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$${{Authors|Mikhail Fedorov|Garaev Timur}} {{Stub}} == Introduction == <wikitex>; \label{sec::intro} In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In $\S$\ref{sec::linking_form} we introduce an invariant of embedding of a $n$-manifold in $(n-1)$-space for even $n$. In $\S$\ref{sec::generalisations} which is independent from $\S$\ref{sec::example}, $\S$\ref{sec::linking_form} and $\S$\ref{sec::classification} we state generalisations of theorems from $\S$\ref{sec::general_theorems} to highly-connected manifolds. For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\S$5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 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. 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. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

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$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

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

The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2].

Inequality in part (b) is sharp, see Proposition 4.1.

Part (b) in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, Corollary 5]. Case $n=1$$n=1$ is clear. Both parts of this theorem are special cases of the Theorem 7.2. Case $n=2$$n=2$ can be proved using the following ideas.

These basic results can be generalized to the highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b

The first proof uses immersions, while an alternative proof does not.

Proof. Proof of theorem 2.1.b. By strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $g|_{X}$$g|_{X}$ is an embedding, because $2(n-1) < 2n-1$$2(n-1) < 2n-1$. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Proof. Alternative proof of theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $M$$M$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial M$$\phi : \partial D^i \times D^{n-i} \to \partial M$, $f : M \to \mathbb R^m$$f : M \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(M)$$f(M)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof.Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ such that the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$, it follows that there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. 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 as $h$$h$ the identity map of $S^k$$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)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\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$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. 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 5.2. Denote

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

where $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 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$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.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

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

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 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

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 5.3 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 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

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 5.4.

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:

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

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$

6 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)$$(n-1)$-sphere. For a free Abelian group $A$$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 6.1. 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 6.2. 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 6.3 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

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

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 6.4. 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

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

is one-to-one.

Lemma 6.5. 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 6.4:

Theorem 6.6. 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]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

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

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'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

