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1 Introduction

This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.


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].

The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):

$\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}$

For an $s$$s$-tuple $(n):=(n_1,\ldots,n_s)$$(n):=(n_1,\ldots,n_s)$ denote $S^{(n)}:=S^{n_1}\sqcup\ldots\sqcup S^{n_s}$$S^{(n)}:=S^{n_1}\sqcup\ldots\sqcup S^{n_s}$. Although $S^{(n)}$$S^{(n)}$ is not a manifold when $n_1,\ldots,n_s$$n_1,\ldots,n_s$ are not all equal, embeddings $S^{(n)}\to S^m$$S^{(n)}\to S^m$ and isotopy between such embeddings are defined analogously to the case of manifolds. Denote by $E^m(S^{(n)})$$E^m(S^{(n)})$ the set of embeddings $S^{(n)}\to S^m$$S^{(n)}\to S^m$ up to isotopy.

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{(n)})$$E^m(S^{(n)})$ for $m-3\ge n_i$$m-3\ge n_i$ [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].

2 Examples

Recall that for each $q$$q$-manifold $N$$N$ and $m\ge2q+2$$m\ge2q+2$, any two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 2.1]. The following example shows that the restriction $m\ge2q+2$$m\ge2q+2$ is sharp for non-connected manifolds.

Example 2.1 (The Hopf Link). For each $n$$n$ there is an embedding $S^q\sqcup S^q\to\Rr^{2q+1}$$S^q\sqcup S^q\to\Rr^{2q+1}$ which is not isotopic to the standard embedding.

For $q=1$$q=1$ the Hopf link is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary $q$$q$ (including $q=1$$q=1$) the image of the Hopf link is the union of two $q$$q$-spheres:

• either $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})$$\partial(D^{q+1}\times D^{q+1})$;
• or given by equations:
$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_q=0\\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.$

This embedding is distinguished from the standard embedding by the linking coefficient ($\S$$\S$3).

Analogously for each $p,q$$p,q$ one constructs an embedding $S^p\sqcup S^q\to\Rr^{p+q+1}$$S^p\sqcup S^q\to\Rr^{p+q+1}$ which is not isotopic to the standard embedding. The image is the union of two spheres:

• either $\partial D^{p+1}\times0$$\partial D^{p+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{p+1}\times D^{q+1})$$\partial(D^{p+1}\times D^{q+1})$.
• or given by equations:
$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$

This embedding is distinguished from the standard embedding also by the linking coefficient ($\S$$\S$3).

Definition 2.2 (The Zeeman map). We define a map

$\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.$

Denote by $i_{m,p}:S^p\to S^m$$i_{m,p}:S^p\to S^m$ the equatorial inclusion. For a map $x:S^p\to S^{m-q-1}$$x:S^p\to S^{m-q-1}$ representing an element of $\pi_p(S^{m-q-1})$$\pi_p(S^{m-q-1})$ let

$\displaystyle \overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,$

where $i$$i$ is the standard embedding [Skopenkov2006, Figure 3.2]. We have $\overline\zeta_x(S^p)\cap i_{m,q}(S^q)\subset i(S^{m-q-1}\times S^p)\cap i(0_{m-q}\times S^q) =\emptyset$$\overline\zeta_x(S^p)\cap i_{m,q}(S^q)\subset i(S^{m-q-1}\times S^p)\cap i(0_{m-q}\times S^q) =\emptyset$. Let $\zeta[x]:=[\overline\zeta_x\sqcup i_{m,q}]$$\zeta[x]:=[\overline\zeta_x\sqcup i_{m,q}]$.

One can easily check that $\zeta$$\zeta$ is well-defined and is a homomorphism.

Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.

