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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.

We describe classification of embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$$== Introduction == ; This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings. We describe classification of embeddings S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m for m-3\ge n_i. For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[\SS^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$$m-3\ge n_i$.

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].

## 8 References

, $\S]{Skopenkov2016c}. The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem \ref{lkhaze} below): $$\begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}$$ For an$s$-tuple$(n):=(n_1,\ldots,n_s)$denote$S^{(n)}:=S^{n_1}\sqcup\ldots\sqcup S^{n_s}$. Although$S^{(n)}$is not a manifold when$n_1,\ldots,n_s$are not all equal, embeddings$S^{(n)}\to S^m$and isotopy between such embeddings are defined analogously to [[Isotopy|the case of manifolds]]. Denote by$E^m(S^{(n)})$the set of embeddings$S^{(n)}\to S^m$up to isotopy. A component-wise version of [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Embedded_connected_sum|embedded connected sum]] \cite[$\S]{Skopenkov2016c} defines a commutative group structure on the set $E^m(S^{(n)})$ for $m-3\ge n_i$ \cite{Haefliger1966}, \cite{Haefliger1966a}, \cite[Group Structure Lemma 2.2 and Remark 2.3.a]{Skopenkov2015}, see \cite[Figure 3.3]{Skopenkov2006}. == Examples == ; [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Recall that]] for each $q$-manifold $N$ and $m\ge2q+2$, any two embeddings $N\to\Rr^m$ are isotopic \cite[General Position Theorem 2.1]{Skopenkov2016c}. The following example shows that the restriction $m\ge2q+2$ is sharp for non-connected manifolds. {{beginthm|Example|(The Hopf Link)}}\label{hopf} For each $n$ there is an embedding $S^q\sqcup S^q\to\Rr^{2q+1}$ which is not isotopic to the standard embedding. {{endthm}} For $q=1$ the Hopf link is shown in \cite[Figure 2.1.a]{Skopenkov2006}. For arbitrary $q$ (including $q=1$) the image of the Hopf link is the union of two $q$-spheres: * either $\partial D^{q+1}\times0$ and $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$$m-3\ge n_i$.

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].

== The linking coefficient == ;\label{s:inv} Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls. {{beginthm|Definition|(The linking coefficient)}}\label{dl} We define a map $$\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.$$ Take an embedding $f:S^p\sqcup S^q\to S^m$ representing an element $[f]\in E^m(S^p\sqcup S^q)$. Take an embedding $g:D^{m-q}\to S^m$ such that $gD^{m-q}$ intersects $fS^q$ transversely at exactly one point with positive sign \cite[Figure 3.1]{Skopenkov2006}. Then the restriction $h':S^{m-q-1}\to S^m-fS^q$ of $g$ to $\partial D^{m-q}$ is a homotopy equivalence. (Indeed, since $m\ge q+3$, by general position the complement $S^m-fS^q$ is simply-connected. By Alexander duality, $h'$ induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems $h'$ is a homotopy equivalence.) Let $h$ be a homotopy inverse of $h'$. Define $$\lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$$ {{endthm}} {{beginthm|Remark}}\label{lkrem} (a) Clearly, $\lambda[f]$ is indeed independent of $g,h',h$. One can check that $\lambda$ is a homomorphism. (b) For $m=p+q+1$ or $m=q+2$ there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|Remark 5.3.f]] of \cite{Skopenkov2016e}. (c) Analogously one can define $\lambda_{21}(f)\in\pi_q(S^{m-p-1})$ for $m\ge p+3$, by exchanging $p$ and $q$ in the above definition. (d) This definition extends to the case $m=q+2$ when $S^m-fS^q$ is simply-connected (or, equivalently for $q>4$, if the restriction of $f$ to $S^q$ is unknotted). (e) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$, even for $m=q+2$ (see Definition \ref{dz}). So $\lambda$ is surjective and $\zeta$ is injective. {{endthm}} By the Freudenthal Suspension Theorem $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$. The stabilization of the linking coefficient can be described as follows. {{beginthm|Definition|(The $\alpha$-invariant)}} We define a map $E^m(S^p\sqcup S^q)\to\pi^S_{p+q+1-m}$ for $p,q\le m-2$. Take an embedding $f:S^p\sqcup S^q\to S^m$ representing an element $[f]\in E^m(S^p\sqcup S^q)$. Define a map \cite[Figure 3.1]{Skopenkov2006} $$\widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.$$ For $p,q\le m-2$ define the $\alpha$-invariant by $$\alpha(f)=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.$$ The second isomorphism in this formula is given by the Freudenthal Suspension Theorem. The map $v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q}$ is the quotient map. See \cite[Figure 3.4]{Skopenkov2006}. The map $v^*$ is an isomorphism for $m\ge q+2$. (For $m\ge q+3$ this follows by general position and for $m=q+2$ by the cofibration Barratt-Puppe exact sequence of pair $(S^p\times S^q,S^p\vee S^q)$ and by the existence of a retraction $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.) {{endthm}} {{beginthm|Lemma|\cite[Lemma 5.1]{Kervaire1959a}}} We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$. {{endthm}} Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$. Note that $\alpha$-invariant can be defined in more general situations \cite{Koschorke1988}, \cite[$\S]{Skopenkov2006}. ==Classification in the metastable range== ; {{beginthm|The Haefliger-Zeeman Theorem}}\label{lkhaze} (D) If S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m for $m-3\ge n_i$$m-3\ge n_i$. For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\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. ## 3 The linking coefficient 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$$\S$2.5], [Skopenkov2016k]. ## 8 References \le p\le q$, then both $\lambda$ and $\zeta$ are isomorphisms for $m\ge\frac{3q}2+2$ in the smooth category. (PL) If S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m for $m-3\ge n_i$$m-3\ge n_i$.

