High codimension links

This page has been accepted for publication in the Bulletin of the Manifold Atlas.

This page has not been refereed. The information given here might be incomplete or provisional.

1 Introduction

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

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$${{Stub}} == Introduction == <wikitex>; This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings. For a [[High_codimension_embeddings#Introduction|general introduction to embeddings]] as well as the [[High_codimension_embeddings#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\S$1, $\S$$\S$2].

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ 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].

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

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

2 Examples

Recall that for each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction $m\ge2n+2$$m\ge2n+2$ is sharp for non-connected manifolds.

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

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

• either $\partial D^{n+1}\times0$$\partial D^{n+1}\times0$ and $0\times\partial D^{n+1}$$0\times\partial D^{n+1}$ in $\partial(D^{n+1}\times D^{n+1})$$\partial(D^{n+1}\times D^{n+1})$;

• or given by equations:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$$

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:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$

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

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

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

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

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

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

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$$m\ge q+3$, the complement $S^m-fS^q$$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

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

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]

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

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

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

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\ge3n+4$$2m\ge3n+4$ and $n$$n$-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:

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

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 Two component links of spheres beyond the metastable range

Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. For some information about this group see [Skopenkov2006, $\S$$\S$3.3].

The Haefliger Theorem 6.1. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$

(b) [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

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

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 6.2. (a) The Haefliger Theorem 6.1(b) implies that for $l\ge2$$l\ge2$ we have an isomorphism

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

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

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

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 6.1(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].

7 General links of spheres beyond the metastable range

For integers $n_1, \dots, n_s$$n_1, \dots, n_s$ with $m \geq n_i + 3$$m \geq n_i + 3$ set $k_i = m-n_i-1$$k_i = m-n_i-1$ and denote $(n) = (n_1, \dots n_s)$$(n) = (n_1, \dots n_s)$ and $(k) = (k_1, \dots, k_s)$$(k) = (k_1, \dots, k_s)$. We call the kernel of the homomorphism

$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$ $$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$$

the group of $1$$1$-Brunnian links and denote it $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$. Evidently $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$. In this subsection we describe an exact sequence of Haefliger [Haefliger1966a] which for fixed $(k)$$(k)$ places the groups of $1$$1$-Brunnian links $\Lambda_{(n)}^{(k)}$$\Lambda_{(n)}^{(k)}$ is an exact sequence.

By general position, the complement $C_f$$C_f$ of an embedding $f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$$f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$ is simply connected. By Alexander duality $C_f$$C_f$ has the homology of the wedge of sphers $\vee_{i=1}^s S^{k_i}$$\vee_{i=1}^s S^{k_i}$ and so by Whitehead's Theoorem there is a homotopy equivalence $C_f \simeq \vee_{i=1}^s S^{k_i}$$C_f \simeq \vee_{i=1}^s S^{k_i}$. For each $j$$j$, $1 \leq j \leq n$$1 \leq j \leq n$ there are natural homomorphisms $p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$$p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$ obtained by projecting to the $j$$j$-component of the wedge. Following Heafliger, we define the groups

Tex syntax error

$$\displaystyle \Lambda_{n_j}^{(k)} := {\rm Ker}\left(p_j \colon \pi_{n_j}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \pi_{n_j}(S^{k_j}) \right) \quad \text{and} \quad \Lambda_{(n)}^{(k)} : = \bigoplus_{j=1}^s \Lambda_{n_j}^{(k)}$$

and

Tex syntax error

$$\displaystyle \Pi_{m-1}^{(k)} := {\rm Ker}\left( \bigoplus_{i=1}^s p_j \colon \pi_{m_1}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \bigoplus_{i=1}^s \pi_{m-1}(S^{k_j}) \right).$$

For $w_j \in \pi_{n_j}(S^{n_j})$$w_j \in \pi_{n_j}(S^{n_j})$ the class of the identity, taking the Whitehead product with $w_j$$w_j$ defines a homomorphism $w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$$w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$ and we define $w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$$w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$.

Theorem 7.1. (a) [Haefliger1966a, Theorem 1.3] Let $(n-1) = (n_1 -1, \dots, n_s-1)$$(n-1) = (n_1 -1, \dots, n_s-1)$. There are homomorphisms $\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$$\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$ and $\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$$\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$ which fit into a long exact exact sequence of abelian groups

$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$ $$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$$

(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers $\Qq$$\Qq$,

Tex syntax error

$\mu \otimes {\rm Id}_{\Qq} = 0$, and so the Haefliger sequence in (a) above splits into short exact sequencs

Tex syntax error

$$\displaystyle 0 \to L_{(n)}^{(k)} \otimes \Qq \xrightarrow{~\lambda \otimes {\rm Id}_{\Qq}~} \Lambda_{(n)}^{(k)} \otimes \Qq \xrightarrow{~w \otimes {\rm Id}_{\Qq}~} \Pi_{m-2}^{(k)} \otimes \Qq \to 0.$$

Remark 7.2. Here is some more information about $1$$1$-Brunnian links.

1. Haefliger's description of $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$ works equally well in the PL-category. Thus as a consequence of [Haefliger1966a, Theorem 1.3] and Zeeman unknotting -cite, there is an isomorphism $E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$$E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$: see [Crowley&Ferry&Skopenkov2011, Section 1.5].
2. An effective procedure for computing the rank of the group $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ can be found in [Crowley&Ferry&Skopenkov2011, Section 1.3].
3. Necessary and sufficient conditions on $p$$p$ and $q$$q$ for when $E^m(S^p \sqcup S^q)$$E^m(S^p \sqcup S^q)$ is finite can be found in [Crowley&Ferry&Skopenkov2011, Section 1.2].

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

8 References

• [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
• [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
• [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
• [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069

, $\S]{Skopenkov2016c}. Although $S^{n_1}\sqcup\ldots\sqcup S^{n_s}$ is not a manifold when $n_1,\ldots,n_s$ are not all equal, embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ and isotopy between such embeddings are defined analogously to [[Isotopy|the case of manifolds]]. Denote by $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ the set of embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ up to isotopy. A component-wise version of [[High_codimension_embeddings#Embedded_connected_sum|embedded connected sum]] \cite[$\S]{Skopenkov2016c} defines a commutative group structure on the set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ 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}. The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem \ref{lkhaze} below): $$\begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}$$ == Examples == ; [[High_codimension_embeddings:_classification#Unknotting_theorems|Recall that]] for each $n$-manifold $N$ and $m\ge2n+2$, any two embeddings $N\to\Rr^m$ are isotopic \cite[General Position Theorem 3.1]{Skopenkov2016c}. The following example shows that the restriction $m\ge2n+2$ is sharp for non-connected manifolds. {{beginthm|Example|(The Hopf Link)}}\label{hopf} For each $n$ there is an embedding $S^n\sqcup S^n\to\Rr^{2n+1}$ which is not isotopic to the standard embedding. {{endthm}} For $n=1$ the Hopf link is shown in \cite[Figure 2.1.a]{Skopenkov2006}. For arbitrary $n$ (including $n=1$) the image of the Hopf link is the union of two $n$-spheres: * either $\partial D^{n+1}\times0$ and $\S$1, $\S$$\S$2].

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ 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].

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

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

2 Examples

Recall that for each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction $m\ge2n+2$$m\ge2n+2$ is sharp for non-connected manifolds.

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

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

• either $\partial D^{n+1}\times0$$\partial D^{n+1}\times0$ and $0\times\partial D^{n+1}$$0\times\partial D^{n+1}$ in $\partial(D^{n+1}\times D^{n+1})$$\partial(D^{n+1}\times D^{n+1})$;

• or given by equations:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$$

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:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$

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

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

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

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

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

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

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$$m\ge q+3$, the complement $S^m-fS^q$$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

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

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]

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

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

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

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\ge3n+4$$2m\ge3n+4$ and $n$$n$-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:

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

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 Two component links of spheres beyond the metastable range

Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. For some information about this group see [Skopenkov2006, $\S$$\S$3.3].

