High codimension links

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1 Introduction

Most of 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.

On this page we describe readily calculable classifications of embeddings for closed disconnected manifolds into $\Rr^m$${{Authors|Askopenkov}} == Introduction == <wikitex>; Most of this page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings. On this page we describe readily calculable classifications of embeddings for closed ''disconnected'' manifolds into $\Rr^m$. Known cases are embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$, embeddings $N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$ for $m\ge 2n_i+1$ for every $i$, and embeddings of some disconnected 3-manifolds in $S^6$. For a related classification of knotted tori see \cite{Skopenkov2016k}. For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\Rr^m$. Known cases are embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$$S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$$m-3\ge n_i$, embeddings $N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$$N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$ for $m\ge 2n_i+1$$m\ge 2n_i+1$ for every $i$$i$, and embeddings of some disconnected 3-manifolds in $S^6$$S^6$. For a related classification of knotted tori see [Skopenkov2016k].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3].

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

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

Figure 1: Embedded connected sum

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{(n)})$$E^m(S^{(n)})$ for $m-3\ge n_i$$m-3\ge n_i$ [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see Figure 1.

The standard embedding $S^k\to D^m$$S^k\to D^m$ is defined by $(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$$(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$. Fix $s$$s$ pairwise disjoint $m$$m$-discs $D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$$D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$. The standard embedding $S^{(n)}\to S^m$$S^{(n)}\to S^m$ is defined by taking the union of the compositions of the standard embeddings $S^{n_k} \to D^m_k$$S^{n_k} \to D^m_k$ with the fixed inclusions $D^m_k \to S^m$$D^m_k \to S^m$.

2 Examples

Recall that for any $q$$q$-manifold $N$$N$ and $m\ge2q+2$$m\ge2q+2$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction $m\ge2q+2$$m\ge2q+2$ is sharp for non-connected manifolds.

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

Figure 2: The Hopf link (a) and the trivial link (b)

For $q=1$$q=1$ the Hopf link is shown in Figure 2. For all $q$$q$ the image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows:

• either the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$;

• or they are given as the sets of points in $\Rr^{2q+1}$$\Rr^{2q+1}$ satisfying the following equations:

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

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

Analogously for any $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 which can be described as follows:

• either the spheres are $\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})\cong S^{p+q+1}$$\partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}$.

• or they are given as the points in $\Rr^{p+q+1}$$\Rr^{p+q+1}$ satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see $\S$$\S$3).

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient $\pm 1$$\pm 1$

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

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

Tex syntax error

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

see Figure 3. We have

Tex syntax error

$\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let

Tex syntax error

$\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$.

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$$g:D^{m-q}\to S^m$ such that $g(D^{m-q})$$g(D^{m-q})$ intersects $f(S^q)$$f(S^q)$ transversely at exactly one point with positive sign; see Figure 4.

Figure 4: The disc $gD^{m-q}$$gD^{m-q}$ and Gauss map $\widetilde f$$\widetilde f$

Then the restriction $h':S^{m-q-1}\to S^m-fS^q$$h':S^{m-q-1}\to S^m-fS^q$ of $g$$g$ to $\partial D^{m-q}$$\partial D^{m-q}$ is a homotopy equivalence.

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$$\lambda[f]$ is well-defined, i.e. is independent of the choices of $g,h',h$$g,h',h$ and of the representative $f$$f$ of $[f]$$[f]$. One can check that $\lambda$$\lambda$ is a homomorphism.

(b) 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.

(c) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$$\lambda\zeta=\id\pi_p(S^{m-q-1})$ for the Zeman map $\zeta$$\zeta$. So $\lambda$$\lambda$ is surjective and $\zeta$$\zeta$ is injective.

(d) For $m=p+q+1$$m=p+q+1$ there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].

Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism $\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 stable suspension 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 the Gauss map (see Figure 4)

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

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 the suspension isomorphism. 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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

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)$.)

One can easily check that $\alpha$$\alpha$ is well-defined and for $p,q\le m-3$$p,q\le m-3$ is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$$p,q\le m-3$.

Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$$\lambda_{12} \neq \pm \lambda_{21}$ as we explain in $\S$$\S$8.

Note that the $\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, each of the same dimension [Haefliger1962t, Theorem at the end of $\S$$\S$5], [Haefliger1966a]. Let $(q) = (q, \dots, q)$$(q) = (q, \dots, q)$ be the $s$$s$-tuple consisting entirely of some positive integer $q$$q$.

Theorem 4.2. The collection of pairwise linking coefficients

$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$ $$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$$

is bijective for $m\ge\frac{3q}2+2$$m\ge\frac{3q}2+2$.

Assume that $N_1,\ldots N_s$$N_1,\ldots N_s$ are closed manifolds, $m=2\dim N_i+1>2\dim N_k+1$$m=2\dim N_i+1>2\dim N_k+1$ for every $1\le i\le t<k\le s$$1\le i\le t<k\le s$ and $N_1,\ldots N_t$$N_1,\ldots N_t$ are orientable. Then for an embedding $N_1,\ldots N_s$$N_1,\ldots N_s$ and every $1\le i<j\le t$$1\le i<j\le t$ one can define the linking coefficient $\lambda_{ij}(f)\in\Z$$\lambda_{ij}(f)\in\Z$, see Remark 3.2.e.

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.

Example 5.1 (Borromean rings). 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' are the three spheres given by the following three systems of equations:

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

Figure 6: The Borromean rings

See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.

An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, $\S$$\S$4.5 `Triple linking modulo 2' and $\S$$\S$4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.

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

Figure 7: The Borromean rings and Whitehead link for $l = 1$$l = 1$

Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, $\S$$\S$3, $\S$$\S$4]; see also the Wikipedia article on the Whitehead link. (For $l=1$$l=1$ the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

We have $\lambda_{12}(w)=0$$\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]). For $l\ne1,3,7$$l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)$$\lambda_{21}(w)$ equal to the Whitehead square $[\iota_l,\iota_l]\ne0$$[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$$\iota_l\in\pi_l(S^l)$. This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14]. For $l=1,3,7$$l=1,3,7$ the Whitehead link 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.

For some results on links $S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link [Skopenkov2015a, $\S$$\S$2.5].