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, $\S]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated. If the category is omitted, then we assume the smooth (DIFF) category. Denote $\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$ up to isotopy. We denote by $\mathrm{lk}$ the linking coefficient \cite[$\S$ 77]{Seifert&Threlfall1980} of two disjoint cycles. 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}}\label{thm::embed} Assume that $N$ is a compact connected $n$-manifold. (a) Then $N$ embeds into $\mathbb R^{2n}$. (b) If $N$ has non-empty boundary, then $N$ embeds into $\mathbb R^{2n-1}$. {{endthm}} ''Part (a)'' is well-known [[Wikipedia:Whitney_embedding_theorem|strong Whitney embedding theorem]]. {{beginthm|Theorem}}\label{thm::unknotting} Assume that $N$ is a compact connected $n$-manifold and either (a) $m \ge 2n+1 \ge 5$ or (b) $N$ has non-empty boundary and $m\geq 2n$. Then any two embeddings of $N$ into $\mathbb R^m$ are isotopic. {{endthm}} The part (a) is Whitney-Wu Unknotting Theorem, see \cite[$\S$ 2, Theorems 2.1, 2.2]{Skopenkov2016c}. Inequality in part (b) is sharp, see Proposition \ref{exm::linked_boundary}. Part (b) 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::k_connect_unknot}. Case $n=2$ can be proved using the following ideas. These basic results can be generalized to the highly-connected manifolds (see $\S$\ref{sec::generalisations}). All stated theorems of $\S$\ref{sec::general_theorems} and $\S$\ref{sec::generalisations} for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas. == Proofs of Theorem \ref{thm::embed}.b == ; \label{sec::pt} The first proof uses immersions, while an alternative proof does not. {{beginproof}} ''Proof of theorem \ref{thm::embed}.b.'' By [[Wikipedia:Whitney_immersion_theorem|strong Whitney immersion theorem]] there exist an immersion $g\colon N\to\mathbb R^{2n-1}$. Since $N$ is connected and has non-empty boundary, it follows that $N$ collapses to an $(n-1)$-dimensional subcomplex $X\subset N$ of some triangulation of $N$. By general position we may assume that $g|_{X}$ is an embedding, because (n-1) < 2n-1$. Since $g$ is an immersion, it follows that $X$ has a sufficiently small regular neighbourhood $M\supset X$ such that $g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$. The composition $g\circ h$ is an embedding of $N$. {{endproof}} This proof is essentially contained in \cite[Theorem 4.6]{Hirsch1961a} for the Diff case and in references for Theorem \ref{thm::k_connect_embeds} below or in \cite[Theorem 5.2]{Horvatic1971} for the PL case. For the alternative proof we need some lemmas. {{beginthm|Lemma}}\label{p:handle-decompose} \cite{Wall1966} Assume that $N$ is a closed connected smooth $n$-manifold. Then $N_0$ have handle decomposition with indices of attaching map at most $n-1$. {{endthm}} {{beginthm|Lemma}}\label{p:handle-add} Assume that $N$ is a closed smooth $n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i Since the index $i$ of $\phi_l$ is smaller than $n$, by lemma \ref{p:handle-add} there is extension of $F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$. {{endproof}} {{beginthm|Lemma}}\label{p:handle-field} Let $M$ be a closed smooth $n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial M$, $f : M \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$ on $\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$ there is a field of $n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$ is tangent to $f(M)$. Then $f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$. {{endthm}} {{beginproof}}''Proof of Lemma \ref{p:handle-add} using Lemma \ref{p:handle-field}.'' Since i+1\leq 2n-1$, it follows that there is an extension of $f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$ the $(n-i)\times n$ matrix such that the rightest $(n-i)\times (n-i)$ block is identity matrix and other elements equal to zero. Denote by $v$ the field of $n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$ such that the $k$-th vector have coordinates equal to the $k$-th row in $V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$. Denote by $v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the normal space to $g(\partial D^i\times 0)$ and tangent space to $\partial N$. Since $\pi_{i-1}(V_{2n-1, n-i})=0$, it follows that there is an extension of $v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$. Then by Lemma \ref{p:handle-field} there is an extension of $f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$. {{endproof}} == Example of non-isotopic embeddings == ; \label{sec::example} The following example is folklore. {{beginthm|Example}} \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 $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$. (c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$. {{endthm}} {{beginproof}} ''Proof of part (b).'' Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let $h\colon S^k\to S^k$ be a map of degree $a$. (To prove part (a) it is sufficient to take as $h$ 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 $g(x, t) = (x, h(x)t)$. Let $f=\mathrm i\circ g$, 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 $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$. ''Proof of part (c).'' Clearly $l$ is well-defined. By (b) $l$ is surjective. Now take any two embeddings $f_1, f_2$ such that $l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields. Since $k\geqslant 2$ [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem]] implies that there exists an isotopy of $f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$ and $f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$ and $f_2$ are isotopic. {{endproof}} [[Image:Punctured_torus_embeddings.svg|thumb|450px|(a): Embeddings $f_1$ (top) and $f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$, $i=1,2$, so the ends of the vector field define the section $s_i$; (c): embedding $s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$.]] Denote \S5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 we state generalisations of theorems from $\S$$\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. 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. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

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$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

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

The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2].

Inequality in part (b) is sharp, see Proposition 4.1.

Part (b) in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, Corollary 5]. Case $n=1$$n=1$ is clear. Both parts of this theorem are special cases of the Theorem 7.2. Case $n=2$$n=2$ can be proved using the following ideas.

These basic results can be generalized to the highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b

The first proof uses immersions, while an alternative proof does not.

Proof. Proof of theorem 2.1.b. By strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $g|_{X}$$g|_{X}$ is an embedding, because $2(n-1) < 2n-1$$2(n-1) < 2n-1$. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Proof. Alternative proof of theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $M$$M$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial M$$\phi : \partial D^i \times D^{n-i} \to \partial M$, $f : M \to \mathbb R^m$$f : M \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(M)$$f(M)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof.Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ such that the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$, it follows that there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. 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 as $h$$h$ the identity map of $S^k$$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)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\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$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. 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 5.2. Denote

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

where $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 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$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.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

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

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 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

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 5.3 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 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

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 5.4.