Definition 3.1 (The linking coefficient). We define a map

$\displaystyle \lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.$

Take an embedding $f:S^p\sqcup S^q\to S^m$$f:S^p\sqcup S^q\to S^m$ representing an element $[f]\in E^m(S^p\sqcup S^q)$$[f]\in E^m(S^p\sqcup S^q)$. Take an embedding $g:D^{m-q}\to S^m$$g:D^{m-q}\to S^m$ such that $gD^{m-q}$$gD^{m-q}$ intersects $fS^q$$fS^q$ transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1]. Then the restriction $h':S^{m-q-1}\to S^m-fS^q$$h':S^{m-q-1}\to S^m-fS^q$ of $g$$g$ to $\partial D^{m-q}$$\partial D^{m-q}$ is a homotopy equivalence.

(Indeed, since $m\ge q+3$$m\ge q+3$, by general position the complement $S^m-fS^q$$S^m-fS^q$ is simply-connected. By Alexander duality, $h'$$h'$ induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems $h'$$h'$ is a homotopy equivalence.)

Let $h$$h$ be a homotopy inverse of $h'$$h'$. Define

$\displaystyle \lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$

Remark 3.2. (a) Clearly, $\lambda[f]$$\lambda[f]$ is indeed independent of $g,h',h$$g,h',h$. One can check that $\lambda$$\lambda$ is a homomorphism.

(b) For $m=p+q+1$$m=p+q+1$ or $m=q+2$$m=q+2$ there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].

(c) Analogously one can define $\lambda_{21}(f)\in\pi_q(S^{m-p-1})$$\lambda_{21}(f)\in\pi_q(S^{m-p-1})$ for $m\ge p+3$$m\ge p+3$, by exchanging $p$$p$ and $q$$q$ in the above definition.

(d) This definition extends to the case $m=q+2$$m=q+2$ when $S^m-fS^q$$S^m-fS^q$ is simply-connected (or, equivalently for $q>4$$q>4$, if the restriction of $f$$f$ to $S^q$$S^q$ is unknotted).

(e) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$$\lambda\zeta=\id\pi_p(S^{m-q-1})$, even for $m=q+2$$m=q+2$ (see Definition 2.2). So $\lambda$$\lambda$ is surjective and $\zeta$$\zeta$ is injective.

By the Freudenthal Suspension Theorem $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$$\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$$m\ge\frac p2+q+2$. The stabilization of the linking coefficient can be described as follows.

Definition 3.3 (The $\alpha$$\alpha$-invariant). We define a map $E^m(S^p\sqcup S^q)\to\pi^S_{p+q+1-m}$$E^m(S^p\sqcup S^q)\to\pi^S_{p+q+1-m}$ for $p,q\le m-2$$p,q\le m-2$. Take an embedding $f:S^p\sqcup S^q\to S^m$$f:S^p\sqcup S^q\to S^m$ representing an element $[f]\in E^m(S^p\sqcup S^q)$$[f]\in E^m(S^p\sqcup S^q)$. Define a map [Skopenkov2006, Figure 3.1]

$\displaystyle \widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.$

For $p,q\le m-2$$p,q\le m-2$ define the $\alpha$$\alpha$-invariant by

$\displaystyle \alpha(f)=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.$

The second isomorphism in this formula is given by the Freudenthal Suspension Theorem. The map $v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q}$$v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q}$ is the quotient map. See [Skopenkov2006, Figure 3.4]. The map $v^*$$v^*$ is an isomorphism for $m\ge q+2$$m\ge q+2$. (For $m\ge q+3$$m\ge q+3$ this follows by general position and for $m=q+2$$m=q+2$ by the cofibration Barratt-Puppe exact sequence of pair $(S^p\times S^q,S^p\vee S^q)$$(S^p\times S^q,S^p\vee S^q)$ and by the existence of a retraction $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$$\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.)

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$.

Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$.

Note that $\alpha$$\alpha$-invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, $\S$$\S$5].

4 Classification in the metastable range

The Haefliger-Zeeman Theorem 4.1. (D) If $1\le p\le q$$1\le p\le q$, then both $\lambda$$\lambda$ and $\zeta$$\zeta$ are isomorphisms for $m\ge\frac{3q}2+2$$m\ge\frac{3q}2+2$ in the smooth category.