The Haefliger Theorem 6.1. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$

(b) [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

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

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 6.2. (a) The Haefliger Theorem 6.1(b) implies that for $l\ge2$$l\ge2$ we have an isomorphism

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

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

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

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 6.1(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].

7 General links of spheres beyond the metastable range

For integers $n_1, \dots, n_s$$n_1, \dots, n_s$ with $m \geq n_i + 3$$m \geq n_i + 3$ set $k_i = m-n_i-1$$k_i = m-n_i-1$ and denote $(n) = (n_1, \dots n_s)$$(n) = (n_1, \dots n_s)$ and $(k) = (k_1, \dots, k_s)$$(k) = (k_1, \dots, k_s)$. We call the kernel of the homomorphism

$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$ $$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$$

the group of $1$$1$-Brunnian links and denote it $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$. Evidently $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$. In this subsection we describe an exact sequence of Haefliger [Haefliger1966a] which for fixed $(k)$$(k)$ places the groups of $1$$1$-Brunnian links $\Lambda_{(n)}^{(k)}$$\Lambda_{(n)}^{(k)}$ is an exact sequence.

By general position, the complement $C_f$$C_f$ of an embedding $f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$$f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$ is simply connected. By Alexander duality $C_f$$C_f$ has the homology of the wedge of sphers $\vee_{i=1}^s S^{k_i}$$\vee_{i=1}^s S^{k_i}$ and so by Whitehead's Theoorem there is a homotopy equivalence $C_f \simeq \vee_{i=1}^s S^{k_i}$$C_f \simeq \vee_{i=1}^s S^{k_i}$. For each $j$$j$, $1 \leq j \leq n$$1 \leq j \leq n$ there are natural homomorphisms $p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$$p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$ obtained by projecting to the $j$$j$-component of the wedge. Following Heafliger, we define the groups

Tex syntax error

$$\displaystyle \Lambda_{n_j}^{(k)} := {\rm Ker}\left(p_j \colon \pi_{n_j}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \pi_{n_j}(S^{k_j}) \right) \quad \text{and} \quad \Lambda_{(n)}^{(k)} : = \bigoplus_{j=1}^s \Lambda_{n_j}^{(k)}$$

and

Tex syntax error

$$\displaystyle \Pi_{m-1}^{(k)} := {\rm Ker}\left( \bigoplus_{i=1}^s p_j \colon \pi_{m_1}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \bigoplus_{i=1}^s \pi_{m-1}(S^{k_j}) \right).$$

For $w_j \in \pi_{n_j}(S^{n_j})$$w_j \in \pi_{n_j}(S^{n_j})$ the class of the identity, taking the Whitehead product with $w_j$$w_j$ defines a homomorphism $w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$$w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$ and we define $w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$$w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$.

Theorem 7.1. (a) [Haefliger1966a, Theorem 1.3] Let $(n-1) = (n_1 -1, \dots, n_s-1)$$(n-1) = (n_1 -1, \dots, n_s-1)$. There are homomorphisms $\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$$\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$ and $\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$$\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$ which fit into a long exact exact sequence of abelian groups

$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$ $$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$$

(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers $\Qq$$\Qq$,

Tex syntax error

$\mu \otimes {\rm Id}_{\Qq} = 0$, and so the Haefliger sequence in (a) above splits into short exact sequencs

Tex syntax error

$$\displaystyle 0 \to L_{(n)}^{(k)} \otimes \Qq \xrightarrow{~\lambda \otimes {\rm Id}_{\Qq}~} \Lambda_{(n)}^{(k)} \otimes \Qq \xrightarrow{~w \otimes {\rm Id}_{\Qq}~} \Pi_{m-2}^{(k)} \otimes \Qq \to 0.$$

Remark 7.2. Here is some more information about $1$$1$-Brunnian links.

1. Haefliger's description of $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$ works equally well in the PL-category. Thus as a consequence of [Haefliger1966a, Theorem 1.3] and Zeeman unknotting -cite, there is an isomorphism $E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$$E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$: see [Crowley&Ferry&Skopenkov2011, Section 1.5].
2. An effective procedure for computing the rank of the group $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ can be found in [Crowley&Ferry&Skopenkov2011, Section 1.3].
3. Necessary and sufficient conditions on $p$$p$ and $q$$q$ for when $E^m(S^p \sqcup S^q)$$E^m(S^p \sqcup S^q)$ is finite can be found in [Crowley&Ferry&Skopenkov2011, Section 1.2].

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

8 References

• [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
• [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
• [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
• [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069

\times\partial D^{n+1}$ in $\partial(D^{n+1}\times D^{n+1})$; * or given by equations: $$\left\{\begin{array}{c} x_1=\dots=x_n=0\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$$ This embedding is distinguished from the standard embedding by [[#Invariants|the linking coefficient]] ($\S$\ref{s:inv}). Analogously for each $p,q$ one constructs an embedding $S^p\sqcup S^q\to\Rr^{p+q+1}$ which is not isotopic to the standard embedding. The image is the union of two spheres: * either $\partial D^{p+1}\times0$ and $\S$1, $\S$$\S$2].

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ 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].

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

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

2 Examples

Recall that for each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction $m\ge2n+2$$m\ge2n+2$ is sharp for non-connected manifolds.

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

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

• either $\partial D^{n+1}\times0$$\partial D^{n+1}\times0$ and $0\times\partial D^{n+1}$$0\times\partial D^{n+1}$ in $\partial(D^{n+1}\times D^{n+1})$$\partial(D^{n+1}\times D^{n+1})$;

• or given by equations:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$$

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:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$

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

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

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

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

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

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

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$$m\ge q+3$, the complement $S^m-fS^q$$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

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

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]

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

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

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

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\ge3n+4$$2m\ge3n+4$ and $n$$n$-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:

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

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 Two component links of spheres beyond the metastable range

Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. For some information about this group see [Skopenkov2006, $\S$$\S$3.3].

The Haefliger Theorem 6.1. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$

(b) [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

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

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 6.2. (a) The Haefliger Theorem 6.1(b) implies that for $l\ge2$$l\ge2$ we have an isomorphism

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

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

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

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 6.1(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].

7 General links of spheres beyond the metastable range

For integers $n_1, \dots, n_s$$n_1, \dots, n_s$ with $m \geq n_i + 3$$m \geq n_i + 3$ set $k_i = m-n_i-1$$k_i = m-n_i-1$ and denote $(n) = (n_1, \dots n_s)$$(n) = (n_1, \dots n_s)$ and $(k) = (k_1, \dots, k_s)$$(k) = (k_1, \dots, k_s)$. We call the kernel of the homomorphism

$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$ $$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$$

the group of $1$$1$-Brunnian links and denote it $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$. Evidently $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$. In this subsection we describe an exact sequence of Haefliger [Haefliger1966a] which for fixed $(k)$$(k)$ places the groups of $1$$1$-Brunnian links $\Lambda_{(n)}^{(k)}$$\Lambda_{(n)}^{(k)}$ is an exact sequence.