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:

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

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$

6 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)$$(n-1)$-sphere. For a free Abelian group $A$$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 6.1. 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 6.2. 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 6.3 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

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

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 6.4. 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

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

is one-to-one.

Lemma 6.5. 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 6.4:

Theorem 6.6. 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]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

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

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'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

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_k:=(1,0,\ldots,0)\in S^k$. {{beginthm|Example}} Let $N=S^k\times S^1$. Assume $k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$. {{endthm}} The surjectivity of $l$ is given analogously to Proposition \ref{exm::linked_boundary}(b). The injectivity of $l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$ and a cylinder. This example shows that Theorem \ref{thm::k_connect_classif} fails for $k=0$. {{beginthm|Example}} Let $N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$. {{endthm}} To prove the surjectivity of $l$ it is sufficient to take linked $k$-spheres in $\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres. {{beginthm|Example}} (a) Let $N_0$ be the punctured 2-torus containing the meridian $x$ and the parallel $y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$ denote by $s$ the normal field of $\epsilon$-length vectors to $f(N_0)$ defined by orientation on $N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$. (b) Let $f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$ and $l(f_2)=0$ which proves the intuitive fact that $f_1$ and $f_2$ are not isotopic. (Notice that the restrictions of $f_1$ and $f_2$ on $x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$, the values of $l(f_1)$ and $l(f_2)$ will change but will still be different (see figure (d)). {{endthm}} == Seifert linking form == ; \label{sec::linking_form} In this section assume that * $N$ is any closed orientable connected $n$-manifold, * $f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding, * if the (co)homology coefficients are omitted, then they are $\mathbb Z$, * $n$ is even and $H_1(N)$ is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}). By $N_0$ we denote the closure of the complement in $N$ to an closed $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. The following folklore result holds. {{beginthm|Lemma}} 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}} 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} 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. {{beginthm|Lemma}}\label{lmm::saeki} 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} 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 presented in \cite{Fedorov2021} 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. 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} For simplicity in this paragraph we consider only punctured manifolds, see $\S$\ref{prt::arbitrary_boundary} for a generalization. Denote by $N$ a closed $n$-manifold. By $N_0$ denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. {{beginthm|Theorem}}\label{thm::k_connect_embeds} Assume that $N$ is a closed $k$-connected $n$-manifold. (a) If $n\geq 2k+3$, then $N$ embeds into $\mathbb R^{2n-k}$. (b) If $n\geq 2k+2$ and $k\geq0$, then $N_0$ embeds into $\mathbb R^{2n-k-1}$. {{endthm}} Part (a) is proved in \cite[Existence Theorem (a)]{Haefliger1961} for the Diff case and in \cite[Theorem 1.1]{Penrose&Whitehead&Zeeman1961}, \cite[Corollary 1.3]{Irwin1965} for PL case. Part (b) is proved in \cite[Corollary 4.2]{Hirsch1961a} for the Diff case and in \cite[Theorem 1.2]{Penrose&Whitehead&Zeeman1961} for the PL case. {{beginthm|Theorem}}\label{thm::k_connect_unknot} Assume that $N$ is a closed $k$-connected $n$-manifold. (a) If $m \ge 2n - k + 1$ and $n\ge2k + 2$, then any two embeddings of $N$ into $\mathbb R^m$ are isotopic. (b) If $m \ge 2n - k$ and $n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$ into $\mathbb R^m$ are isotopic. {{endthm}} Part (a) is [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.4]] of the survey \cite[$\S$ 2]{Skopenkov2016c}, and is proved in \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}. Part (b) is proved in \cite[Theorem 10.3]{Hudson1969} for the PL case, using [[Isotopy#Concordance|concordance implies isotopy theorem]]. For $k>1$ part (b) is a corollary of Theorem \ref{thm::k_connect_classif} below. For $k=0$ part (b) coincides with Theorem \ref{thm::unknotting}b.