6 Linked 3-manifolds in 6-space

An embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$$k,m,n$ such that $m-n$$m-n$ is even, one can explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ so that the following theorem holds.

Theorem 6.1. [Avvakumov2016] Any Brunnian embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is isotopic to fk,m,n for some integers k,m,n such that $m-n$$m-n$ is even. Two embeddings fk,m,n and fk′,m′,n′ are isotopic if and only if $k=k′$$k=k′$ and both $m-m′$$m-m′$ and $n-n′$$n-n′$ are divisible by $2k$$2k$.

The proof uses classification of embeddings $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ (the Haefliger Theorem 8.1 for $m=2p=2q=6$$m=2p=2q=6$). The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ is not trivial.

Corollary 6.2. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $g,g':S^3\sqcup S^3\to\Rr^6$$g,g':S^3\sqcup S^3\to\Rr^6$ such that the componentwise embedded connected sum $f\#g$$f\#g$ is isotopic to $f\#g'$$f\#g'$ but $g$$g$ is not isotopic to $g'$$g'$.

See an unpublished generalization in [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define $E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$$E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$ to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding $S^{n_i} \to S^m$$S^{n_i} \to S^m$. We remark that $E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$$E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$ [Haefliger1966a, $\S\S$$\S\S$ 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.5].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$$s$-tuple of the isotopy classes of its components:

$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$ $$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$$

Take $s$$s$ pairwise disjoint $m$$m$-discs in $S^m$$S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$$g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$. Define

$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$ $$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$$

Then $j$$j$ is a right inverse of the restriction homomorphism $r$$r$, i.e. $r\circ j=\mathrm{id}$$r\circ j=\mathrm{id}$. The unknotting homomorphism $u$$u$ is defined to be the homomorphism

$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$ $$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$$

Informally, the homomorphism $u$$u$ is obtained by taking embedded connected sums of components with knots $h_i:S^{n_i}\to S^m$$h_i:S^{n_i}\to S^m$ representing the elements of $E^m_D(S^{n_i})$$E^m_D(S^{n_i})$ inverse to the components, whose images $h_i(S^{n_i})$$h_i(S^{n_i})$ are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For $n_1,\ldots,n_s\le m-3$$n_1,\ldots,n_s\le m-3$, the homomorphism

$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$ $$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$$

is an isomorphism.

For some information on the groups $E^m_D(S^q)$$E^m_D(S^q)$ see [Skopenkov2016s], [Skopenkov2006, $\S$$\S$3.3].

8 Classification beyond the metastable range

The Haefliger Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of $\beta$$\beta$ see [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5], cf. [Skopenkov2007, $\S$$\S$2] and [Crowley&Skopenkov2016, $\S$$\S$2.3].

Remark 8.2. (a) The Haefliger Theorem 8.1 implies that for any $l\ge2$$l\ge2$ we have an isomorphism

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

(b) For any $l\not\in\{1,3,7\}$$l\not\in\{1,3,7\}$, 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+(-1)^lb)=0\}$$\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$.

For $l\ge4$$l\ge4$ see [Haefliger1962t, $\S$$\S$6]. The following proof for $l=2$$l=2$ and general remark are intended for specialists.

For $l=2$$l=2$ there is an exact sequence $\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$$\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$, where $\lambda_{12}$$\lambda_{12}$ is $\lambda_{12,PL}$$\lambda_{12,PL}$ not $\lambda_{12,D}$$\lambda_{12,D}$ [Haefliger1966a, Corollary 10.3]. We have $\pi_3(S^2)\cong\Zz$$\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$$\pi_4(S^3)\cong\Zz_2$ and $\Sigma$$\Sigma$ is the reduction modulo 2. By the exactness, $\text{im}\lambda_{21}=2\Z$$\text{im}\lambda_{21}=2\Z$. By (a) $\ker\lambda_{12}\cong\Z$$\ker\lambda_{12}\cong\Z$. Hence $\lambda_{21}|_{\ker\lambda_{12}}$$\lambda_{21}|_{\ker\lambda_{12}}$ is injective. We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$$\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of $\beta$$\beta$ [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5] and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$$\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ [Koschorke&Sanderson1977] one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$$\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$. Then (b) would follow.

(c) For any $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, $\S$$\S$6].

(d) For any $l\ge4$$l\ge4$, $l\ne7$$l\ne7$ we have an isomorphism

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

which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number ($\S$$\S$8). This follows from [Haefliger1962t, $\S$$\S$6]. We conjecture that this result holds also for $l=2,3,7$$l=2,3,7$.

9 Classification in codimension at least 3

In this section we assume that $(n)$$(n)$ is an $s$$s$-tuple such that $m\ge n_i+3$$m\ge n_i+3$. For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$$E^m_{PL}(S^{(n)})\otimes\Qq$ is obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.2] one are necessary and sufficient conditions on $(n)$$(n)$ when $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers $m,n_1,\ldots,n_s>0$$m,n_1,\ldots,n_s>0$

(a) calculate

Tex syntax error

${\rm rk}E^m_{PL}(S^{(n)})$.

(b) find out whether $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite.

Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups

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

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. In the above sequence the $s$$s$-tuples $m-n_1,\ldots,m-n_s$$m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$$W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$. For any $j=1,\ldots,s$$j=1,\ldots,s$ and positive integer $k$$k$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$$p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$$j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$$(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$$\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$.

Analogous to Definition 3.1 there is a canonical homotopy equivalence $C_f \sim W$$C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$$S^{(n)} \to S^m$ has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement. One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$$\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$$\lambda_j$ is a generalisation of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$$\lambda:= \oplus_{j=1}^s \lambda_j$.

