emb}^{m}N_0$$\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$$N_0$ into $\mathbb R^{m}$$\mathbb R^{m}$ up to isotopy. Theorem 6.5. Assume $N$$N$ is a closed orientable $k$$k$-connected manifold embeddable into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Then for each $k\ge1$$k\ge1$ there exists a bijection $\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$ where $\mathbb Z_{(s)}$$\mathbb Z_{(s)}$ denote $\mathbb Z$$\mathbb Z$ for $s$$s$ even and $\mathbb Z_2$$\mathbb Z_2$ for $s$$s$ odd. For definition of $W_0'$$W_0'$ and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2($W_0'$$W_0'$)]. Latter Theorem is essetialy known result. Compare to the Theorem 5.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case. ## 7 References-connected, $m \ge 2n - 1\ge3$. Then any two embeddings of $N$ into $\mathbb R^m$ are isotopic. {{endthm}} Part (a) of this theorem in case $n>2$ can be found in \cite[$\S$ 4, Corollary 5]{Edwards1968}. Case $n=1$ is clear. Both parts of this theorem are special cases of the Theorem \ref{thm::highly_connected_boundary_unknotting}. Inequality in part (a) is sharp, see Proposition \ref{exm::linked_boundary}. These basic results can be generalized to the highly-connected manifolds (see $\S$\ref{sec::generalisations}).
== Example on non-isotopic embeddings == ; \label{sec::example} Denote by $\mathrm{lk}$ the linking coefficient of two disjoint cycles with integer coefficient. The following example is folklore. {{beginthm|Proposition}} \label{exm::linked_boundary} Let $N=S^k\times [0, 1]$ be the cylinder over $S^k$. (a) Then there exist non-isotopic embeddings of $N$ into $\mathbb R^{2k+1}$. (b) Then for each $a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$. {{endthm}} {{beginproof}} Let $h\colon S^k\to S^k$ be a map of degree $a$. To prove part (a) it is sufficient to take the identity map of $S^k$ as a map of degree one and the constant map as a map of degree zero. Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$, where \S4 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$6 which is independent from $\S$$\S$3, $\S$$\S$4 and $\S$$\S$5 we state generalisations of theorems from $\S$$\S$2 to highly-connected manifolds.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category.

We state the simplest results. These results 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.

We do not claim the references we give are references to original proofs.

## 2 Embedding and unknotting theorems

Theorem 2.1. Assume that $N$$N$ is a closed compact $n$$n$-manifold. Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb 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$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

This theorem is a corollary of strong Whitney immersion theorem. For the Diff case of this result see [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$$N$ into $\mathbb R^m$$\mathbb 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, which is shown by the construction of the Hopf link.

Theorem 2.4. 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$$N$ into $\mathbb R^m$$\mathbb 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. Both parts of this theorem are special cases of the Theorem 6.4.

Inequality in part (a) is sharp, see Proposition 3.1.

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

## 3 Example on non-isotopic embeddings

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

The following example is folklore.

Proposition 3.1. Let $N=S^k\times [0, 1]$$N=S^k\times [0, 1]$ be the cylinder over $S^k$$S^k$. (a) Then there exist non-isotopic embeddings of $N$$N$ into $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$.

(b) Then for each $a\in\mathbb Z$$a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$$f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$$\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$.

Proof. Let $h\colon S^k\to S^k$$h\colon S^k\to S^k$ be a map of degree $a$$a$. To prove part (a) it is sufficient to take the identity map of $S^k$$S^k$ as a map of degree one and the constant map as a map of degree zero. Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$$g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$$g(x, t) = (x, h(x)t)$, where $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Let $f=\mathrm ig$$f=\mathrm ig$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is the standard embedding.Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$$\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$

$\square$$\square$

## 4 Seifert linking form

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. If the (co)homology coefficients are omitted, then we assume them to be $\mathbb Z$$\mathbb Z$.

Denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

Example 4.1. For $N=S^k\times S^1$$N=S^k\times S^1$ and each $k\ge2$$k\ge2$ there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ given by $\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given by Proposition 3.1(b).

The following folklore result holds.