By general position, the complement $C_f$$C_f$ of an embedding $f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$$f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$ is simply connected. By Alexander duality $C_f$$C_f$ has the homology of the wedge of sphers $\vee_{i=1}^s S^{k_i}$$\vee_{i=1}^s S^{k_i}$ and so by Whitehead's Theoorem there is a homotopy equivalence $C_f \simeq \vee_{i=1}^s S^{k_i}$$C_f \simeq \vee_{i=1}^s S^{k_i}$. For each $j$$j$, $1 \leq j \leq n$$1 \leq j \leq n$ there are natural homomorphisms $p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$$p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$ obtained by projecting to the $j$$j$-component of the wedge. Following Heafliger, we define the groups

Tex syntax error

$$\displaystyle \Lambda_{n_j}^{(k)} := {\rm Ker}\left(p_j \colon \pi_{n_j}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \pi_{n_j}(S^{k_j}) \right) \quad \text{and} \quad \Lambda_{(n)}^{(k)} : = \bigoplus_{j=1}^s \Lambda_{n_j}^{(k)}$$

and

Tex syntax error

$$\displaystyle \Pi_{m-1}^{(k)} := {\rm Ker}\left( \bigoplus_{i=1}^s p_j \colon \pi_{m_1}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \bigoplus_{i=1}^s \pi_{m-1}(S^{k_j}) \right).$$

For $w_j \in \pi_{n_j}(S^{n_j})$$w_j \in \pi_{n_j}(S^{n_j})$ the class of the identity, taking the Whitehead product with $w_j$$w_j$ defines a homomorphism $w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$$w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$ and we define $w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$$w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$.

Theorem 7.1. (a) [Haefliger1966a, Theorem 1.3] Let $(n-1) = (n_1 -1, \dots, n_s-1)$$(n-1) = (n_1 -1, \dots, n_s-1)$. There are homomorphisms $\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$$\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$ and $\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$$\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$ which fit into a long exact exact sequence of abelian groups

$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$ $$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$$

(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers $\Qq$$\Qq$,

Tex syntax error

$\mu \otimes {\rm Id}_{\Qq} = 0$, and so the Haefliger sequence in (a) above splits into short exact sequencs

Tex syntax error

$$\displaystyle 0 \to L_{(n)}^{(k)} \otimes \Qq \xrightarrow{~\lambda \otimes {\rm Id}_{\Qq}~} \Lambda_{(n)}^{(k)} \otimes \Qq \xrightarrow{~w \otimes {\rm Id}_{\Qq}~} \Pi_{m-2}^{(k)} \otimes \Qq \to 0.$$

Remark 7.2. Here is some more information about $1$$1$-Brunnian links.

1. Haefliger's description of $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$ works equally well in the PL-category. Thus as a consequence of [Haefliger1966a, Theorem 1.3] and Zeeman unknotting -cite, there is an isomorphism $E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$$E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$: see [Crowley&Ferry&Skopenkov2011, Section 1.5].
2. An effective procedure for computing the rank of the group $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ can be found in [Crowley&Ferry&Skopenkov2011, Section 1.3].
3. Necessary and sufficient conditions on $p$$p$ and $q$$q$ for when $E^m(S^p \sqcup S^q)$$E^m(S^p \sqcup S^q)$ is finite can be found in [Crowley&Ferry&Skopenkov2011, Section 1.2].

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

8 References

• [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
• [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
• [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
• [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069

\times\partial D^{q+1}$ in $\partial(D^{p+1}\times D^{q+1})$. * or given by equations: $$\left\{\begin{array}{c} x_1=\dots=x_p=0\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$ This embedding is distinguished from the standard embedding also by [[#Invariants|the linking coefficient]] ($\S$\ref{s:inv}). {{beginthm|Definition|(The Zeeman map)}}\label{dz} We define a map $$\zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.$$ Denote by $i_{m,p}:S^p\to S^m$ the equatorial inclusion. For a map $x:S^p\to S^{m-q-1}$ representing an element of $\pi_p(S^{m-q-1})$ let $$\overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,$$ where $i$ is the standard embedding \cite[Figure 3.2]{Skopenkov2006}. We have $\overline\zeta_x(S^p)\cap i_{m,q}(S^q)\subset i(S^{m-q-1}\times S^p)\cap i(0_{m-q}\times S^q) =\emptyset$. Let $\zeta[x]:=[\overline\zeta_x\sqcup i_{m,q}]$. {{endthm}} One can easily check that $\zeta$ is well-defined and is a homomorphism. == 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 $m\ge q+3$, 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 \S1, $\S$$\S$2].

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ 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].

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

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

2 Examples

Recall that for each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction $m\ge2n+2$$m\ge2n+2$ is sharp for non-connected manifolds.

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

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

• either $\partial D^{n+1}\times0$$\partial D^{n+1}\times0$ and $0\times\partial D^{n+1}$$0\times\partial D^{n+1}$ in $\partial(D^{n+1}\times D^{n+1})$$\partial(D^{n+1}\times D^{n+1})$;

• or given by equations:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$$

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:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$

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

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

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

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

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

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

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$$m\ge q+3$, the complement $S^m-fS^q$$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

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

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]

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

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

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

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\ge3n+4$$2m\ge3n+4$ and $n$$n$-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:

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

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 Two component links of spheres beyond the metastable range

Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. For some information about this group see [Skopenkov2006, $\S$$\S$3.3].

The Haefliger Theorem 6.1. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$

(b) [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

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

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 6.2. (a) The Haefliger Theorem 6.1(b) implies that for $l\ge2$$l\ge2$ we have an isomorphism

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

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

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

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 6.1(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].

7 General links of spheres beyond the metastable range

For integers $n_1, \dots, n_s$$n_1, \dots, n_s$ with $m \geq n_i + 3$$m \geq n_i + 3$ set $k_i = m-n_i-1$$k_i = m-n_i-1$ and denote $(n) = (n_1, \dots n_s)$$(n) = (n_1, \dots n_s)$ and $(k) = (k_1, \dots, k_s)$$(k) = (k_1, \dots, k_s)$. We call the kernel of the homomorphism

$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$ $$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$$

the group of $1$$1$-Brunnian links and denote it $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$. Evidently $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$. In this subsection we describe an exact sequence of Haefliger [Haefliger1966a] which for fixed $(k)$$(k)$ places the groups of $1$$1$-Brunnian links $\Lambda_{(n)}^{(k)}$$\Lambda_{(n)}^{(k)}$ is an exact sequence.

By general position, the complement $C_f$$C_f$ of an embedding $f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$$f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$ is simply connected. By Alexander duality $C_f$$C_f$ has the homology of the wedge of sphers $\vee_{i=1}^s S^{k_i}$$\vee_{i=1}^s S^{k_i}$ and so by Whitehead's Theoorem there is a homotopy equivalence $C_f \simeq \vee_{i=1}^s S^{k_i}$$C_f \simeq \vee_{i=1}^s S^{k_i}$. For each $j$$j$, $1 \leq j \leq n$$1 \leq j \leq n$ there are natural homomorphisms $p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$$p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$ obtained by projecting to the $j$$j$-component of the wedge. Following Heafliger, we define the groups

Tex syntax error

$$\displaystyle \Lambda_{n_j}^{(k)} := {\rm Ker}\left(p_j \colon \pi_{n_j}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \pi_{n_j}(S^{k_j}) \right) \quad \text{and} \quad \Lambda_{(n)}^{(k)} : = \bigoplus_{j=1}^s \Lambda_{n_j}^{(k)}$$

and

Tex syntax error

$$\displaystyle \Pi_{m-1}^{(k)} := {\rm Ker}\left( \bigoplus_{i=1}^s p_j \colon \pi_{m_1}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \bigoplus_{i=1}^s \pi_{m-1}(S^{k_j}) \right).$$

For $w_j \in \pi_{n_j}(S^{n_j})$$w_j \in \pi_{n_j}(S^{n_j})$ the class of the identity, taking the Whitehead product with $w_j$$w_j$ defines a homomorphism $w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$$w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$ and we define $w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$$w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$.