'''Proof of Theorem \ref{thm::k_connect_unknot}(b) for $k=1$.''' By Theorem \ref{thm::isotop_unknot} below every two immersions of $N_0$ into $\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$ for each $x\in N_0$. It follows from Theorem \ref{thm::k_connect_is_spine} that $N_0$ collapses to an $(n-2)$-dimensional subcomplex $X\subset N_0$ of some triangulation of $N_0$. By general position we may assume that $F|_{X\times[0,1]}$ is an embedding, because (n-1) < 2n$. Since $F$ is an immersion, it follows that $X$ has a sufficiently small regular neighbourhood $M\supset X$ such that $F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$. It is clear that $f$ is isotopic to $f\circ h$ and $g$ is isotopic to $g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$ is a concordance of $f\circ h$ and $g\circ h$. By [[Isotopy#Concordance|concordance implies isotopy Theorem]] $f$ and $g$ are isotopic.{{endproof}} {{beginthm|Conjecture}} Assume that $N$ is a closed \S5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 we state generalisations of theorems from $\S$$\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. 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. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

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$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

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

The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2].

Inequality in part (b) is sharp, see Proposition 4.1.

Part (b) in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, Corollary 5]. Case $n=1$$n=1$ is clear. Both parts of this theorem are special cases of the Theorem 7.2. Case $n=2$$n=2$ can be proved using the following ideas.

These basic results can be generalized to the highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b

The first proof uses immersions, while an alternative proof does not.

Proof. Proof of theorem 2.1.b. By strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $g|_{X}$$g|_{X}$ is an embedding, because $2(n-1) < 2n-1$$2(n-1) < 2n-1$. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Proof. Alternative proof of theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $M$$M$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial M$$\phi : \partial D^i \times D^{n-i} \to \partial M$, $f : M \to \mathbb R^m$$f : M \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(M)$$f(M)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof.Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ such that the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$, it follows that there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. 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 as $h$$h$ the identity map of $S^k$$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)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\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$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. 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 5.2. Denote

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

where $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 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$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.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

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

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 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

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 5.3 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 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

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 5.4.

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:

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

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$

6 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)$$(n-1)$-sphere. For a free Abelian group $A$$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 6.1. 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 6.2. 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 6.3 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

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

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 6.4. 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

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

is one-to-one.

Lemma 6.5. 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 6.4:

Theorem 6.6. 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]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

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

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'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