Lemma 4.2. Assume $N$$N$ is a closed orientable connected $n$$n$-manifold, $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free. Then for 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)\cong H_1(N_0, \partial N_0)\cong H_1(N)$$\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$$f(N_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$. Since $H_1(N)$$H_1(N)$ is torsion free, it follows that $\bar e=0$$\bar e=0$.

Since $N_0$$N_0$ has non-empty boundary, we have that $N_0$$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)$$f(N_0)$. Since $\bar e=0$$\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

Denote by $x, y$$x, y$ two disjoint $(n-1)$$(n-1)$-cycles in $N_0$$N_0$ with integer coefficients.

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

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

where $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the results of the shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Lemma 4.4 ($L$$L$ is well-defined). For even $n$$n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ the integer $L(f)(x, y)$$L(f)(x, y)$:

• is well-defined, i.e. does not change when $s$$s$ is replaced by $s'$$s'$,
• does not change when $x$$x$ or $y$$y$ are changed to homologous cycles and,
• does not change when $f$$f$ is changed to an isotopic embedding.

The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.

We will need the following supporting lemma.

Lemma 4.5. Let $f:N_0\to \mathbb R^{2n-1}$$f:N_0\to \mathbb R^{2n-1}$ be an embedding. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

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

where $s(y)$$s(y)$ is the result of the shift of $f(y)$$f(y)$ by $s$$s$, and $d(s,s')\in H_2(N_0)$$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 nowhere vanishing vector fields.

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

Proof of Lemma 4.4. The first bullet point follows because:
\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}

Here the second equality follows from Lemma 4.5.

For each two homologous $(n-1)$$(n-1)$-cycles $x, x'$$x, x'$ in $N_0$$N_0$, the image of the homology between $x$$x$ and $x'$$x'$ is a $n$$n$-chain $X$$X$ of $f(N_0)$$f(N_0)$ such that $\partial X = f(x) - f(x')$$\partial X = f(x) - f(x')$. Since $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$, this implies that the supports of $s(y)$$s(y)$ and $X$$X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$$\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.

Since isotopy of $f$$f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$$f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy.

$\square$$\square$

Lemma 4.4 implies that $L(f)$$L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$$H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter.

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

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

Lemma 4.6. Let $f:N_0\to \mathbb R^{2n-1}$$f:N_0\to \mathbb R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 4.6. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get
\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \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 congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$$s(X)\sqcup f(Y)$ by $-s$$-s$, we get the link $f(X)\sqcup -s(Y)$$f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 4.5.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$$s'$ a general perturbation of $s$$s$. We get:

$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$

The first equality holds because $s'$$s'$ and $s$$s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$$s'$ and $-s$$-s$ degenerates only at those points $x$$x$ where $s'(x)=s(x)$$s'(x)=s(x)$. These points $x$$x$ are exactly points where $s'(x)$$s'(x)$ and $s(x)$$s(x)$ are linearly dependent. All those point $x$$x$ form a $2$$2$-cycle modulo two in $N_0$$N_0$. The homotopy class of this $2$$2$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$$\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class.

$\square$$\square$

## 5 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. Denote $\mathrm{Emb}^mN$$\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$$f\colon N\to\mathbb R^m$ up to isotopy. For a free Abelian group $A$$A$, let $B_n^∗A$$B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$$\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$$\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$$\phi(x, x)$ is even for each $x$$x$ (the second condition automatically holds for n odd).

Definition 5.1. For each even $n$$n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$$W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$$g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$$f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$$W\Lambda([f]) = W(g)$, where $W$$W$ is Whitney invariant, [Skopenkov2016e, $\S$$\S$5].

Lemma 5.2. The invariant $W\Lambda$$W\Lambda$ is well-defined for $n\ge4$$n\ge4$.

Proof. Note that Unknotting Spheres Theorem implies that $\partial N_0$$\partial N_0$ unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus $f|_{\partial N_0}$$f|_{\partial N_0}$ can be extended to embedding of an $n$$n$-ball $B^n$$B^n$ into $\mathbb R^{2n}$$\mathbb R^{2n}$. Unknotting Spheres Theorem implies that $n$$n$-sphere unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus all extensions of $f$$f$ are isotopic in PL category. Note also that if $f$$f$ and $g$$g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$$W$ is invariant for PL embeddings.