Theorem 7.1. (a) [Haefliger1966a, Theorem 1.3] Let $(n-1) = (n_1 -1, \dots, n_s-1)$$(n-1) = (n_1 -1, \dots, n_s-1)$. There are homomorphisms $\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$$\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$ and $\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$$\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$ which fit into a long exact exact sequence of abelian groups

$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$ $$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$$

(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers $\Qq$$\Qq$,

Tex syntax error

$\mu \otimes {\rm Id}_{\Qq} = 0$, and so the Haefliger sequence in (a) above splits into short exact sequencs

Tex syntax error

$$\displaystyle 0 \to L_{(n)}^{(k)} \otimes \Qq \xrightarrow{~\lambda \otimes {\rm Id}_{\Qq}~} \Lambda_{(n)}^{(k)} \otimes \Qq \xrightarrow{~w \otimes {\rm Id}_{\Qq}~} \Pi_{m-2}^{(k)} \otimes \Qq \to 0.$$

Remark 7.2. Here is some more information about $1$$1$-Brunnian links.

1. Haefliger's description of $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$ works equally well in the PL-category. Thus as a consequence of [Haefliger1966a, Theorem 1.3] and Zeeman unknotting -cite, there is an isomorphism $E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$$E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$: see [Crowley&Ferry&Skopenkov2011, Section 1.5].
2. An effective procedure for computing the rank of the group $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ can be found in [Crowley&Ferry&Skopenkov2011, Section 1.3].
3. Necessary and sufficient conditions on $p$$p$ and $q$$q$ for when $E^m(S^p \sqcup S^q)$$E^m(S^p \sqcup S^q)$ is finite can be found in [Crowley&Ferry&Skopenkov2011, Section 1.2].

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

8 References

• [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
• [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
• [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
• [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069

\le p\le q$, then both $\lambda$ and $\zeta$ are isomorphisms for $m\ge\frac{3q}2+2$ in the smooth category. (PL) If \S1, $\S$$\S$2].

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ 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].

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

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

2 Examples

Recall that for each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction $m\ge2n+2$$m\ge2n+2$ is sharp for non-connected manifolds.

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

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

• either $\partial D^{n+1}\times0$$\partial D^{n+1}\times0$ and $0\times\partial D^{n+1}$$0\times\partial D^{n+1}$ in $\partial(D^{n+1}\times D^{n+1})$$\partial(D^{n+1}\times D^{n+1})$;

• or given by equations:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$$

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:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$

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

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

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

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

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

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

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$$m\ge q+3$, the complement $S^m-fS^q$$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

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

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]

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

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

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

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\ge3n+4$$2m\ge3n+4$ and $n$$n$-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:

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

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 Two component links of spheres beyond the metastable range

Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. For some information about this group see [Skopenkov2006, $\S$$\S$3.3].

The Haefliger Theorem 6.1. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$

(b) [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

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

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 6.2. (a) The Haefliger Theorem 6.1(b) implies that for $l\ge2$$l\ge2$ we have an isomorphism

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

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

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

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 6.1(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].

7 General links of spheres beyond the metastable range

For integers $n_1, \dots, n_s$$n_1, \dots, n_s$ with $m \geq n_i + 3$$m \geq n_i + 3$ set $k_i = m-n_i-1$$k_i = m-n_i-1$ and denote $(n) = (n_1, \dots n_s)$$(n) = (n_1, \dots n_s)$ and $(k) = (k_1, \dots, k_s)$$(k) = (k_1, \dots, k_s)$. We call the kernel of the homomorphism

$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$ $$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$$

the group of $1$$1$-Brunnian links and denote it $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$. Evidently $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$. In this subsection we describe an exact sequence of Haefliger [Haefliger1966a] which for fixed $(k)$$(k)$ places the groups of $1$$1$-Brunnian links $\Lambda_{(n)}^{(k)}$$\Lambda_{(n)}^{(k)}$ is an exact sequence.

By general position, the complement $C_f$$C_f$ of an embedding $f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$$f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$ is simply connected. By Alexander duality $C_f$$C_f$ has the homology of the wedge of sphers $\vee_{i=1}^s S^{k_i}$$\vee_{i=1}^s S^{k_i}$ and so by Whitehead's Theoorem there is a homotopy equivalence $C_f \simeq \vee_{i=1}^s S^{k_i}$$C_f \simeq \vee_{i=1}^s S^{k_i}$. For each $j$$j$, $1 \leq j \leq n$$1 \leq j \leq n$ there are natural homomorphisms $p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$$p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$ obtained by projecting to the $j$$j$-component of the wedge. Following Heafliger, we define the groups

Tex syntax error

$$\displaystyle \Lambda_{n_j}^{(k)} := {\rm Ker}\left(p_j \colon \pi_{n_j}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \pi_{n_j}(S^{k_j}) \right) \quad \text{and} \quad \Lambda_{(n)}^{(k)} : = \bigoplus_{j=1}^s \Lambda_{n_j}^{(k)}$$

and

Tex syntax error

$$\displaystyle \Pi_{m-1}^{(k)} := {\rm Ker}\left( \bigoplus_{i=1}^s p_j \colon \pi_{m_1}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \bigoplus_{i=1}^s \pi_{m-1}(S^{k_j}) \right).$$

For $w_j \in \pi_{n_j}(S^{n_j})$$w_j \in \pi_{n_j}(S^{n_j})$ the class of the identity, taking the Whitehead product with $w_j$$w_j$ defines a homomorphism $w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$$w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$ and we define $w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$$w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$.

Theorem 7.1. (a) [Haefliger1966a, Theorem 1.3] Let $(n-1) = (n_1 -1, \dots, n_s-1)$$(n-1) = (n_1 -1, \dots, n_s-1)$. There are homomorphisms $\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$$\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$ and $\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$$\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$ which fit into a long exact exact sequence of abelian groups

$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$ $$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$$

(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers $\Qq$$\Qq$,

Tex syntax error

$\mu \otimes {\rm Id}_{\Qq} = 0$, and so the Haefliger sequence in (a) above splits into short exact sequencs

Tex syntax error

$$\displaystyle 0 \to L_{(n)}^{(k)} \otimes \Qq \xrightarrow{~\lambda \otimes {\rm Id}_{\Qq}~} \Lambda_{(n)}^{(k)} \otimes \Qq \xrightarrow{~w \otimes {\rm Id}_{\Qq}~} \Pi_{m-2}^{(k)} \otimes \Qq \to 0.$$

Remark 7.2. Here is some more information about $1$$1$-Brunnian links.

1. Haefliger's description of $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$ works equally well in the PL-category. Thus as a consequence of [Haefliger1966a, Theorem 1.3] and Zeeman unknotting -cite, there is an isomorphism $E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$$E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$: see [Crowley&Ferry&Skopenkov2011, Section 1.5].
2. An effective procedure for computing the rank of the group $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ can be found in [Crowley&Ferry&Skopenkov2011, Section 1.3].
3. Necessary and sufficient conditions on $p$$p$ and $q$$q$ for when $E^m(S^p \sqcup S^q)$$E^m(S^p \sqcup S^q)$ is finite can be found in [Crowley&Ferry&Skopenkov2011, Section 1.2].

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

8 References

• [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
• [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
• [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
• [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069

\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\ge3n+4$ and $n$-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 [[High_codimension_embeddings#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. == Two component links of spheres beyond the metastable range == ; Let $C_q^{m-q}:=E^m_D(S^q)$. For [[Knots,_i.e._embeddings_of_spheres|some information about this group]] see \cite[$\S.3]{Skopenkov2006}. {{beginthm|The Haefliger Theorem}}\label{belmethae} (a) \cite{Haefliger1966a} If $p,q\le m-3$, then $$E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$ (b) \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 $l\ge2$ we have an isomorphism $$\lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{(l)}.$$ (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}. {{endthm}} == General links of spheres beyond the metastable range == ; For integers $n_1, \dots, n_s$ with $m \geq n_i + 3$ set $k_i = m-n_i-1$ and denote $(n) = (n_1, \dots n_s)$ and $(k) = (k_1, \dots, k_s)$. We call the kernel of the homomorphism $$ E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i} $$ the group of ''\S1, $\S$$\S$2].

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ 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].