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

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

Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

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

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

In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving the Haefliger Theorem 8.1 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

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• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
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, $\S]{Skopenkov2016c}. The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem \ref{lkhaze} below): $$\begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}$$ [[Image:EmbeddedConnectedSum.jpg|thumb|350px|Figure 1: Embedded connected sum]] For an $s$-tuple $(n):=(n_1,\ldots,n_s)$ denote $S^{(n)}:=S^{n_1}\sqcup\ldots\sqcup S^{n_s}$. Although $S^{(n)}$ is not a manifold when $n_1,\ldots,n_s$ are not all equal, embeddings $S^{(n)}\to S^m$ and isotopy between such embeddings are defined analogously to [[Isotopy|the case of manifolds]]. Denote by $E^m(S^{(n)})$ the set of embeddings $S^{(n)}\to S^m$ up to isotopy. A component-wise version of [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Embedded_connected_sum|embedded connected sum]] \cite[$\S]{Skopenkov2016c} defines a commutative group structure on the set $E^m(S^{(n)})$ for $m-3\ge n_i$ \cite{Haefliger1966}, \cite{Haefliger1966a}, \cite[Group Structure Lemma 2.2 and Remark 2.3.a]{Skopenkov2015}, see Figure 1. The ''standard embedding'' $S^k\to D^m$ is defined by $(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$. Fix $s$ pairwise disjoint $m$-discs $D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$. The ''standard embedding'' $S^{(n)}\to S^m$ is defined by taking the union of the compositions of the standard embeddings $S^{n_k} \to D^m_k$ with the fixed inclusions $D^m_k \to S^m$. == Examples == ; [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Recall that]] for any $q$-manifold $N$ and $m\ge2q+2$, every two embeddings $N\to\Rr^m$ are isotopic \cite[General Position Theorem 2.1]{Skopenkov2016c}, \cite[General Position Theorem 2.1]{Skopenkov2006}. The following example shows that the restriction $m\ge2q+2$ is sharp for non-connected manifolds. {{beginthm|Example|(The Hopf Link)}}\label{hopf} For every positive integer $q$ there is an embedding $S^q\sqcup S^q\to\Rr^{2q+1}$, which is not isotopic to the standard embedding. {{endthm}} [[Image:HopfLink-and-TrivialLink.jpg|thumb|350px|Figure 2: The Hopf link (a) and the trivial link (b)]] For $q=1$ the Hopf link is shown in Figure 2. For all $q$ the image of the Hopf link is the union of two $q$-spheres which can be described as follows: * either the spheres are $\partial D^{q+1}\times0$ and $\Rr^m$. Known cases are embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$$S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$$m-3\ge n_i$, embeddings $N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$$N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$ for $m\ge 2n_i+1$$m\ge 2n_i+1$ for every $i$$i$, and embeddings of some disconnected 3-manifolds in $S^6$$S^6$. For a related classification of knotted tori see [Skopenkov2016k].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3].

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

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

Figure 1: Embedded connected sum

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{(n)})$$E^m(S^{(n)})$ for $m-3\ge n_i$$m-3\ge n_i$ [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see Figure 1.

The standard embedding $S^k\to D^m$$S^k\to D^m$ is defined by $(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$$(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$. Fix $s$$s$ pairwise disjoint $m$$m$-discs $D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$$D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$. The standard embedding $S^{(n)}\to S^m$$S^{(n)}\to S^m$ is defined by taking the union of the compositions of the standard embeddings $S^{n_k} \to D^m_k$$S^{n_k} \to D^m_k$ with the fixed inclusions $D^m_k \to S^m$$D^m_k \to S^m$.

2 Examples

Recall that for any $q$$q$-manifold $N$$N$ and $m\ge2q+2$$m\ge2q+2$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction $m\ge2q+2$$m\ge2q+2$ is sharp for non-connected manifolds.

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

Figure 2: The Hopf link (a) and the trivial link (b)

For $q=1$$q=1$ the Hopf link is shown in Figure 2. For all $q$$q$ the image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows:

• either the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$;

• or they are given as the sets of points in $\Rr^{2q+1}$$\Rr^{2q+1}$ satisfying the following equations:

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

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

Analogously for any $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 which can be described as follows:

• either the spheres are $\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})\cong S^{p+q+1}$$\partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}$.

• or they are given as the points in $\Rr^{p+q+1}$$\Rr^{p+q+1}$ satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see $\S$$\S$3).

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient $\pm 1$$\pm 1$

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

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

Tex syntax error

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

see Figure 3. We have

Tex syntax error

$\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let

Tex syntax error

$\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$.

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$$g:D^{m-q}\to S^m$ such that $g(D^{m-q})$$g(D^{m-q})$ intersects $f(S^q)$$f(S^q)$ transversely at exactly one point with positive sign; see Figure 4.

Figure 4: The disc $gD^{m-q}$$gD^{m-q}$ and Gauss map $\widetilde f$$\widetilde f$

Then the restriction $h':S^{m-q-1}\to S^m-fS^q$$h':S^{m-q-1}\to S^m-fS^q$ of $g$$g$ to $\partial D^{m-q}$$\partial D^{m-q}$ is a homotopy equivalence.

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$$\lambda[f]$ is well-defined, i.e. is independent of the choices of $g,h',h$$g,h',h$ and of the representative $f$$f$ of $[f]$$[f]$. One can check that $\lambda$$\lambda$ is a homomorphism.

(b) 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.

(c) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$$\lambda\zeta=\id\pi_p(S^{m-q-1})$ for the Zeman map $\zeta$$\zeta$. So $\lambda$$\lambda$ is surjective and $\zeta$$\zeta$ is injective.

(d) For $m=p+q+1$$m=p+q+1$ there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].

Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism $\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 stable suspension 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 the Gauss map (see Figure 4)

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

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 the suspension isomorphism. 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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

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)$.)

One can easily check that $\alpha$$\alpha$ is well-defined and for $p,q\le m-3$$p,q\le m-3$ is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$$p,q\le m-3$.

Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$$\lambda_{12} \neq \pm \lambda_{21}$ as we explain in $\S$$\S$8.

Note that the $\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, each of the same dimension [Haefliger1962t, Theorem at the end of $\S$$\S$5], [Haefliger1966a]. Let $(q) = (q, \dots, q)$$(q) = (q, \dots, q)$ be the $s$$s$-tuple consisting entirely of some positive integer $q$$q$.

Theorem 4.2. The collection of pairwise linking coefficients

$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$ $$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$$

is bijective for $m\ge\frac{3q}2+2$$m\ge\frac{3q}2+2$.

Assume that $N_1,\ldots N_s$$N_1,\ldots N_s$ are closed manifolds, $m=2\dim N_i+1>2\dim N_k+1$$m=2\dim N_i+1>2\dim N_k+1$ for every $1\le i\le t<k\le s$$1\le i\le t<k\le s$ and $N_1,\ldots N_t$$N_1,\ldots N_t$ are orientable. Then for an embedding $N_1,\ldots N_s$$N_1,\ldots N_s$ and every $1\le i<j\le t$$1\le i<j\le t$ one can define the linking coefficient $\lambda_{ij}(f)\in\Z$$\lambda_{ij}(f)\in\Z$, see Remark 3.2.e.