$\square$$\square$

Definition 5.3 of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion-free. Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$$\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$$W\Lambda(f_z)=z$. For each $f$$f$ such that $W\Lambda(f)=z$$W\Lambda(f)=z$ define

$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$

where $x,y\in H_{n-1}(N_0)$$x,y\in H_{n-1}(N_0)$.

Note also that $G$$G$ depends on choice of collection $\{f_z\}$$\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$$\{f_z\}$.

Theorem 5.4. 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. The map

$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$

is one-to-one.

Lemma 5.5. For each even $n\in H_{n-1}(N)$$n\in H_{n-1}(N)$ and each $x$$x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$$W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$.

An equivalemt statement of Theorem 5.4:

Theorem 5.6. 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)= \bar w_2(N_0)\cap\rho_2(x\cap y)$$\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class.

This is the main Theorem of [Tonkonog2010]

## 6 A generalization to highly-connected manifolds

Theorem 6.1. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n\geq2k+2$$n\geq2k+2$. Then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [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 $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

For the Diff case see [Haefliger1961, $\S$$\S$ 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result). For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].

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 $\mathbb R^m$$\mathbb R^m$ are isotopic.

See Theorem 2.4 of the 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 $\mathbb R^m$$\mathbb R^m$ are isotopic.

For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. Denote by $\mathrm{Emb}^{m}N_0$$\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$$N_0$ into $\mathbb R^{m}$$\mathbb R^{m}$ up to isotopy.

Theorem 6.5. Assume $N$$N$ is a closed orientable $k$$k$-connected manifold embeddable into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Then for each $k\ge1$$k\ge1$ there exists a bijection

$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$

where $\mathbb Z_{(s)}$$\mathbb Z_{(s)}$ denote $\mathbb Z$$\mathbb Z$ for $s$$s$ even and $\mathbb Z_2$$\mathbb Z_2$ for $s$$s$ odd.

For definition of $W_0'$$W_0'$ and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2($W_0'$$W_0'$)]. Latter Theorem is essetialy known result. Compare to the Theorem 5.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