(PL) If $1\le p\le q$$1\le p\le q$, then both $\lambda$$\lambda$ and $\zeta$$\zeta$ are isomorphisms for $m\ge\frac p2+q+2$$m\ge\frac p2+q+2$ in the PL category.

The surjectivity of $\lambda$$\lambda$ (or the injectivity of $\zeta$$\zeta$) follows from $\lambda\zeta=\id$$\lambda\zeta=\id$. The injectivity of $\lambda$$\lambda$ (or the surjectivity of $\zeta$$\zeta$) is proved in [Haefliger1962t], [Zeeman1962] (or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).

An analogue of this result holds for links with many components [Haefliger1966a].

Theorem 4.2. The collection of pairwise linking coefficients is bijective for $2m\ge3q+4$$2m\ge3q+4$ and $q$$q$-dimensional links in $\R^m$$\R^m$.

5 Examples beyond the metastable range

We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.

Borromean rings example 5.1. The embedding defined below is a non-trivial embedding $S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t].

Denote coordinates in $\Rr^{3l}$$\Rr^{3l}$ by $(x,y,z)=(x_1,\dots,x_l,y_1,\dots,y_l,z_1,\dots,z_l)$$(x,y,z)=(x_1,\dots,x_l,y_1,\dots,y_l,z_1,\dots,z_l)$. The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i are the three spheres given by the following three systems of equations:

$\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1,\end{array}\right. \qquad \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1. \end{array}\right.$

The required embedding is any embedding whose image consists of Borromean rings.

This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because joining the three components with two tubes, i.e. linked analogue' of embedded connected sum of the components [Skopenkov2016c], yields a non-trivial knot [Haefliger1962], cf. the Haefliger Trefoil knot [Skopenkov2016t]).

For $l=1$$l=1$ this and other results of this section are parts of low-dimensional link theory and so were known well before given references.

Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.

Whitehead link example 5.2. There is a non-trivial embedding $w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ whose linking coefficient $\lambda_{12}(w)$$\lambda_{12}(w)$ is trivial.

The (higher-dimensional) Whitehead link is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by linked analogue' of embedded connected sum of the components [Skopenkov2016c].

We have $\lambda_{12}(w)=0$$\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in [Skopenkov2006a]). For $l\ne1,3,7$$l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$$\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$. This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18]. For $l=1,3,7$$l=1,3,7$ the Whitehead link is is distinguished from the standard embedding by more complicated invariants [Skopenkov2006a], [Haefliger1962t, $\S$$\S$3].

This example (in higher dimensions, i.e. for $l>1$$l>1$) seems to have been discovered by Whitehead, in connection with Whitehead product.

6 Classification beyond the metastable range

Theorem 6.1. [Haefliger1966a] If $n_1,\ldots,n_s\le m-3$$n_1,\ldots,n_s\le m-3$, then

$\displaystyle E^m_D(S^{(n)})\cong E^m_{PL}(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i}).$

Thus $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is isomorphic to the kernel of the restriction homomorphism $E^m_D(S^{(n)})\to\oplus_{i=1}^s E^m_D(S^{n_i})$$E^m_D(S^{(n)})\to\oplus_{i=1}^s E^m_D(S^{n_i})$ and to the group of $1$$1$-Brunnian links (i.e. of links whose restrictions to the components are unknotted). For some information on the groups $E^m_D(S^q)$$E^m_D(S^q)$ see [Skopenkov2006, $\S$$\S$3.3].

The Haefliger Theorem 6.2. [Haefliger1966a, Theorem 10.7], [Skopenkov2009] If $p\le q\le m-3$$p\le q\le m-3$ and $3m\ge2p+2q+6$$3m\ge2p+2q+6$, then there is a homomorphism $\beta$$\beta$ for which the following map is an isomorphism

$\displaystyle \lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).$

The map $\beta$$\beta$ and its right inverse $\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$$\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$ are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].