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.

Example 5.1 (Borromean rings). 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' are the three spheres given by the following three systems of equations:

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

Figure 6: The Borromean rings

See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.

An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, $\S$$\S$4.5 `Triple linking modulo 2' and $\S$$\S$4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.

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

Figure 7: The Borromean rings and Whitehead link for $l = 1$$l = 1$

Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, $\S$$\S$3, $\S$$\S$4]; see also the Wikipedia article on the Whitehead link. (For $l=1$$l=1$ the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

We have $\lambda_{12}(w)=0$$\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]). For $l\ne1,3,7$$l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)$$\lambda_{21}(w)$ equal to the Whitehead square $[\iota_l,\iota_l]\ne0$$[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$$\iota_l\in\pi_l(S^l)$. This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14]. For $l=1,3,7$$l=1,3,7$ the Whitehead link 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.

For some results on links $S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link [Skopenkov2015a, $\S$$\S$2.5].

6 Linked 3-manifolds in 6-space

An embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$$k,m,n$ such that $m-n$$m-n$ is even, one can explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ so that the following theorem holds.

Theorem 6.1. [Avvakumov2016] Any Brunnian embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is isotopic to fk,m,n for some integers k,m,n such that $m-n$$m-n$ is even. Two embeddings fk,m,n and fk′,m′,n′ are isotopic if and only if $k=k′$$k=k′$ and both $m-m′$$m-m′$ and $n-n′$$n-n′$ are divisible by $2k$$2k$.

The proof uses classification of embeddings $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ (the Haefliger Theorem 8.1 for $m=2p=2q=6$$m=2p=2q=6$). The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ is not trivial.

Corollary 6.2. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $g,g':S^3\sqcup S^3\to\Rr^6$$g,g':S^3\sqcup S^3\to\Rr^6$ such that the componentwise embedded connected sum $f\#g$$f\#g$ is isotopic to $f\#g'$$f\#g'$ but $g$$g$ is not isotopic to $g'$$g'$.

See an unpublished generalization in [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define $E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$$E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$ to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding $S^{n_i} \to S^m$$S^{n_i} \to S^m$. We remark that $E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$$E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$ [Haefliger1966a, $\S\S$$\S\S$ 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.5].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$$s$-tuple of the isotopy classes of its components:

$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$ $$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$$

Take $s$$s$ pairwise disjoint $m$$m$-discs in $S^m$$S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$$g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$. Define

$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$ $$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$$

Then $j$$j$ is a right inverse of the restriction homomorphism $r$$r$, i.e. $r\circ j=\mathrm{id}$$r\circ j=\mathrm{id}$. The unknotting homomorphism $u$$u$ is defined to be the homomorphism

$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$ $$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$$

Informally, the homomorphism $u$$u$ is obtained by taking embedded connected sums of components with knots $h_i:S^{n_i}\to S^m$$h_i:S^{n_i}\to S^m$ representing the elements of $E^m_D(S^{n_i})$$E^m_D(S^{n_i})$ inverse to the components, whose images $h_i(S^{n_i})$$h_i(S^{n_i})$ are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For $n_1,\ldots,n_s\le m-3$$n_1,\ldots,n_s\le m-3$, the homomorphism

$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$ $$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$$

is an isomorphism.

For some information on the groups $E^m_D(S^q)$$E^m_D(S^q)$ see [Skopenkov2016s], [Skopenkov2006, $\S$$\S$3.3].

8 Classification beyond the metastable range

The Haefliger Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of $\beta$$\beta$ see [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5], cf. [Skopenkov2007, $\S$$\S$2] and [Crowley&Skopenkov2016, $\S$$\S$2.3].

Remark 8.2. (a) The Haefliger Theorem 8.1 implies that for any $l\ge2$$l\ge2$ we have an isomorphism

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

(b) For any $l\not\in\{1,3,7\}$$l\not\in\{1,3,7\}$, 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+(-1)^lb)=0\}$$\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$.

For $l\ge4$$l\ge4$ see [Haefliger1962t, $\S$$\S$6]. The following proof for $l=2$$l=2$ and general remark are intended for specialists.

For $l=2$$l=2$ there is an exact sequence $\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$$\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$, where $\lambda_{12}$$\lambda_{12}$ is $\lambda_{12,PL}$$\lambda_{12,PL}$ not $\lambda_{12,D}$$\lambda_{12,D}$ [Haefliger1966a, Corollary 10.3]. We have $\pi_3(S^2)\cong\Zz$$\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$$\pi_4(S^3)\cong\Zz_2$ and $\Sigma$$\Sigma$ is the reduction modulo 2. By the exactness, $\text{im}\lambda_{21}=2\Z$$\text{im}\lambda_{21}=2\Z$. By (a) $\ker\lambda_{12}\cong\Z$$\ker\lambda_{12}\cong\Z$. Hence $\lambda_{21}|_{\ker\lambda_{12}}$$\lambda_{21}|_{\ker\lambda_{12}}$ is injective. We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$$\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of $\beta$$\beta$ [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5] and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$$\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ [Koschorke&Sanderson1977] one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$$\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$. Then (b) would follow.

(c) For any $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, $\S$$\S$6].

(d) For any $l\ge4$$l\ge4$, $l\ne7$$l\ne7$ we have an isomorphism

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

which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number ($\S$$\S$8). This follows from [Haefliger1962t, $\S$$\S$6]. We conjecture that this result holds also for $l=2,3,7$$l=2,3,7$.

9 Classification in codimension at least 3

In this section we assume that $(n)$$(n)$ is an $s$$s$-tuple such that $m\ge n_i+3$$m\ge n_i+3$. For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$$E^m_{PL}(S^{(n)})\otimes\Qq$ is obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.2] one are necessary and sufficient conditions on $(n)$$(n)$ when $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers $m,n_1,\ldots,n_s>0$$m,n_1,\ldots,n_s>0$

(a) calculate

Tex syntax error

${\rm rk}E^m_{PL}(S^{(n)})$.