## 7 References

_k:=(1,0,\ldots,0)\in S^k$. Let$f=\mathrm ig$, where$\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$is [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation_and_conventions|the standard embedding]].Thus$\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a${{endproof}} == Seifert linking form == ; \label{sec::linking_form} Let$N$be a closed orientable connected$n$-manifold. By$N_0$we denote the complement in$N$to an open$n$-ball. Thus$\partial N_0$is the$(n-1)$-sphere. If the (co)homology coefficients are omitted, then we assume them to be$\mathbb Z$. Denote by$\mathrm{lk}$the linking coefficient \cite[$\S$77]{Seifert&Threlfall1980} of two disjoint cycles. {{beginthm|Example}} For$N=S^k\times S^1$and each$k\ge2$there exists a bijection$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$given by$\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$. {{endthm}} The surjectivity of$l$is given by Proposition \ref{exm::linked_boundary}(b). The following folklore result holds. {{beginthm|Lemma}} Assume$N$is a closed orientable connected$n$-manifold,$n$is even and$H_1(N)$is torsion free. Then for each embedding$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)\cong H_1(N_0, \partial N_0)\cong H_1(N)$to existence of a nowhere vanishing normal vector field to$f(N_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$. Since$H_1(N)$is torsion free, it follows that$\bar e=0$. Since$N_0$has non-empty boundary, we have that$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)$. Since$\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to$f(N_0)$. {{endproof}} Denote by$x, y$two disjoint$(n-1)$-cycles in$N_0$with integer coefficients. {{beginthm|Definition}} For even$n$and every embedding$f\colon N_0\to\mathbb R^{2n-1}$denote $$L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$ where$s$is a nowhere vanishing normal field to$f(N_0)$and$s(x), s(y)$are the results of the shift of$f(x), f(y)$by$s$. {{endthm}} {{beginthm|Lemma|($L$is well-defined)}}\label{lmm:L_well_def} For even$n$and every embedding$f\colon N_0\to\mathbb R^{2n-1}$the integer$L(f)(x, y)$: * is well-defined, i.e. does not change when$s$is replaced by$s'$, * does not change when$x$or$y$are changed to homologous cycles and, * does not change when$f$is changed to an isotopic embedding. {{endthm}} The first bullet was stated and proved in unpublished update of \cite{Tonkonog2010}, other two bullets are simple. We will need the following supporting lemma. {{beginthm|Lemma}}\label{lmm::saeki} Let$f:N_0\to \mathbb R^{2n-1}$be an embedding. Let$s,s'$be two nowhere vanishing normal vector fields to$f(N_0)$. Then $$\mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$ where$s(y)$is the result of the shift of$f(y)$by$s$, and$d(s,s')\in H_2(N_0)$is (Poincare dual to) the first obstruction to$s,s'$being homotopic in the class of the nowhere vanishing vector fields. {{endthm}} This Lemma is proved in \cite[Lemma 2.2]{Saeki1999} for$n=3, but the proof is valid in all dimensions. '''Proof of Lemma \ref{lmm:L_well_def}.''' The first bullet point follows because: \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned} Here the second equality follows from Lemma \ref{lmm::saeki}. For each two homologous(n-1)$-cycles$x, x'$in$N_0$, the image of the homology between$x$and$x'$is a$n$-chain$X$of$f(N_0)$such that$\partial X = f(x) - f(x')$. Since$s$is a nowhere vanishing normal field to$f(N_0)$, this implies that the supports of$s(y)$and$X$are disjoint. Hence$\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$. Since isotopy of$f$is a map from$\mathbb R^{2n-1}\times [0, 1]$to$\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link$f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy. {{endproof}} Lemma \ref{lmm:L_well_def} implies that$L(f)$generates a bilinear form$H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$denoted by the same letter. Denote by$\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$the reduction modulo$. 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 to $f(N_0)$ are linearly dependent. {{beginthm|Lemma}} \label{lmm::L_equality} Let $f:N_0\to \mathbb R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0)$ the following equality holds: $$\rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$ {{endthm}} This Lemma was stated in a unpublished update of \cite{Tonkonog2010}, the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in \cite[Lemma 2.2]{Crowley&Skopenkov2016}.