10 References

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$-connected $-manifold. Then any two embeddings of $N_0$ in $\mathbb R^7$ are isotopic. {{endthm}} We may hope to get around the restrictions of Theorem \ref{thm::k_connect_is_spine} using the [[Some_calculations_involving_configuration_spaces_of_distinct_points|deleted product criterion]]. {{beginthm|Theorem}}\label{thm::k_connect_classif} Assume $N$ is a closed $k$-connected $n$-manifold. 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}. The latter theorem was essentially proven in \cite[Theorem 2.1]{Vrabec1989}. Latter Theorem is essentially known result. Compare to the Theorem \ref{thm::punctured_class}, which describes $\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case. == Comments on non-spherical boundary == ; \label{prt::arbitrary_boundary} {{beginthm|Theorem}}\label{thm::arbitarty_k_connect_embeds} Assume that $N$ is a compact $k$-connected $n$-manifold, $\partial N\neq\emptyset$, $(N, \partial N)$ is $k$-connected and $k< n - 3$. Then $N$ embeds into $\mathbb R^{2n-k-1}$. {{endthm}} This is \cite[Theorem on p.567]{Wall1965}. {{beginproof}} By Theorem \ref{thm::k_immersible} below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$. Since $N$ is $k$-connected it follows from Theorem \ref{thm::k_connect_is_spine} that $N$ collapses to an $(n-k-1)$-dimensional subcomplex $X\subset N$ of some triangulation of $N$. By general position we may assume that $f|_X$ is an embedding, because (n-k) < 2n-k-1$. Since $f$ is an immersion, it follows that $X$ has a sufficiently small regular neighbourhood $M\supset X$ such that $f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$. It is clear that $f\circ h$ is an embedding.{{endproof}} {{beginthm|Theorem}}\label{thm::arbitrary_k_connect_unknotting} Assume that $N$ is a $n$-manifold. If $N$ has $(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$, $m \ge 2n - k$, then any two embeddings of $N$ into $\mathbb R^m$ are isotopic. {{endthm}} Proof is similar to the proof of theorem \ref{thm::k_connect_unknot}. For a compact connected $n$-manifold with boundary, the property of having an $(n − k − 1)$-dimensional spine is close to $k$-connectedness. Indeed, the following theorem holds. {{beginthm|Theorem}}\label{thm::k_connect_is_spine} Every compact connected $n$-manifold $N$ with boundary for which $(N, \partial N)$ is $k$-connected, $\pi_1(\partial N)=0$, $k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$-dimensional spine. {{endthm}} For this result see \cite[Theorem 5.5]{Wall1964a} and \cite[Lemma 5.1 and Remark 5.2]{Horvatic1969}. See also valuable remarks in \cite{Levine&Lidman2018} and \cite{Skopenkov2019}. == Comments on immersions == ; {{beginthm|Theorem}}[Smale-Hirsch; \cite{Hirsch1959} and \cite{Haefliger&Poenaru1964}] The space of immersions of a manifold in $\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$ to $\R^m$. {{endthm}} {{beginthm|Theorem}}[\cite[Theorem 6.4]{Hirsch1959}]\label{thm::imm_dec_dim} If $N$ is immersible in $\R^{m+r}$ with a normal $r$-field, then $N$ is immersible in $\R^m$. {{endthm}} {{beginthm|Theorem}} Every $n$-manifold $N$ with non-empty boundary is immersible in $\R^{2n-1}$. {{endthm}} {{beginthm|Theorem}}[Whitney; \cite[Theorem 6.6]{Hirsch1961a}] Every $n$-manifold $N$ is immersible in $\R^{2n-1}$. {{endthm}} Denote by $V_{m,n}$ is Stiefel manifold of $n$-frames in $\R^m$. {{beginthm|Theorem}}\label{thm::k_immersible} Suppose $N$ is a $n$-manifold with non-empty boudary, $(N,\partial N)$ is $k$-connected. Then $N$ is immersible in $\R^m$ for each $m\geq2n-k-1$. {{endthm}} {{beginproof}} It suffices to show that exists an immersion of $N$ in $\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$ to $\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$. It is clear that a linear monomorphism exists on $\S$5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 we state generalisations of theorems from $\S$$\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. 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. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

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$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

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

The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2].

Inequality in part (b) is sharp, see Proposition 4.1.

Part (b) in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, Corollary 5]. Case $n=1$$n=1$ is clear. Both parts of this theorem are special cases of the Theorem 7.2. Case $n=2$$n=2$ can be proved using the following ideas.

These basic results can be generalized to the highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b

The first proof uses immersions, while an alternative proof does not.

Proof. Proof of theorem 2.1.b. By strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $g|_{X}$$g|_{X}$ is an embedding, because $2(n-1) < 2n-1$$2(n-1) < 2n-1$. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Proof. Alternative proof of theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $M$$M$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial M$$\phi : \partial D^i \times D^{n-i} \to \partial M$, $f : M \to \mathbb R^m$$f : M \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(M)$$f(M)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof.Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ such that the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$, it follows that there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. 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 as $h$$h$ the identity map of $S^k$$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)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\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$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. 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 5.2. Denote

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

where $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 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$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.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

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

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 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

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 5.3 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 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

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 5.4.