(b) find out whether $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite.

Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups

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

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. In the above sequence the $s$$s$-tuples $m-n_1,\ldots,m-n_s$$m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$$W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$. For any $j=1,\ldots,s$$j=1,\ldots,s$ and positive integer $k$$k$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$$p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$$j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$$(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$$\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$.

Analogous to Definition 3.1 there is a canonical homotopy equivalence $C_f \sim W$$C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$$S^{(n)} \to S^m$ has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement. One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$$\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$$\lambda_j$ is a generalisation of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$$\lambda:= \oplus_{j=1}^s \lambda_j$.

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

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

Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

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

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

In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving the Haefliger Theorem 8.1 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

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• [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.
• [Koschorke&Sanderson1977] U. Koschorke and B. Sanderson, Geometric interpretation of the generalized Hopf invariant, Math. Scand. 41 (1977) 199--217.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Massey1968] W. S. Massey, Higher order linking numbers, Proc. Conf. on Algebraic Topology, Univ. Illinois, Chicago Circle, Chicago, Ill., (1968) pp. 174--205. MR0254832 (40 #8039), see also MR1625365 (99e:57016) massey.pdf.
• [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.
• [Skopenkov2006b] M. Skopenkov, A formula for the group of links in the 2-metastable dimension, arxiv:math/0610320v1
• [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
• [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.
• [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, 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 $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$; * or they are given as the sets of points in $\Rr^{2q+1}$ satisfying the following equations: $$\left\{\begin{array}{c} x_1=\dots=x_q=0\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.$$ This embedding is distinguished from the standard embedding by [[#The linking coefficient|the linking coefficient]] (see $\S$\ref{s:inv}). Analogously for any $p,q$ 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 which can be described as follows: * either the spheres are $\partial D^{p+1}\times0$ and $\Rr^m$. Known cases are embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$$S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$$m-3\ge n_i$, embeddings $N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$$N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$ for $m\ge 2n_i+1$$m\ge 2n_i+1$ for every $i$$i$, and embeddings of some disconnected 3-manifolds in $S^6$$S^6$. For a related classification of knotted tori see [Skopenkov2016k].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3].

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

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

Figure 1: Embedded connected sum

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{(n)})$$E^m(S^{(n)})$ for $m-3\ge n_i$$m-3\ge n_i$ [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see Figure 1.

The standard embedding $S^k\to D^m$$S^k\to D^m$ is defined by $(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$$(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$. Fix $s$$s$ pairwise disjoint $m$$m$-discs $D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$$D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$. The standard embedding $S^{(n)}\to S^m$$S^{(n)}\to S^m$ is defined by taking the union of the compositions of the standard embeddings $S^{n_k} \to D^m_k$$S^{n_k} \to D^m_k$ with the fixed inclusions $D^m_k \to S^m$$D^m_k \to S^m$.

2 Examples

Recall that for any $q$$q$-manifold $N$$N$ and $m\ge2q+2$$m\ge2q+2$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction $m\ge2q+2$$m\ge2q+2$ is sharp for non-connected manifolds.

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

Figure 2: The Hopf link (a) and the trivial link (b)

For $q=1$$q=1$ the Hopf link is shown in Figure 2. For all $q$$q$ the image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows:

• either the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$;

• or they are given as the sets of points in $\Rr^{2q+1}$$\Rr^{2q+1}$ satisfying the following equations:

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

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

Analogously for any $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 which can be described as follows:

• either the spheres are $\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})\cong S^{p+q+1}$$\partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}$.

• or they are given as the points in $\Rr^{p+q+1}$$\Rr^{p+q+1}$ satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see $\S$$\S$3).

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient $\pm 1$$\pm 1$

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

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

Tex syntax error

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

see Figure 3. We have

Tex syntax error

$\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let

Tex syntax error

$\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$.

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$$g:D^{m-q}\to S^m$ such that $g(D^{m-q})$$g(D^{m-q})$ intersects $f(S^q)$$f(S^q)$ transversely at exactly one point with positive sign; see Figure 4.

Figure 4: The disc $gD^{m-q}$$gD^{m-q}$ and Gauss map $\widetilde f$$\widetilde f$

Then the restriction $h':S^{m-q-1}\to S^m-fS^q$$h':S^{m-q-1}\to S^m-fS^q$ of $g$$g$ to $\partial D^{m-q}$$\partial D^{m-q}$ is a homotopy equivalence.

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$$\lambda[f]$ is well-defined, i.e. is independent of the choices of $g,h',h$$g,h',h$ and of the representative $f$$f$ of $[f]$$[f]$. One can check that $\lambda$$\lambda$ is a homomorphism.

(b) 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.

(c) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$$\lambda\zeta=\id\pi_p(S^{m-q-1})$ for the Zeman map $\zeta$$\zeta$. So $\lambda$$\lambda$ is surjective and $\zeta$$\zeta$ is injective.

(d) For $m=p+q+1$$m=p+q+1$ there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].

Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism $\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 stable suspension 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 the Gauss map (see Figure 4)

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

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 the suspension isomorphism. 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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

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)$.)

One can easily check that $\alpha$$\alpha$ is well-defined and for $p,q\le m-3$$p,q\le m-3$ is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$$p,q\le m-3$.

Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$$\lambda_{12} \neq \pm \lambda_{21}$ as we explain in $\S$$\S$8.

Note that the $\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, each of the same dimension [Haefliger1962t, Theorem at the end of $\S$$\S$5], [Haefliger1966a]. Let $(q) = (q, \dots, q)$$(q) = (q, \dots, q)$ be the $s$$s$-tuple consisting entirely of some positive integer $q$$q$.

Theorem 4.2. The collection of pairwise linking coefficients

$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$ $$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$$

is bijective for $m\ge\frac{3q}2+2$$m\ge\frac{3q}2+2$.

Assume that $N_1,\ldots N_s$$N_1,\ldots N_s$ are closed manifolds, $m=2\dim N_i+1>2\dim N_k+1$$m=2\dim N_i+1>2\dim N_k+1$ for every $1\le i\le t<k\le s$$1\le i\le t<k\le s$ and $N_1,\ldots N_t$$N_1,\ldots N_t$ are orientable. Then for an embedding $N_1,\ldots N_s$$N_1,\ldots N_s$ and every $1\le i<j\le t$$1\le i<j\le t$ one can define the linking coefficient $\lambda_{ij}(f)\in\Z$$\lambda_{ij}(f)\in\Z$, see Remark 3.2.e.