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

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

2 Examples

Recall that for each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction $m\ge2n+2$$m\ge2n+2$ is sharp for non-connected manifolds.

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

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

• either $\partial D^{n+1}\times0$$\partial D^{n+1}\times0$ and $0\times\partial D^{n+1}$$0\times\partial D^{n+1}$ in $\partial(D^{n+1}\times D^{n+1})$$\partial(D^{n+1}\times D^{n+1})$;

• or given by equations:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$$

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:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$

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

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

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

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

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

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

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$$m\ge q+3$, the complement $S^m-fS^q$$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

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

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]

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

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

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

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\ge3n+4$$2m\ge3n+4$ and $n$$n$-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:

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

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 Two component links of spheres beyond the metastable range

Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. For some information about this group see [Skopenkov2006, $\S$$\S$3.3].

The Haefliger Theorem 6.1. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$

(b) [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

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

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 6.2. (a) The Haefliger Theorem 6.1(b) implies that for $l\ge2$$l\ge2$ we have an isomorphism

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

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

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

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 6.1(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].

7 General links of spheres beyond the metastable range

For integers $n_1, \dots, n_s$$n_1, \dots, n_s$ with $m \geq n_i + 3$$m \geq n_i + 3$ set $k_i = m-n_i-1$$k_i = m-n_i-1$ and denote $(n) = (n_1, \dots n_s)$$(n) = (n_1, \dots n_s)$ and $(k) = (k_1, \dots, k_s)$$(k) = (k_1, \dots, k_s)$. We call the kernel of the homomorphism

$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$ $$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$$

the group of $1$$1$-Brunnian links and denote it $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$. Evidently $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$. In this subsection we describe an exact sequence of Haefliger [Haefliger1966a] which for fixed $(k)$$(k)$ places the groups of $1$$1$-Brunnian links $\Lambda_{(n)}^{(k)}$$\Lambda_{(n)}^{(k)}$ is an exact sequence.

By general position, the complement $C_f$$C_f$ of an embedding $f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$$f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$ is simply connected. By Alexander duality $C_f$$C_f$ has the homology of the wedge of sphers $\vee_{i=1}^s S^{k_i}$$\vee_{i=1}^s S^{k_i}$ and so by Whitehead's Theoorem there is a homotopy equivalence $C_f \simeq \vee_{i=1}^s S^{k_i}$$C_f \simeq \vee_{i=1}^s S^{k_i}$. For each $j$$j$, $1 \leq j \leq n$$1 \leq j \leq n$ there are natural homomorphisms $p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$$p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$ obtained by projecting to the $j$$j$-component of the wedge. Following Heafliger, we define the groups

Tex syntax error

$$\displaystyle \Lambda_{n_j}^{(k)} := {\rm Ker}\left(p_j \colon \pi_{n_j}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \pi_{n_j}(S^{k_j}) \right) \quad \text{and} \quad \Lambda_{(n)}^{(k)} : = \bigoplus_{j=1}^s \Lambda_{n_j}^{(k)}$$

and

Tex syntax error

$$\displaystyle \Pi_{m-1}^{(k)} := {\rm Ker}\left( \bigoplus_{i=1}^s p_j \colon \pi_{m_1}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \bigoplus_{i=1}^s \pi_{m-1}(S^{k_j}) \right).$$

For $w_j \in \pi_{n_j}(S^{n_j})$$w_j \in \pi_{n_j}(S^{n_j})$ the class of the identity, taking the Whitehead product with $w_j$$w_j$ defines a homomorphism $w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$$w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$ and we define $w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$$w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$.

Theorem 7.1. (a) [Haefliger1966a, Theorem 1.3] Let $(n-1) = (n_1 -1, \dots, n_s-1)$$(n-1) = (n_1 -1, \dots, n_s-1)$. There are homomorphisms $\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$$\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$ and $\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$$\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$ which fit into a long exact exact sequence of abelian groups

$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$ $$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$$

(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers $\Qq$$\Qq$,

Tex syntax error

$\mu \otimes {\rm Id}_{\Qq} = 0$, and so the Haefliger sequence in (a) above splits into short exact sequencs

Tex syntax error

$$\displaystyle 0 \to L_{(n)}^{(k)} \otimes \Qq \xrightarrow{~\lambda \otimes {\rm Id}_{\Qq}~} \Lambda_{(n)}^{(k)} \otimes \Qq \xrightarrow{~w \otimes {\rm Id}_{\Qq}~} \Pi_{m-2}^{(k)} \otimes \Qq \to 0.$$

Remark 7.2. Here is some more information about $1$$1$-Brunnian links.

1. Haefliger's description of $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$ works equally well in the PL-category. Thus as a consequence of [Haefliger1966a, Theorem 1.3] and Zeeman unknotting -cite, there is an isomorphism $E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$$E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$: see [Crowley&Ferry&Skopenkov2011, Section 1.5].
2. An effective procedure for computing the rank of the group $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ can be found in [Crowley&Ferry&Skopenkov2011, Section 1.3].
3. Necessary and sufficient conditions on $p$$p$ and $q$$q$ for when $E^m(S^p \sqcup S^q)$$E^m(S^p \sqcup S^q)$ is finite can be found in [Crowley&Ferry&Skopenkov2011, Section 1.2].

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

8 References

• [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
• [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
• [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
• [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069

$-Brunnian'' links and denote it $L_{(n)}^{(k)}$. Evidently $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$. In this subsection we describe an exact sequence of Haefliger \cite{Haefliger1966a} which for fixed $(k)$ places the groups of \S1, $\S$$\S$2].

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ 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].

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

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

2 Examples

Recall that for each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction $m\ge2n+2$$m\ge2n+2$ is sharp for non-connected manifolds.

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

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

• either $\partial D^{n+1}\times0$$\partial D^{n+1}\times0$ and $0\times\partial D^{n+1}$$0\times\partial D^{n+1}$ in $\partial(D^{n+1}\times D^{n+1})$$\partial(D^{n+1}\times D^{n+1})$;

• or given by equations:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$$

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:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$

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

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

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

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

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

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

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$$m\ge q+3$, the complement $S^m-fS^q$$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

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

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]

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

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

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

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\ge3n+4$$2m\ge3n+4$ and $n$$n$-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:

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

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 Two component links of spheres beyond the metastable range

Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. For some information about this group see [Skopenkov2006, $\S$$\S$3.3].

The Haefliger Theorem 6.1. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$

(b) [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

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

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 6.2. (a) The Haefliger Theorem 6.1(b) implies that for $l\ge2$$l\ge2$ we have an isomorphism

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

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

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

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 6.1(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].

7 General links of spheres beyond the metastable range

For integers $n_1, \dots, n_s$$n_1, \dots, n_s$ with $m \geq n_i + 3$$m \geq n_i + 3$ set $k_i = m-n_i-1$$k_i = m-n_i-1$ and denote $(n) = (n_1, \dots n_s)$$(n) = (n_1, \dots n_s)$ and $(k) = (k_1, \dots, k_s)$$(k) = (k_1, \dots, k_s)$. We call the kernel of the homomorphism

$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$ $$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$$

the group of $1$$1$-Brunnian links and denote it $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$. Evidently $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$. In this subsection we describe an exact sequence of Haefliger [Haefliger1966a] which for fixed $(k)$$(k)$ places the groups of $1$$1$-Brunnian links $\Lambda_{(n)}^{(k)}$$\Lambda_{(n)}^{(k)}$ is an exact sequence.