'''Proof of Lemma \ref{lmm::L_equality}.''' Let $-s$ be the normal field to $f(N_0)$ opposite to $s$. We get \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \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 congruence is clear. The second equality holds because if we shift the link $s(X)\sqcup f(Y)$ by $-s$, we get the link $f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift. The third equality follows from Lemma \ref{lmm::saeki}. Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$ a general perturbation of $s$. We get: $$\rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$$ The first equality holds because $s'$ and $s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$ and $-s$ degenerates only at those points $x$ where $s'(x)=s(x)$. These points $x$ are exactly points where $s'(x)$ and $s(x)$ are linearly dependent. All those point $x$ form a $-cycle modulo two in$N_0$. The homotopy class of this$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class. {{endproof}} == Classification theorems == ; \label{sec::classification} Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary. Let $N$ be a closed orientable connected $n$-manifold. By $N_0$ we denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. Denote $\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$ up to isotopy. For a free Abelian group $A$, let $B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$ is even for each $x$ (the second condition automatically holds for n odd). {{beginthm|Definition}} For each even $n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$, where $W$ is [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|Whitney invariant]], \cite[$\S]{Skopenkov2016e}. {{endthm}} {{beginthm|Lemma}} The invariant$W\Lambda$is well-defined for$n\ge4$. {{endthm}} {{beginproof}} Note that [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem]] implies that$\partial N_0$unknots in$\mathbb R^{2n}$. Thus$f|_{\partial N_0}$can be extended to embedding of an$n$-ball$B^n$into$\mathbb R^{2n}$. [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem]] implies that$n$-sphere unknots in$\mathbb R^{2n}$. Thus all extensions of$f$are isotopic in PL category. Note also that if$f$and$g$are isotopic then their extensions are isotopic as well. And Whitney invariant$W$is invariant for PL embeddings. {{endproof}} {{beginthm|Definition|of$G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$if$n$is even and$H_1(N)$is torsion-free}}\label{DefG} Take a collection$\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$such that$W\Lambda(f_z)=z$. For each$f$such that$W\Lambda(f)=z$define $$G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$$ where$x,y\in H_{n-1}(N_0)$. {{endthm}} Note also that$G$depends on choice of collection$\{f_z\}$. The following Theorems hold for any choice of$\{f_z\}$. {{beginthm|Theorem}}\label{Tlink} Let$N$be a closed connected orientable$n$-manifold with$H_1(N)$torsion-free,$n\ge 4$,$n$even. The map $$G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$$ is one-to-one. {{endthm}} {{beginthm|Lemma}} For each even$n\in H_{n-1}(N)$and each$x$the following equality holds:$W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$. {{endthm}} An equivalemt statement of Theorem \ref{Tlink}: {{beginthm|Theorem}}\label{thm::punctured_class} 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$\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here$\bar w_2(N_0)$is the normal Stiefel-Whitney class. {{endthm}} This is the main Theorem of \cite{Tonkonog2010} == A generalization to highly-connected manifolds == ; \label{sec::generalisations} {{beginthm|Theorem}} Assume that$N$is a closed compact$k$-connected$n$-manifold and$n\geq2k+2$. Then$N$embeds into$\mathbb R^{2n-k}$. {{endthm}} The Diff case of this result is in \cite[Existence Theorem (a)]{Haefliger1961}, the PL case of this result is in \cite[Theorem 1.1]{Penrose&Whitehead&Zeeman1961}, \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$\mathbb R^{2n-k-1}$. {{endthm}} For the Diff case see \cite[$\S$1.7, remark 2]{Haefliger1961} (there Haefliger proposes to use the deleted product criterion to obtain this result). For the PL case see \cite[Theorem 1.2]{Penrose&Whitehead&Zeeman1961}. {{beginthm|Theorem}}\label{thm::k_connect_closed_unknot} Assume that$N$is a closed$k$-connected$n$-manifold. Then for each$n\ge2k + 2$,$m \ge 2n - k + 1$any two embeddings of$N$into$\mathbb R^m$are isotopic. {{endthm}} See [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.4]] of the survey \cite[$\S$2]{Skopenkov2016c}, or \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}. {{beginthm|Theorem}}\label{thm::highly_connected_boundary_unknotting} Assume that$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$into$\mathbb R^m$are isotopic. {{endthm}} For the PL case of this result see \cite[Theorem 10.3]{Hudson1969}, which is proved using [[Isotopy#Concordance|concordance implies isotopy theorem]]. By$N_0$we denote the complement in$N$to an open$n$-ball. Thus$\partial N_0$is the$(n-1)$-sphere. Denote by$\mathrm{Emb}^{m}N_0$the set embeddings of$N_0$into$\mathbb R^{m}$up to isotopy. {{beginthm|Theorem}} Assume$N$is a closed orientable$k$-connected manifold embeddable into$\mathbb R^{2n-k-1}$. Then for each$k\ge1$there exists a bijection $$W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$$ where$\mathbb Z_{(s)}$denote$\mathbb Z$for$s$even and$\mathbb Z_2$for$s$odd. {{endthm}} For definition of$W_0'$and the proof of the latter Theorem see \cite[Lemma 2.2($W_0'$)]{Skopenkov2010}. Latter Theorem is essetialy known result. Compare to the Theorem \ref{thm::punctured_class}, which describes$\mathrm{Emb}^{2n-1}(N_0)\$ and differs from the general case. == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S4 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$6 which is independent from $\S$$\S$3, $\S$$\S$4 and $\S$$\S$5 we state generalisations of theorems from $\S$$\S$2 to highly-connected manifolds.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category.