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.

Example 5.1 (Borromean rings). 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' are the three spheres given by the following three systems of equations:

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

Figure 6: The Borromean rings

See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.

An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, $\S$$\S$4.5 `Triple linking modulo 2' and $\S$$\S$4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.

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

Figure 7: The Borromean rings and Whitehead link for $l = 1$$l = 1$

Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, $\S$$\S$3, $\S$$\S$4]; see also the Wikipedia article on the Whitehead link. (For $l=1$$l=1$ the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

We have $\lambda_{12}(w)=0$$\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]). For $l\ne1,3,7$$l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)$$\lambda_{21}(w)$ equal to the Whitehead square $[\iota_l,\iota_l]\ne0$$[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$$\iota_l\in\pi_l(S^l)$. This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14]. For $l=1,3,7$$l=1,3,7$ the Whitehead link 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.

For some results on links $S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link [Skopenkov2015a, $\S$$\S$2.5].

6 Linked 3-manifolds in 6-space

An embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$$k,m,n$ such that $m-n$$m-n$ is even, one can explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ so that the following theorem holds.

Theorem 6.1. [Avvakumov2016] Any Brunnian embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is isotopic to fk,m,n for some integers k,m,n such that $m-n$$m-n$ is even. Two embeddings fk,m,n and fk′,m′,n′ are isotopic if and only if $k=k′$$k=k′$ and both $m-m′$$m-m′$ and $n-n′$$n-n′$ are divisible by $2k$$2k$.

The proof uses classification of embeddings $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ (the Haefliger Theorem 8.1 for $m=2p=2q=6$$m=2p=2q=6$). The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ is not trivial.

Corollary 6.2. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $g,g':S^3\sqcup S^3\to\Rr^6$$g,g':S^3\sqcup S^3\to\Rr^6$ such that the componentwise embedded connected sum $f\#g$$f\#g$ is isotopic to $f\#g'$$f\#g'$ but $g$$g$ is not isotopic to $g'$$g'$.

See an unpublished generalization in [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define $E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$$E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$ to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding $S^{n_i} \to S^m$$S^{n_i} \to S^m$. We remark that $E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$$E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$ [Haefliger1966a, $\S\S$$\S\S$ 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.5].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$$s$-tuple of the isotopy classes of its components:

$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$ $$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$$

Take $s$$s$ pairwise disjoint $m$$m$-discs in $S^m$$S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$$g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$. Define

$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$ $$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$$

Then $j$$j$ is a right inverse of the restriction homomorphism $r$$r$, i.e. $r\circ j=\mathrm{id}$$r\circ j=\mathrm{id}$. The unknotting homomorphism $u$$u$ is defined to be the homomorphism

$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$ $$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$$

Informally, the homomorphism $u$$u$ is obtained by taking embedded connected sums of components with knots $h_i:S^{n_i}\to S^m$$h_i:S^{n_i}\to S^m$ representing the elements of $E^m_D(S^{n_i})$$E^m_D(S^{n_i})$ inverse to the components, whose images $h_i(S^{n_i})$$h_i(S^{n_i})$ are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For $n_1,\ldots,n_s\le m-3$$n_1,\ldots,n_s\le m-3$, the homomorphism

$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$ $$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$$

is an isomorphism.

For some information on the groups $E^m_D(S^q)$$E^m_D(S^q)$ see [Skopenkov2016s], [Skopenkov2006, $\S$$\S$3.3].

8 Classification beyond the metastable range

The Haefliger Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of $\beta$$\beta$ see [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5], cf. [Skopenkov2007, $\S$$\S$2] and [Crowley&Skopenkov2016, $\S$$\S$2.3].

Remark 8.2. (a) The Haefliger Theorem 8.1 implies that for any $l\ge2$$l\ge2$ we have an isomorphism

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

(b) For any $l\not\in\{1,3,7\}$$l\not\in\{1,3,7\}$, 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+(-1)^lb)=0\}$$\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$.

For $l\ge4$$l\ge4$ see [Haefliger1962t, $\S$$\S$6]. The following proof for $l=2$$l=2$ and general remark are intended for specialists.

For $l=2$$l=2$ there is an exact sequence $\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$$\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$, where $\lambda_{12}$$\lambda_{12}$ is $\lambda_{12,PL}$$\lambda_{12,PL}$ not $\lambda_{12,D}$$\lambda_{12,D}$ [Haefliger1966a, Corollary 10.3]. We have $\pi_3(S^2)\cong\Zz$$\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$$\pi_4(S^3)\cong\Zz_2$ and $\Sigma$$\Sigma$ is the reduction modulo 2. By the exactness, $\text{im}\lambda_{21}=2\Z$$\text{im}\lambda_{21}=2\Z$. By (a) $\ker\lambda_{12}\cong\Z$$\ker\lambda_{12}\cong\Z$. Hence $\lambda_{21}|_{\ker\lambda_{12}}$$\lambda_{21}|_{\ker\lambda_{12}}$ is injective. We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$$\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of $\beta$$\beta$ [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5] and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$$\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ [Koschorke&Sanderson1977] one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$$\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$. Then (b) would follow.

(c) For any $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, $\S$$\S$6].

(d) For any $l\ge4$$l\ge4$, $l\ne7$$l\ne7$ we have an isomorphism

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

which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number ($\S$$\S$8). This follows from [Haefliger1962t, $\S$$\S$6]. We conjecture that this result holds also for $l=2,3,7$$l=2,3,7$.

9 Classification in codimension at least 3

In this section we assume that $(n)$$(n)$ is an $s$$s$-tuple such that $m\ge n_i+3$$m\ge n_i+3$. For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$$E^m_{PL}(S^{(n)})\otimes\Qq$ is obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.2] one are necessary and sufficient conditions on $(n)$$(n)$ when $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers $m,n_1,\ldots,n_s>0$$m,n_1,\ldots,n_s>0$

(a) calculate

Tex syntax error

${\rm rk}E^m_{PL}(S^{(n)})$.

(b) find out whether $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite.

Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups

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

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. In the above sequence the $s$$s$-tuples $m-n_1,\ldots,m-n_s$$m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$$W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$. For any $j=1,\ldots,s$$j=1,\ldots,s$ and positive integer $k$$k$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$$p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$$j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$$(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$$\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$.

Analogous to Definition 3.1 there is a canonical homotopy equivalence $C_f \sim W$$C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$$S^{(n)} \to S^m$ has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement. One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$$\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$$\lambda_j$ is a generalisation of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$$\lambda:= \oplus_{j=1}^s \lambda_j$.

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

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

Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

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

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

In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving the Haefliger Theorem 8.1 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

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• [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.
• [Koschorke&Sanderson1977] U. Koschorke and B. Sanderson, Geometric interpretation of the generalized Hopf invariant, Math. Scand. 41 (1977) 199--217.
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Massey1968] W. S. Massey, Higher order linking numbers, Proc. Conf. on Algebraic Topology, Univ. Illinois, Chicago Circle, Chicago, Ill., (1968) pp. 174--205. MR0254832 (40 #8039), see also MR1625365 (99e:57016) massey.pdf.
• [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.
• [Skopenkov2006b] M. Skopenkov, A formula for the group of links in the 2-metastable dimension, arxiv:math/0610320v1
• [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
• [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.
• [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, 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 $\partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}$. * or they are given as the points in $\Rr^{p+q+1}$ satisfying the following 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 also distinguished from the standard embedding by [[#The linking coefficient|the linking coefficient]] (see $\S$\ref{s:inv}). {{beginthm|Definition|(A link with prescribed [[#The linking coefficient|linking coefficient]])}}\label{dz} We define the `Zeeman' map [[Image:LinkingCoefficient.jpg|thumb|350px|Figure 3: A link with linking coefficient $\pm 1$]] $$\zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.$$ For a map $x:S^p\to S^{m-q-1}$ representing an element of $\pi_p(S^{m-q-1})$ let $$\overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times{\rm i}}\to S^{m-q-1}\times S^q\overset{{\rm i}_{m,q}}\to S^m,$$ see Figure 3. We have $\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let $\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$. {{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 $g(D^{m-q})$ intersects $f(S^q)$ transversely at exactly one point with positive sign; see Figure 4. [[Image:GaussMap-and-MeridianDisc.jpg|thumb|450px|Figure 4: The disc $gD^{m-q}$ and Gauss map $\widetilde f$]] Then the restriction $h':S^{m-q-1}\to S^m-fS^q$ of $g$ to $\partial D^{m-q}$ is a homotopy equivalence. (Indeed, since $m\ge q+3$, by general position the complement $S^m-fS^q$ is simply-connected. By Alexander duality, $h'$ induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems $h'$ is a homotopy equivalence.) Let $h$ be a homotopy inverse of $h'$. Define $$\lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$$ {{endthm}} {{beginthm|Remark}}\label{lkrem} (a) Clearly, $\lambda[f]$ is well-defined, i.e. is independent of the choices of $g,h',h$ and of the representative $f$ of $[f]$. One can check that $\lambda$ is a homomorphism. (b) 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. (c) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$ for the [[#Examples|Zeman map]] $\zeta$. So $\lambda$ is surjective and $\zeta$ is injective. (d) For $m=p+q+1$ there is a simpler alternative definitions using homological ideas. That definition 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}. {{endthm}} Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$. The stable suspension 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 the Gauss map (see Figure 4) $$\widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.$$ 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 the suspension isomorphism. 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 Figure 5. [[Image:33.jpg|thumb|450px|Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere]] 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}} One can easily check that $\alpha$ is well-defined and for $p,q\le m-3$ is a homomorphism. {{beginthm|Lemma|\cite[Lemma 5.1]{Kervaire1959a}}} We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$. {{endthm}} Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$ as we explain in $\S$\ref{s:ebmr}. Note that the $\alpha$-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 \Rr^m. Known cases are embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$$S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$$m-3\ge n_i$, embeddings $N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$$N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$ for $m\ge 2n_i+1$$m\ge 2n_i+1$ for every $i$$i$, and embeddings of some disconnected 3-manifolds in $S^6$$S^6$. For a related classification of knotted tori see [Skopenkov2016k].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3].

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

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

Figure 1: Embedded connected sum

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{(n)})$$E^m(S^{(n)})$ for $m-3\ge n_i$$m-3\ge n_i$ [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see Figure 1.

The standard embedding $S^k\to D^m$$S^k\to D^m$ is defined by $(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$$(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$. Fix $s$$s$ pairwise disjoint $m$$m$-discs $D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$$D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$. The standard embedding $S^{(n)}\to S^m$$S^{(n)}\to S^m$ is defined by taking the union of the compositions of the standard embeddings $S^{n_k} \to D^m_k$$S^{n_k} \to D^m_k$ with the fixed inclusions $D^m_k \to S^m$$D^m_k \to S^m$.

2 Examples

Recall that for any $q$$q$-manifold $N$$N$ and $m\ge2q+2$$m\ge2q+2$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction $m\ge2q+2$$m\ge2q+2$ is sharp for non-connected manifolds.

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

Figure 2: The Hopf link (a) and the trivial link (b)

For $q=1$$q=1$ the Hopf link is shown in Figure 2. For all $q$$q$ the image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows:

• either the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$;

• or they are given as the sets of points in $\Rr^{2q+1}$$\Rr^{2q+1}$ satisfying the following equations:

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

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

Analogously for any $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 which can be described as follows:

• either the spheres are $\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})\cong S^{p+q+1}$$\partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}$.

• or they are given as the points in $\Rr^{p+q+1}$$\Rr^{p+q+1}$ satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see $\S$$\S$3).

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient $\pm 1$$\pm 1$

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

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

Tex syntax error

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

see Figure 3. We have

Tex syntax error

$\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let

Tex syntax error

$\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$.

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$$g:D^{m-q}\to S^m$ such that $g(D^{m-q})$$g(D^{m-q})$ intersects $f(S^q)$$f(S^q)$ transversely at exactly one point with positive sign; see Figure 4.

Figure 4: The disc $gD^{m-q}$$gD^{m-q}$ and Gauss map $\widetilde f$$\widetilde f$

Then the restriction $h':S^{m-q-1}\to S^m-fS^q$$h':S^{m-q-1}\to S^m-fS^q$ of $g$$g$ to $\partial D^{m-q}$$\partial D^{m-q}$ is a homotopy equivalence.