By general position, the complement $C_f$$C_f$ of an embedding $f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$$f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$ is simply connected. By Alexander duality $C_f$$C_f$ has the homology of the wedge of sphers $\vee_{i=1}^s S^{k_i}$$\vee_{i=1}^s S^{k_i}$ and so by Whitehead's Theoorem there is a homotopy equivalence $C_f \simeq \vee_{i=1}^s S^{k_i}$$C_f \simeq \vee_{i=1}^s S^{k_i}$. For each $j$$j$, $1 \leq j \leq n$$1 \leq j \leq n$ there are natural homomorphisms $p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$$p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$ obtained by projecting to the $j$$j$-component of the wedge. Following Heafliger, we define the groups

Tex syntax error

$$\displaystyle \Lambda_{n_j}^{(k)} := {\rm Ker}\left(p_j \colon \pi_{n_j}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \pi_{n_j}(S^{k_j}) \right) \quad \text{and} \quad \Lambda_{(n)}^{(k)} : = \bigoplus_{j=1}^s \Lambda_{n_j}^{(k)}$$

and

Tex syntax error

$$\displaystyle \Pi_{m-1}^{(k)} := {\rm Ker}\left( \bigoplus_{i=1}^s p_j \colon \pi_{m_1}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \bigoplus_{i=1}^s \pi_{m-1}(S^{k_j}) \right).$$

For $w_j \in \pi_{n_j}(S^{n_j})$$w_j \in \pi_{n_j}(S^{n_j})$ the class of the identity, taking the Whitehead product with $w_j$$w_j$ defines a homomorphism $w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$$w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$ and we define $w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$$w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$.

Theorem 7.1. (a) [Haefliger1966a, Theorem 1.3] Let $(n-1) = (n_1 -1, \dots, n_s-1)$$(n-1) = (n_1 -1, \dots, n_s-1)$. There are homomorphisms $\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$$\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$ and $\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$$\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$ which fit into a long exact exact sequence of abelian groups

$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$ $$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$$

(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers $\Qq$$\Qq$,

Tex syntax error

$\mu \otimes {\rm Id}_{\Qq} = 0$, and so the Haefliger sequence in (a) above splits into short exact sequencs

Tex syntax error

$$\displaystyle 0 \to L_{(n)}^{(k)} \otimes \Qq \xrightarrow{~\lambda \otimes {\rm Id}_{\Qq}~} \Lambda_{(n)}^{(k)} \otimes \Qq \xrightarrow{~w \otimes {\rm Id}_{\Qq}~} \Pi_{m-2}^{(k)} \otimes \Qq \to 0.$$

Remark 7.2. Here is some more information about $1$$1$-Brunnian links.

1. Haefliger's description of $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$ works equally well in the PL-category. Thus as a consequence of [Haefliger1966a, Theorem 1.3] and Zeeman unknotting -cite, there is an isomorphism $E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$$E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$: see [Crowley&Ferry&Skopenkov2011, Section 1.5].
2. An effective procedure for computing the rank of the group $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ can be found in [Crowley&Ferry&Skopenkov2011, Section 1.3].
3. Necessary and sufficient conditions on $p$$p$ and $q$$q$ for when $E^m(S^p \sqcup S^q)$$E^m(S^p \sqcup S^q)$ is finite can be found in [Crowley&Ferry&Skopenkov2011, Section 1.2].

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

8 References

• [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
• [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
• [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
• [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069

$-Brunnian links $\Lambda_{(n)}^{(k)}$ is an exact sequence. By general position, the complement $C_f$ of an embedding $f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$ is simply connected. By Alexander duality $C_f$ has the homology of the wedge of sphers $\vee_{i=1}^s S^{k_i}$ and so by Whitehead's Theoorem there is a homotopy equivalence $C_f \simeq \vee_{i=1}^s S^{k_i}$. For each $j$, \S1, $\S$$\S$2].

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ 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].

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

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

2 Examples

Recall that for each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction $m\ge2n+2$$m\ge2n+2$ is sharp for non-connected manifolds.

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

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

• either $\partial D^{n+1}\times0$$\partial D^{n+1}\times0$ and $0\times\partial D^{n+1}$$0\times\partial D^{n+1}$ in $\partial(D^{n+1}\times D^{n+1})$$\partial(D^{n+1}\times D^{n+1})$;

• or given by equations:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$$

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:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$

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

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

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

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

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

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

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$$m\ge q+3$, the complement $S^m-fS^q$$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

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

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]

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

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

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

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\ge3n+4$$2m\ge3n+4$ and $n$$n$-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:

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

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 Two component links of spheres beyond the metastable range

Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. For some information about this group see [Skopenkov2006, $\S$$\S$3.3].

The Haefliger Theorem 6.1. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$

(b) [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

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

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 6.2. (a) The Haefliger Theorem 6.1(b) implies that for $l\ge2$$l\ge2$ we have an isomorphism

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

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

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

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 6.1(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].

7 General links of spheres beyond the metastable range

For integers $n_1, \dots, n_s$$n_1, \dots, n_s$ with $m \geq n_i + 3$$m \geq n_i + 3$ set $k_i = m-n_i-1$$k_i = m-n_i-1$ and denote $(n) = (n_1, \dots n_s)$$(n) = (n_1, \dots n_s)$ and $(k) = (k_1, \dots, k_s)$$(k) = (k_1, \dots, k_s)$. We call the kernel of the homomorphism

$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$ $$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$$

the group of $1$$1$-Brunnian links and denote it $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$. Evidently $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$. In this subsection we describe an exact sequence of Haefliger [Haefliger1966a] which for fixed $(k)$$(k)$ places the groups of $1$$1$-Brunnian links $\Lambda_{(n)}^{(k)}$$\Lambda_{(n)}^{(k)}$ is an exact sequence.

By general position, the complement $C_f$$C_f$ of an embedding $f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$$f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$ is simply connected. By Alexander duality $C_f$$C_f$ has the homology of the wedge of sphers $\vee_{i=1}^s S^{k_i}$$\vee_{i=1}^s S^{k_i}$ and so by Whitehead's Theoorem there is a homotopy equivalence $C_f \simeq \vee_{i=1}^s S^{k_i}$$C_f \simeq \vee_{i=1}^s S^{k_i}$. For each $j$$j$, $1 \leq j \leq n$$1 \leq j \leq n$ there are natural homomorphisms $p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$$p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$ obtained by projecting to the $j$$j$-component of the wedge. Following Heafliger, we define the groups

Tex syntax error

$$\displaystyle \Lambda_{n_j}^{(k)} := {\rm Ker}\left(p_j \colon \pi_{n_j}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \pi_{n_j}(S^{k_j}) \right) \quad \text{and} \quad \Lambda_{(n)}^{(k)} : = \bigoplus_{j=1}^s \Lambda_{n_j}^{(k)}$$

and

Tex syntax error

$$\displaystyle \Pi_{m-1}^{(k)} := {\rm Ker}\left( \bigoplus_{i=1}^s p_j \colon \pi_{m_1}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \bigoplus_{i=1}^s \pi_{m-1}(S^{k_j}) \right).$$

For $w_j \in \pi_{n_j}(S^{n_j})$$w_j \in \pi_{n_j}(S^{n_j})$ the class of the identity, taking the Whitehead product with $w_j$$w_j$ defines a homomorphism $w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$$w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$ and we define $w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$$w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$.

Theorem 7.1. (a) [Haefliger1966a, Theorem 1.3] Let $(n-1) = (n_1 -1, \dots, n_s-1)$$(n-1) = (n_1 -1, \dots, n_s-1)$. There are homomorphisms $\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$$\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$ and $\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$$\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$ which fit into a long exact exact sequence of abelian groups

$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$ $$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$$

(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers $\Qq$$\Qq$,

Tex syntax error

$\mu \otimes {\rm Id}_{\Qq} = 0$, and so the Haefliger sequence in (a) above splits into short exact sequencs

Tex syntax error

$$\displaystyle 0 \to L_{(n)}^{(k)} \otimes \Qq \xrightarrow{~\lambda \otimes {\rm Id}_{\Qq}~} \Lambda_{(n)}^{(k)} \otimes \Qq \xrightarrow{~w \otimes {\rm Id}_{\Qq}~} \Pi_{m-2}^{(k)} \otimes \Qq \to 0.$$

Remark 7.2. Here is some more information about $1$$1$-Brunnian links.