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].

## 8 References

\le p\le q$, then both$\lambda$and$\zeta$are isomorphisms for$m\ge\frac p2+q+2$in the PL category. {{endthm}} The surjectivity of$\lambda$(or the injectivity of$\zeta$) follows from$\lambda\zeta=\id$. The injectivity of$\lambda$(or the surjectivity of$\zeta$) is proved in \cite{Haefliger1962t}, \cite{Zeeman1962} (or follows from \cite[the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]{Skopenkov2006}). An analogue of this result holds for links with many components \cite{Haefliger1966a}. {{beginthm|Theorem}}\label{t:lkmany} The collection of pairwise linking coefficients is bijective for m\ge3q+4$ and $q$-dimensional links in $\R^m$. {{endthm}}
==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 \ref{t:lkmany}. {{beginthm|Borromean rings example}}\label{belmetbor} The embedding defined below is a non-trivial embedding $S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ whose restrictions to each 2-component sublink is trivial \cite[4.1]{Haefliger1962}, \cite{Haefliger1962t}. Denote coordinates in $\Rr^{3l}$ by $(x,y,z)=(x_1,\dots,x_l,y_1,\dots,y_l,z_1,\dots,z_l)$. The (higher-dimensional) ''Borromean rings'' \cite[Figures 3.5 and 3.6]{Skopenkov2006}, i are the three spheres given by the following three systems of equations: $$\left\{\begin{array}{c} x=0\ |y|^2+2|z|^2=1,\end{array}\right. \qquad \left\{\begin{array}{c} y=0\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\ |x|^2+2|y|^2=1. \end{array}\right.$$ The required embedding is any embedding whose image consists of Borromean rings. {{endthm}} This embedding is distinguished from the standard embedding by the well-known ''Massey invariant'' \cite{Skopenkov2017} (or because joining the three components with two tubes, i.e. linked analogue' of [[High_codimension_embeddings#Embedded_connected_sum|embedded connected sum]] of the components \cite{Skopenkov2016c}, yields a non-trivial knot \cite{Haefliger1962}, cf. [[3-manifolds_in_6-space#Examples|the Haefliger Trefoil knot]] \cite{Skopenkov2016t}). For $l=1$ 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 \ref{lkhaze}. {{beginthm|Whitehead link example}}\label{belmetwhi} There is a non-trivial embedding $w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ whose linking coefficient $\lambda_{12}(w)$ 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 [[Embeddings_just_below_the_stable_range:_classification#Embedded_connected_sum|embedded connected sum]] of the components \cite{Skopenkov2016c}. {{endthm}} We have $\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in \cite{Skopenkov2006a}). For $l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$. This fact should be well-known, but I do not know a published proof except \cite[Lemma 2.18]{Skopenkov2015a}. For $l=1,3,7$ the Whitehead link is is distinguished from the standard embedding by more complicated invariants \cite{Skopenkov2006a}, \cite[$\S]{Haefliger1962t}. This example (in higher dimensions, i.e. for$l>1$) seems to have been discovered by Whitehead, in connection with Whitehead product. == Classification beyond the metastable range == ; {{beginthm|Theorem}}\label{dpl} \cite{Haefliger1966a} If$n_1,\ldots,n_s\le m-3$, then $$E^m_D(S^{(n)})\cong E^m_{PL}(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i}).$$ {{endthm}} Thus$E^m_{PL}(S^{(n)})$is isomorphic to the kernel of the restriction homomorphism$E^m_D(S^{(n)})\to\oplus_{i=1}^s E^m_D(S^{n_i})$and to the group of ''S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m for $m-3\ge n_i$$m-3\ge n_i$. For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\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. ## 3 The linking coefficient 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$$\S$2.5], [Skopenkov2016k]. ## 8 References$-Brunnian'' links (i.e. of links whose restrictions to the components are unknotted). For [[Knots,_i.e._embeddings_of_spheres|some information on the groups $E^m_D(S^q)$]] see \cite[$\S.3]{Skopenkov2006}. {{beginthm|The Haefliger Theorem}}\label{belmethae} \cite[Theorem 10.7]{Haefliger1966a}, \cite{Skopenkov2009} If$p\le q\le m-3$and m\ge2p+2q+6$, then there is a homomorphism $\beta$ for which the following map is an isomorphism $$\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}).