We state the simplest results. These results 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.

We do not claim the references we give are references to original proofs.

## 2 Embedding and unknotting theorems

Theorem 2.1. Assume that $N$$N$ is a closed compact $n$$n$-manifold. Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb 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$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

This theorem is a corollary of strong Whitney immersion theorem. For the Diff case of this result see [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$$N$ into $\mathbb R^m$$\mathbb 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, which is shown by the construction of the Hopf link.

Theorem 2.4. 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$$N$ into $\mathbb R^m$$\mathbb 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. Both parts of this theorem are special cases of the Theorem 6.4.

Inequality in part (a) is sharp, see Proposition 3.1.

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

## 3 Example on non-isotopic embeddings

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

The following example is folklore.

Proposition 3.1. Let $N=S^k\times [0, 1]$$N=S^k\times [0, 1]$ be the cylinder over $S^k$$S^k$. (a) Then there exist non-isotopic embeddings of $N$$N$ into $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$.

(b) Then for each $a\in\mathbb Z$$a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$$f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$$\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$.

Proof. Let $h\colon S^k\to S^k$$h\colon S^k\to S^k$ be a map of degree $a$$a$. To prove part (a) it is sufficient to take the identity map of $S^k$$S^k$ as a map of degree one and the constant map as a map of degree zero. Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$$g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$$g(x, t) = (x, h(x)t)$, where $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Let $f=\mathrm ig$$f=\mathrm ig$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is the standard embedding.Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$$\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$

$\square$$\square$

## 4 Seifert linking form

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. If the (co)homology coefficients are omitted, then we assume them to be $\mathbb Z$$\mathbb Z$.

Denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

Example 4.1. For $N=S^k\times S^1$$N=S^k\times S^1$ and each $k\ge2$$k\ge2$ there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ given by $\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given by Proposition 3.1(b).

The following folklore result holds.

Lemma 4.2. Assume $N$$N$ is a closed orientable connected $n$$n$-manifold, $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free. Then for 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)\cong H_1(N_0, \partial N_0)\cong H_1(N)$$\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$$f(N_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$. Since $H_1(N)$$H_1(N)$ is torsion free, it follows that $\bar e=0$$\bar e=0$.

Since $N_0$$N_0$ has non-empty boundary, we have that $N_0$$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)$$f(N_0)$. Since $\bar e=0$$\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

Denote by $x, y$$x, y$ two disjoint $(n-1)$$(n-1)$-cycles in $N_0$$N_0$ with integer coefficients.

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

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

where $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the results of the shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Lemma 4.4 ($L$$L$ is well-defined). For even $n$$n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ the integer $L(f)(x, y)$$L(f)(x, y)$:

• is well-defined, i.e. does not change when $s$$s$ is replaced by $s'$$s'$,
• does not change when $x$$x$ or $y$$y$ are changed to homologous cycles and,
• does not change when $f$$f$ is changed to an isotopic embedding.

The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.

We will need the following supporting lemma.

Lemma 4.5. Let $f:N_0\to \mathbb R^{2n-1}$$f:N_0\to \mathbb R^{2n-1}$ be an embedding. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

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

where $s(y)$$s(y)$ is the result of the shift of $f(y)$$f(y)$ by $s$$s$, and $d(s,s')\in H_2(N_0)$$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 nowhere vanishing vector fields.

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

Proof of Lemma 4.4. The first bullet point follows because:
\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}

Here the second equality follows from Lemma 4.5.

For each two homologous $(n-1)$$(n-1)$-cycles $x, x'$$x, x'$ in $N_0$$N_0$, the image of the homology between $x$$x$ and $x'$$x'$ is a $n$$n$-chain $X$$X$ of $f(N_0)$$f(N_0)$ such that $\partial X = f(x) - f(x')$$\partial X = f(x) - f(x')$. Since $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$, this implies that the supports of $s(y)$$s(y)$ and $X$$X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$$\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.

Since isotopy of $f$$f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$$f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy.

$\square$$\square$

Lemma 4.4 implies that $L(f)$$L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$$H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter.