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:

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

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$

6 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)$$(n-1)$-sphere. For a free Abelian group $A$$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 6.1. 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 6.2. 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 6.3 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

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

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 6.4. 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

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

is one-to-one.

Lemma 6.5. 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 6.4:

Theorem 6.6. 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]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

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

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'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

10 References

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$-skeleton of $N$. The obstruction to extend the linear monomorphism from $(r-1)$-skeleton to $r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$. For $r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$ is $k$-connected and has non-empty boundary. Thus the obstruction is always zero and such linear monomorphism exists. {{endproof}} {{beginthm|Theorem}}\label{thm::isotop_unknot} Suppose $N$ is a connected $n$-manifold with non-empty boudary, $(N, \partial N)$ is $k$-connected and $m\geq2n-k$. Then every two immersions of $N$ in $\R^m$ are regulary homotopic. {{endthm}} {{beginproof}} It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$ to $\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$-skeleton of $N$. It is clear that homotopy exists on $\S$5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 we state generalisations of theorems from $\S$$\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. 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. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

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$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

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

The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2].

Inequality in part (b) is sharp, see Proposition 4.1.

Part (b) in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, Corollary 5]. Case $n=1$$n=1$ is clear. Both parts of this theorem are special cases of the Theorem 7.2. Case $n=2$$n=2$ can be proved using the following ideas.

These basic results can be generalized to the highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b

The first proof uses immersions, while an alternative proof does not.

Proof. Proof of theorem 2.1.b. By strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $g|_{X}$$g|_{X}$ is an embedding, because $2(n-1) < 2n-1$$2(n-1) < 2n-1$. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Proof. Alternative proof of theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $M$$M$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial M$$\phi : \partial D^i \times D^{n-i} \to \partial M$, $f : M \to \mathbb R^m$$f : M \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(M)$$f(M)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof.Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ such that the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$, it follows that there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. 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 as $h$$h$ the identity map of $S^k$$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)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\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$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. 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 5.2. Denote

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

where $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 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$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.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

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

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 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

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 5.3 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 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

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 5.4.

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:

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

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$

6 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)$$(n-1)$-sphere. For a free Abelian group $A$$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 6.1. 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 6.2. 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 6.3 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

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

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 6.4. 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

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

is one-to-one.

Lemma 6.5. 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 6.4:

Theorem 6.6. 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]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

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

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'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

10 References

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$-skeleton of $N$. The obstruction to extend the homotopy from $(r-1)$-skeleton to $r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$. For $r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$ is $k$-connected and $N$ has non-empty boundary. Thus the obstruction is always zero and such homotopy of linear monomorphisms exists. {{endproof}} == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 we state generalisations of theorems from $\S$$\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. 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. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

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$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

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

The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2].

Inequality in part (b) is sharp, see Proposition 4.1.

Part (b) in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, Corollary 5]. Case $n=1$$n=1$ is clear. Both parts of this theorem are special cases of the Theorem 7.2. Case $n=2$$n=2$ can be proved using the following ideas.

These basic results can be generalized to the highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b

The first proof uses immersions, while an alternative proof does not.

Proof. Proof of theorem 2.1.b. By strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $g|_{X}$$g|_{X}$ is an embedding, because $2(n-1) < 2n-1$$2(n-1) < 2n-1$. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Proof. Alternative proof of theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $M$$M$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial M$$\phi : \partial D^i \times D^{n-i} \to \partial M$, $f : M \to \mathbb R^m$$f : M \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(M)$$f(M)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof.Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ such that the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$, it follows that there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. 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 as $h$$h$ the identity map of $S^k$$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)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\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$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. 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 5.2. Denote

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

where $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 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$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.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

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

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 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

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 5.3 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 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

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 5.4.

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:

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

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$

6 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)$$(n-1)$-sphere. For a free Abelian group $A$$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 6.1. 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 6.2. 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 6.3 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

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

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 6.4. 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

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

is one-to-one.

Lemma 6.5. 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 6.4:

Theorem 6.6. 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]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

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

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'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

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