1. Haefliger's description of $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$ works equally well in the PL-category. Thus as a consequence of [Haefliger1966a, Theorem 1.3] and Zeeman unknotting -cite, there is an isomorphism $E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$$E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$: see [Crowley&Ferry&Skopenkov2011, Section 1.5].
2. An effective procedure for computing the rank of the group $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ can be found in [Crowley&Ferry&Skopenkov2011, Section 1.3].
3. Necessary and sufficient conditions on $p$$p$ and $q$$q$ for when $E^m(S^p \sqcup S^q)$$E^m(S^p \sqcup S^q)$ is finite can be found in [Crowley&Ferry&Skopenkov2011, Section 1.2].

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

8 References

• [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
• [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
• [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
• [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069

\leq j \leq n$ there are natural homomorphisms $p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$ obtained by projecting to the $j$-component of the wedge. Following Heafliger, we define the groups $$ \Lambda_{n_j}^{(k)} := {\rm Ker}\left(p_j \colon \pi_{n_j}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \pi_{n_j}(S^{k_j}) \right) \quad \text{and} \quad \Lambda_{(n)}^{(k)} : = \bigoplus_{j=1}^s \Lambda_{n_j}^{(k)} $$ and $$\Pi_{m-1}^{(k)} := {\rm Ker}\left( \bigoplus_{i=1}^s p_j \colon \pi_{m_1}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \bigoplus_{i=1}^s \pi_{m-1}(S^{k_j}) \right).$$ For $w_j \in \pi_{n_j}(S^{n_j})$ the class of the identity, taking the Whitehead product with $w_j$ defines a homomorphism $w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$ and we define $w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$. {{beginthm|Theorem}}\label{thm:hclink} (a) \cite[Theorem 1.3]{Haefliger1966a} Let $(n-1) = (n_1 -1, \dots, n_s-1)$. There are homomorphisms $\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$ and $\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$ which fit into a long exact exact sequence of abelian groups $$ \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$$ (b) \cite[Lemma 1.3]{Crowley&Ferry&Skopenkov2011} After tensoring with the rational numbers $\Qq$, $\mu \otimes {\rm Id}_{\Qq} = 0$, and so the Haefliger sequence in (a) above splits into short exact sequencs $$ 0 \to L_{(n)}^{(k)} \otimes \Qq \xrightarrow{~\lambda \otimes {\rm Id}_{\Qq}~} \Lambda_{(n)}^{(k)} \otimes \Qq \xrightarrow{~w \otimes {\rm Id}_{\Qq}~} \Pi_{m-2}^{(k)} \otimes \Qq \to 0.$$ {{endthm}} {{beginrem|Remark}} Here is some more information about \S1, $\S$$\S$2].

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ 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].

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

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

2 Examples

Recall that for each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction $m\ge2n+2$$m\ge2n+2$ is sharp for non-connected manifolds.

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

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

• either $\partial D^{n+1}\times0$$\partial D^{n+1}\times0$ and $0\times\partial D^{n+1}$$0\times\partial D^{n+1}$ in $\partial(D^{n+1}\times D^{n+1})$$\partial(D^{n+1}\times D^{n+1})$;

• or given by equations:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$$

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:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$

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

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

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

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

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

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

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$$m\ge q+3$, the complement $S^m-fS^q$$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

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

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]

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

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

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

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\ge3n+4$$2m\ge3n+4$ and $n$$n$-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:

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

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 Two component links of spheres beyond the metastable range

Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. For some information about this group see [Skopenkov2006, $\S$$\S$3.3].

The Haefliger Theorem 6.1. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$

(b) [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

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

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 6.2. (a) The Haefliger Theorem 6.1(b) implies that for $l\ge2$$l\ge2$ we have an isomorphism

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

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

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

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 6.1(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].

7 General links of spheres beyond the metastable range

For integers $n_1, \dots, n_s$$n_1, \dots, n_s$ with $m \geq n_i + 3$$m \geq n_i + 3$ set $k_i = m-n_i-1$$k_i = m-n_i-1$ and denote $(n) = (n_1, \dots n_s)$$(n) = (n_1, \dots n_s)$ and $(k) = (k_1, \dots, k_s)$$(k) = (k_1, \dots, k_s)$. We call the kernel of the homomorphism

$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$ $$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$$

the group of $1$$1$-Brunnian links and denote it $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$. Evidently $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$. In this subsection we describe an exact sequence of Haefliger [Haefliger1966a] which for fixed $(k)$$(k)$ places the groups of $1$$1$-Brunnian links $\Lambda_{(n)}^{(k)}$$\Lambda_{(n)}^{(k)}$ is an exact sequence.

By general position, the complement $C_f$$C_f$ of an embedding $f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$$f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$ is simply connected. By Alexander duality $C_f$$C_f$ has the homology of the wedge of sphers $\vee_{i=1}^s S^{k_i}$$\vee_{i=1}^s S^{k_i}$ and so by Whitehead's Theoorem there is a homotopy equivalence $C_f \simeq \vee_{i=1}^s S^{k_i}$$C_f \simeq \vee_{i=1}^s S^{k_i}$. For each $j$$j$, $1 \leq j \leq n$$1 \leq j \leq n$ there are natural homomorphisms $p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$$p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$ obtained by projecting to the $j$$j$-component of the wedge. Following Heafliger, we define the groups

Tex syntax error

$$\displaystyle \Lambda_{n_j}^{(k)} := {\rm Ker}\left(p_j \colon \pi_{n_j}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \pi_{n_j}(S^{k_j}) \right) \quad \text{and} \quad \Lambda_{(n)}^{(k)} : = \bigoplus_{j=1}^s \Lambda_{n_j}^{(k)}$$

and

Tex syntax error

$$\displaystyle \Pi_{m-1}^{(k)} := {\rm Ker}\left( \bigoplus_{i=1}^s p_j \colon \pi_{m_1}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \bigoplus_{i=1}^s \pi_{m-1}(S^{k_j}) \right).$$

For $w_j \in \pi_{n_j}(S^{n_j})$$w_j \in \pi_{n_j}(S^{n_j})$ the class of the identity, taking the Whitehead product with $w_j$$w_j$ defines a homomorphism $w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$$w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$ and we define $w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$$w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$.

Theorem 7.1. (a) [Haefliger1966a, Theorem 1.3] Let $(n-1) = (n_1 -1, \dots, n_s-1)$$(n-1) = (n_1 -1, \dots, n_s-1)$. There are homomorphisms $\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$$\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$ and $\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$$\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$ which fit into a long exact exact sequence of abelian groups

$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$ $$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$$

(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers $\Qq$$\Qq$,

Tex syntax error

$\mu \otimes {\rm Id}_{\Qq} = 0$, and so the Haefliger sequence in (a) above splits into short exact sequencs

Tex syntax error

$$\displaystyle 0 \to L_{(n)}^{(k)} \otimes \Qq \xrightarrow{~\lambda \otimes {\rm Id}_{\Qq}~} \Lambda_{(n)}^{(k)} \otimes \Qq \xrightarrow{~w \otimes {\rm Id}_{\Qq}~} \Pi_{m-2}^{(k)} \otimes \Qq \to 0.$$

Remark 7.2. Here is some more information about $1$$1$-Brunnian links.

1. Haefliger's description of $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$ works equally well in the PL-category. Thus as a consequence of [Haefliger1966a, Theorem 1.3] and Zeeman unknotting -cite, there is an isomorphism $E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$$E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$: see [Crowley&Ferry&Skopenkov2011, Section 1.5].
2. An effective procedure for computing the rank of the group $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ can be found in [Crowley&Ferry&Skopenkov2011, Section 1.3].
3. Necessary and sufficient conditions on $p$$p$ and $q$$q$ for when $E^m(S^p \sqcup S^q)$$E^m(S^p \sqcup S^q)$ is finite can be found in [Crowley&Ferry&Skopenkov2011, Section 1.2].