$$ {{endthm}} The map $\beta$ and its right inverse $\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$ are constructed in \cite{Haefliger1966} and \cite{Haefliger1966a}, cf. \cite{Skopenkov2009}. {{beginthm|Remark}}\label{r:bel} (a) The Haefliger Theorem \ref{belmethae}(b) implies that for each $l\ge2$ we have an isomorphism $$\lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{\varepsilon(l)}.$$ (b) When $l \geq 2$ but $l \neq 3, 7$, the map $$\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)$$ is injective and its image is $\{(a,b)\ :\ \Sigma a=\Sigma b\}$. This is proved in \cite{Haefliger1962t} and also follows from Theorem \ref{belmethae}(b). (c) For $l\in\{1,3,7\}$ the map $\lambda_{12}\oplus\lambda_{21}$ in (b) above is not injective \cite{Haefliger1962t}. (d) \cite{Haefliger1962t} For each $l\ge2$ we have an isomorphism $$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.$$ which is the sum of 3 pairwise linking coefficients, 3 pairwise $\beta$-invariants and triplewise Massey invariant. {{endthm}}
== Classification in codimension 3 == ; In this subsection we assume that $(n)$ is an $s$-tuple such that $m\ge n_i+3$. {{beginthm|Definition|(The Haefliger link sequence)}}\label{d:hclink} We define the following long sequence of abelian groups $$\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~.$$ In the above sequence $s$-tuples $m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W:=\vee_{i=1}^s S^{m-n_i-1}$. For each $j=1,\ldots,s$ and integer $k>0$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$. Analogously to Definition \ref{dl} there is a canonical homotopy equivalence $C_f \sim W$. The homotopy class of one component in the complement of the others gives then a map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$, see details in \cite[1.4]{Haefliger1966a} (this is a generalization of linking coefficient). Define $\lambda:= \oplus_{j=1}^s \lambda_j$. Taking the Whitehead product with the class of the identity in $\pi_{n_j}(S^{n_j})$ defines a homomorphism $w_j \colon \ker p_{n_j,j}\to \Pi_{m-2}^{(n-1)}$. Define $w:=\oplus_{j=1}^s w_j$. The definition of the homomorphism $\mu$ is sketched in \cite[1.5]{Haefliger1966a}. {{endthm}} {{beginthm|Theorem}}\label{thm:hclink} (a) \cite[Theorem 1.3]{Haefliger1966a} The Haefliger link sequence is exact. (b) \cite{Crowley&Ferry&Skopenkov2011} The map $\lambda\otimes\Qq \colon E^m_{PL}(S^{(n)})\otimes\Qq \to \ker(w\otimes\Qq)$ is an isomorphism. {{endthm}} Part (b) follows because $\mu \otimes \Qq = 0$ \cite[Lemma 1.3]{Crowley&Ferry&Skopenkov2011}, and so the link Haefliger sequence splits into short exact sequences.
$-Brunnian links. Haefliger's description of$L_{(n)}^{(k)}$works equally well in the PL-category. Thus as a consequence of \cite[Theorem 1.3]{Haefliger1966a} and Zeeman unknotting -cite, there is an isomorphism$E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$: see \cite[Section 1.5]{Crowley&Ferry&Skopenkov2011}. {{endrem}} --> In \cite[$\SS^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m for $m-3\ge n_i$$m-3\ge n_i$.

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].

.2, $\SS^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m for $m-3\ge n_i$$m-3\ge n_i$. For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\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. ## 3 The linking coefficient 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$$\S$2.5], [Skopenkov2016k]. ## 8 References .3]{Crowley&Ferry&Skopenkov2011} one can find necessary and sufficient conditions on$p$and$q$when$E^m_{PL}(S^p\sqcup S^q)$is finite, as well as an effective procedure for computing the rank of the group$E^m_{PL}(S^{(n)})$. For more results related to high codimension links we refer the reader to \cite{Skopenkov2009}, \cite{Avvakumov2016}, \cite[$\S.5]{Skopenkov2015a}, \cite{Skopenkov2016k}.
== References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m for $m-3\ge n_i$$m-3\ge n_i$.

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].