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

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

Lemma 4.6. Let $f:N_0\to \mathbb R^{2n-1}$$f:N_0\to \mathbb R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 4.6. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get
\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \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 congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$$s(X)\sqcup f(Y)$ by $-s$$-s$, we get the link $f(X)\sqcup -s(Y)$$f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 4.5.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$$s'$ a general perturbation of $s$$s$. We get:

$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$

The first equality holds because $s'$$s'$ and $s$$s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$$s'$ and $-s$$-s$ degenerates only at those points $x$$x$ where $s'(x)=s(x)$$s'(x)=s(x)$. These points $x$$x$ are exactly points where $s'(x)$$s'(x)$ and $s(x)$$s(x)$ are linearly dependent. All those point $x$$x$ form a $2$$2$-cycle modulo two in $N_0$$N_0$. The homotopy class of this $2$$2$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$$\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class.

$\square$$\square$

## 5 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. Denote $\mathrm{Emb}^mN$$\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$$f\colon N\to\mathbb R^m$ up to isotopy. For a free Abelian group $A$$A$, let $B_n^∗A$$B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$$\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$$\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$$\phi(x, x)$ is even for each $x$$x$ (the second condition automatically holds for n odd).

Definition 5.1. For each even $n$$n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$$W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$$g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$$f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$$W\Lambda([f]) = W(g)$, where $W$$W$ is Whitney invariant, [Skopenkov2016e, $\S$$\S$5].

Lemma 5.2. The invariant $W\Lambda$$W\Lambda$ is well-defined for $n\ge4$$n\ge4$.

Proof. Note that Unknotting Spheres Theorem implies that $\partial N_0$$\partial N_0$ unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus $f|_{\partial N_0}$$f|_{\partial N_0}$ can be extended to embedding of an $n$$n$-ball $B^n$$B^n$ into $\mathbb R^{2n}$$\mathbb R^{2n}$. Unknotting Spheres Theorem implies that $n$$n$-sphere unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus all extensions of $f$$f$ are isotopic in PL category. Note also that if $f$$f$ and $g$$g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$$W$ is invariant for PL embeddings.

$\square$$\square$

Definition 5.3 of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion-free. Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$$\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$$W\Lambda(f_z)=z$. For each $f$$f$ such that $W\Lambda(f)=z$$W\Lambda(f)=z$ define

$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$

where $x,y\in H_{n-1}(N_0)$$x,y\in H_{n-1}(N_0)$.

Note also that $G$$G$ depends on choice of collection $\{f_z\}$$\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$$\{f_z\}$.

Theorem 5.4. 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. The map

$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$

is one-to-one.

Lemma 5.5. For each even $n\in H_{n-1}(N)$$n\in H_{n-1}(N)$ and each $x$$x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$$W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$.

An equivalemt statement of Theorem 5.4:

Theorem 5.6. 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)= \bar w_2(N_0)\cap\rho_2(x\cap y)$$\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class.

This is the main Theorem of [Tonkonog2010]

## 6 A generalization to highly-connected manifolds

Theorem 6.1. Assume that $N$$N$ is a closed compact $k$$k$-connected $n$$n$-manifold and $n\geq2k+2$$n\geq2k+2$. Then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [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 $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

For the Diff case see [Haefliger1961, $\S$$\S$ 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result). For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].

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 $\mathbb R^m$$\mathbb R^m$ are isotopic.

See Theorem 2.4 of the 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 $\mathbb R^m$$\mathbb R^m$ are isotopic.

For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. Denote by $\mathrm{Emb}^{m}N_0$$\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$$N_0$ into $\mathbb R^{m}$$\mathbb R^{m}$ up to isotopy.

Theorem 6.5. Assume $N$$N$ is a closed orientable $k$$k$-connected manifold embeddable into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Then for each $k\ge1$$k\ge1$ there exists a bijection

$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$

where $\mathbb Z_{(s)}$$\mathbb Z_{(s)}$ denote $\mathbb Z$$\mathbb Z$ for $s$$s$ even and $\mathbb Z_2$$\mathbb Z_2$ for $s$$s$ odd.

For definition of $W_0'$$W_0'$ and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2($W_0'$$W_0'$)]. Latter Theorem is essetialy known result. Compare to the Theorem 5.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.