Remark 6.3. (a) The Haefliger Theorem 6.2(b) implies that for each $l\ge2$$l\ge2$ we have an isomorphism

$\displaystyle \lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{\varepsilon(l)}.$

(b) When $l \geq 2$$l \geq 2$ but $l \neq 3, 7$$l \neq 3, 7$, the map

$\displaystyle \lambda_{12}\oplus\lambda_{21}:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\pi_{2l-1}(S^l)$

is injective and its image is $\{(a,b)\ :\ \Sigma a=\Sigma b\}$$\{(a,b)\ :\ \Sigma a=\Sigma b\}$. This is proved in [Haefliger1962t] and also follows from Theorem 6.2(b).

(c) For $l\in\{1,3,7\}$$l\in\{1,3,7\}$ the map $\lambda_{12}\oplus\lambda_{21}$$\lambda_{12}\oplus\lambda_{21}$ in (b) above is not injective [Haefliger1962t].

(d) [Haefliger1962t] For each $l\ge2$$l\ge2$ we have an isomorphism

$\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{\varepsilon(l)}^4.$

which is the sum of 3 pairwise linking coefficients, 3 pairwise $\beta$$\beta$-invariants and triplewise Massey invariant.

7 Classification in codimension 3

In this subsection we assume that $(n)$$(n)$ is an $s$$s$-tuple such that $m\ge n_i+3$$m\ge n_i+3$.

Definition 7.1 (The Haefliger link sequence). We define the following long sequence of abelian groups

$\displaystyle \dots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \dots~.$

In the above sequence $s$$s$-tuples $m-n_1,\ldots,m-n_s$$m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W:=\vee_{i=1}^s S^{m-n_i-1}$$W:=\vee_{i=1}^s S^{m-n_i-1}$. For each $j=1,\ldots,s$$j=1,\ldots,s$ and integer $k>0$$k>0$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$$p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$$j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$$(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$$\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$.

Analogously to Definition 3.1 there is a canonical homotopy equivalence $C_f \sim W$$C_f \sim W$. The homotopy class of one component in the complement of the others gives then a map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$$\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$, see details in [Haefliger1966a, 1.4] (this is a generalization of linking coefficient). Define $\lambda:= \oplus_{j=1}^s \lambda_j$$\lambda:= \oplus_{j=1}^s \lambda_j$.

Taking the Whitehead product with the class of the identity in $\pi_{n_j}(S^{n_j})$$\pi_{n_j}(S^{n_j})$ defines a homomorphism $w_j \colon \ker p_{n_j,j}\to \Pi_{m-2}^{(n-1)}$$w_j \colon \ker p_{n_j,j}\to \Pi_{m-2}^{(n-1)}$. Define $w:=\oplus_{j=1}^s w_j$$w:=\oplus_{j=1}^s w_j$.

The definition of the homomorphism $\mu$$\mu$ is sketched in [Haefliger1966a, 1.5].

Theorem 7.2. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

(b) [Crowley&Ferry&Skopenkov2011] The map $\lambda\otimes\Qq \colon E^m_{PL}(S^{(n)})\otimes\Qq \to \ker(w\otimes\Qq)$$\lambda\otimes\Qq \colon E^m_{PL}(S^{(n)})\otimes\Qq \to \ker(w\otimes\Qq)$ is an isomorphism.

Part (b) follows because $\mu \otimes \Qq = 0$$\mu \otimes \Qq = 0$ [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the link Haefliger sequence splits into short exact sequences.

In [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.2, $\S$$\S$1.3] one can find necessary and sufficient conditions on $p$$p$ and $q$$q$ when $E^m_{PL}(S^p\sqcup S^q)$$E^m_{PL}(S^p\sqcup S^q)$ is finite, as well as an effective procedure for computing the rank of the group $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$.

For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Skopenkov2015a, $\S$$\S$2.5], [Skopenkov2016k].