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

8 References

• [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
• [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
• [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
• [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069

$-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}. # An effective procedure for computing the rank of the group $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ can be found in \cite[Section 1.3]{Crowley&Ferry&Skopenkov2011}. # Necessary and sufficient conditions on $p$ and $q$ for when $E^m(S^p \sqcup S^q)$ is finite can be found in \cite[Section 1.2]{Crowley&Ferry&Skopenkov2011}. {{endrem}} For more results related to high codimension links we refer the reader to \cite{Skopenkov2009}, \cite{Avvakumov2016}, \cite[$\S.5]{Skopenkov2015a}, \cite{Skopenkov2016k}.

$. Restricting an element of $SO_n$ to $S^{n-1} \subset \Rr^n$ identifies $SO_n$ as a subspace of $G_n$.--> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S1, $\S$$\S$2].

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ 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].

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

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

2 Examples

Recall that for each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction $m\ge2n+2$$m\ge2n+2$ is sharp for non-connected manifolds.

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

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

• either $\partial D^{n+1}\times0$$\partial D^{n+1}\times0$ and $0\times\partial D^{n+1}$$0\times\partial D^{n+1}$ in $\partial(D^{n+1}\times D^{n+1})$$\partial(D^{n+1}\times D^{n+1})$;

• or given by equations:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1.\end{array}\right.$$

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:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$ $$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$

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

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

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

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

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

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

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$$m\ge q+3$, the complement $S^m-fS^q$$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

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

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]

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

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

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

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\ge3n+4$$2m\ge3n+4$ and $n$$n$-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:

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

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 Two component links of spheres beyond the metastable range

Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. For some information about this group see [Skopenkov2006, $\S$$\S$3.3].

The Haefliger Theorem 6.1. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$

(b) [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

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

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 6.2. (a) The Haefliger Theorem 6.1(b) implies that for $l\ge2$$l\ge2$ we have an isomorphism

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

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

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

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 6.1(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].

7 General links of spheres beyond the metastable range

For integers $n_1, \dots, n_s$$n_1, \dots, n_s$ with $m \geq n_i + 3$$m \geq n_i + 3$ set $k_i = m-n_i-1$$k_i = m-n_i-1$ and denote $(n) = (n_1, \dots n_s)$$(n) = (n_1, \dots n_s)$ and $(k) = (k_1, \dots, k_s)$$(k) = (k_1, \dots, k_s)$. We call the kernel of the homomorphism

$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$ $$\displaystyle E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \to \bigoplus_{i=1}^s C^{k_i+1}_{n_i}$$

the group of $1$$1$-Brunnian links and denote it $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$. Evidently $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s}) \cong L_{(n)}^{(k)} \oplus \bigoplus_{i=1}^s C_{n_i}^{k_i+1}$. In this subsection we describe an exact sequence of Haefliger [Haefliger1966a] which for fixed $(k)$$(k)$ places the groups of $1$$1$-Brunnian links $\Lambda_{(n)}^{(k)}$$\Lambda_{(n)}^{(k)}$ is an exact sequence.

By general position, the complement $C_f$$C_f$ of an embedding $f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$$f \colon S^{n_1} \sqcup \dots \sqcup S^{n_2}$ is simply connected. By Alexander duality $C_f$$C_f$ has the homology of the wedge of sphers $\vee_{i=1}^s S^{k_i}$$\vee_{i=1}^s S^{k_i}$ and so by Whitehead's Theoorem there is a homotopy equivalence $C_f \simeq \vee_{i=1}^s S^{k_i}$$C_f \simeq \vee_{i=1}^s S^{k_i}$. For each $j$$j$, $1 \leq j \leq n$$1 \leq j \leq n$ there are natural homomorphisms $p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$$p_j \colon \pi_*\bigl(\vee_{i=1}^s S^{k_i}\bigr) \to \pi_*(S^{k_j})$ obtained by projecting to the $j$$j$-component of the wedge. Following Heafliger, we define the groups

Tex syntax error

$$\displaystyle \Lambda_{n_j}^{(k)} := {\rm Ker}\left(p_j \colon \pi_{n_j}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \pi_{n_j}(S^{k_j}) \right) \quad \text{and} \quad \Lambda_{(n)}^{(k)} : = \bigoplus_{j=1}^s \Lambda_{n_j}^{(k)}$$

and

Tex syntax error

$$\displaystyle \Pi_{m-1}^{(k)} := {\rm Ker}\left( \bigoplus_{i=1}^s p_j \colon \pi_{m_1}\bigl(\vee_{j=1}^s S^{k_i}\bigr) \to \bigoplus_{i=1}^s \pi_{m-1}(S^{k_j}) \right).$$

For $w_j \in \pi_{n_j}(S^{n_j})$$w_j \in \pi_{n_j}(S^{n_j})$ the class of the identity, taking the Whitehead product with $w_j$$w_j$ defines a homomorphism $w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$$w_j \colon \Lambda_{n_j}^{(k)} \to \Pi_{(n)}^{(k)}$ and we define $w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$$w = \bigoplus_{j=1}^s w_j \colon \Lambda_{(n)}^{(k)} \to \Pi_{m-1}^{(k)}$.

Theorem 7.1. (a) [Haefliger1966a, Theorem 1.3] Let $(n-1) = (n_1 -1, \dots, n_s-1)$$(n-1) = (n_1 -1, \dots, n_s-1)$. There are homomorphisms $\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$$\mu \colon \Pi_{m-1}^{(k)} \to L_{(n)}^{(k)}$ and $\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$$\lambda \colon L_{(n)}^{(k)} \to \Lambda_{(n)}^{(k)}$ which fit into a long exact exact sequence of abelian groups

$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$ $$\displaystyle \dots \to \Pi_{m-1}^{(k)} \xrightarrow{~\mu~} L_{(n)}^{(k)} \xrightarrow{~\lambda~} \Lambda_{(n)}^{(k)} \xrightarrow{~w~} \Pi_{m-2}^{(k)} \xrightarrow{~\mu~} L_{(n-1)}^{(k)} \to \dots~.$$

(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers $\Qq$$\Qq$,

Tex syntax error

$\mu \otimes {\rm Id}_{\Qq} = 0$, and so the Haefliger sequence in (a) above splits into short exact sequencs

Tex syntax error

$$\displaystyle 0 \to L_{(n)}^{(k)} \otimes \Qq \xrightarrow{~\lambda \otimes {\rm Id}_{\Qq}~} \Lambda_{(n)}^{(k)} \otimes \Qq \xrightarrow{~w \otimes {\rm Id}_{\Qq}~} \Pi_{m-2}^{(k)} \otimes \Qq \to 0.$$

Remark 7.2. Here is some more information about $1$$1$-Brunnian links.

1. Haefliger's description of $L_{(n)}^{(k)}$$L_{(n)}^{(k)}$ works equally well in the PL-category. Thus as a consequence of [Haefliger1966a, Theorem 1.3] and Zeeman unknotting -cite, there is an isomorphism $E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$$E^m_{PL}(S^{n_1} \sqcup \dots S^{n_1}) \cong L_{(n)}^{(k)}$: see [Crowley&Ferry&Skopenkov2011, Section 1.5].
2. An effective procedure for computing the rank of the group $E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\ldots\sqcup S^{n_s})$ can be found in [Crowley&Ferry&Skopenkov2011, Section 1.3].
3. Necessary and sufficient conditions on $p$$p$ and $q$$q$ for when $E^m(S^p \sqcup S^q)$$E^m(S^p \sqcup S^q)$ is finite can be found in [Crowley&Ferry&Skopenkov2011, Section 1.2].

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

8 References

• [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
• [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
• [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
• [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069