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$$\lambda[f]$ is well-defined, i.e. is independent of the choices of $g,h',h$$g,h',h$ and of the representative $f$$f$ of $[f]$$[f]$. One can check that $\lambda$$\lambda$ is a homomorphism.

(b) 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.

(c) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$$\lambda\zeta=\id\pi_p(S^{m-q-1})$ for the Zeman map $\zeta$$\zeta$. So $\lambda$$\lambda$ is surjective and $\zeta$$\zeta$ is injective.

(d) For $m=p+q+1$$m=p+q+1$ there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].

Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism $\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 stable suspension 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 the Gauss map (see Figure 4)

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

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 the suspension isomorphism. 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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

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)$.)

One can easily check that $\alpha$$\alpha$ is well-defined and for $p,q\le m-3$$p,q\le m-3$ is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$$p,q\le m-3$.

Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$$\lambda_{12} \neq \pm \lambda_{21}$ as we explain in $\S$$\S$8.

Note that the $\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, each of the same dimension [Haefliger1962t, Theorem at the end of $\S$$\S$5], [Haefliger1966a]. Let $(q) = (q, \dots, q)$$(q) = (q, \dots, q)$ be the $s$$s$-tuple consisting entirely of some positive integer $q$$q$.

Theorem 4.2. The collection of pairwise linking coefficients

$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$ $$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$$

is bijective for $m\ge\frac{3q}2+2$$m\ge\frac{3q}2+2$.

Assume that $N_1,\ldots N_s$$N_1,\ldots N_s$ are closed manifolds, $m=2\dim N_i+1>2\dim N_k+1$$m=2\dim N_i+1>2\dim N_k+1$ for every $1\le i\le t<k\le s$$1\le i\le t<k\le s$ and $N_1,\ldots N_t$$N_1,\ldots N_t$ are orientable. Then for an embedding $N_1,\ldots N_s$$N_1,\ldots N_s$ and every $1\le i<j\le t$$1\le i<j\le t$ one can define the linking coefficient $\lambda_{ij}(f)\in\Z$$\lambda_{ij}(f)\in\Z$, see Remark 3.2.e.

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.

Example 5.1 (Borromean rings). 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' are the three spheres given by the following three systems of equations:

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

Figure 6: The Borromean rings

See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.

An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, $\S$$\S$4.5 `Triple linking modulo 2' and $\S$$\S$4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.

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

Figure 7: The Borromean rings and Whitehead link for $l = 1$$l = 1$

Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, $\S$$\S$3, $\S$$\S$4]; see also the Wikipedia article on the Whitehead link. (For $l=1$$l=1$ the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

We have $\lambda_{12}(w)=0$$\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]). For $l\ne1,3,7$$l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)$$\lambda_{21}(w)$ equal to the Whitehead square $[\iota_l,\iota_l]\ne0$$[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$$\iota_l\in\pi_l(S^l)$. This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14]. For $l=1,3,7$$l=1,3,7$ the Whitehead link 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.

For some results on links $S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link [Skopenkov2015a, $\S$$\S$2.5].

6 Linked 3-manifolds in 6-space

An embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$$k,m,n$ such that $m-n$$m-n$ is even, one can explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ so that the following theorem holds.

Theorem 6.1. [Avvakumov2016] Any Brunnian embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is isotopic to fk,m,n for some integers k,m,n such that $m-n$$m-n$ is even. Two embeddings fk,m,n and fk′,m′,n′ are isotopic if and only if $k=k′$$k=k′$ and both $m-m′$$m-m′$ and $n-n′$$n-n′$ are divisible by $2k$$2k$.

The proof uses classification of embeddings $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ (the Haefliger Theorem 8.1 for $m=2p=2q=6$$m=2p=2q=6$). The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ is not trivial.

Corollary 6.2. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $g,g':S^3\sqcup S^3\to\Rr^6$$g,g':S^3\sqcup S^3\to\Rr^6$ such that the componentwise embedded connected sum $f\#g$$f\#g$ is isotopic to $f\#g'$$f\#g'$ but $g$$g$ is not isotopic to $g'$$g'$.

See an unpublished generalization in [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define $E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$$E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$ to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding $S^{n_i} \to S^m$$S^{n_i} \to S^m$. We remark that $E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$$E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$ [Haefliger1966a, $\S\S$$\S\S$ 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.5].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$$s$-tuple of the isotopy classes of its components:

$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$ $$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$$

Take $s$$s$ pairwise disjoint $m$$m$-discs in $S^m$$S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$$g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$. Define

$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$ $$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$$

Then $j$$j$ is a right inverse of the restriction homomorphism $r$$r$, i.e. $r\circ j=\mathrm{id}$$r\circ j=\mathrm{id}$. The unknotting homomorphism $u$$u$ is defined to be the homomorphism

$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$ $$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$$

Informally, the homomorphism $u$$u$ is obtained by taking embedded connected sums of components with knots $h_i:S^{n_i}\to S^m$$h_i:S^{n_i}\to S^m$ representing the elements of $E^m_D(S^{n_i})$$E^m_D(S^{n_i})$ inverse to the components, whose images $h_i(S^{n_i})$$h_i(S^{n_i})$ are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For $n_1,\ldots,n_s\le m-3$$n_1,\ldots,n_s\le m-3$, the homomorphism

$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$ $$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$$

is an isomorphism.

For some information on the groups $E^m_D(S^q)$$E^m_D(S^q)$ see [Skopenkov2016s], [Skopenkov2006, $\S$$\S$3.3].

8 Classification beyond the metastable range

The Haefliger Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of $\beta$$\beta$ see [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5], cf. [Skopenkov2007, $\S$$\S$2] and [Crowley&Skopenkov2016, $\S$$\S$2.3].

Remark 8.2. (a) The Haefliger Theorem 8.1 implies that for any $l\ge2$$l\ge2$ we have an isomorphism

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

(b) For any $l\not\in\{1,3,7\}$$l\not\in\{1,3,7\}$, 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+(-1)^lb)=0\}$$\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$.

For $l\ge4$$l\ge4$ see [Haefliger1962t, $\S$$\S$6]. The following proof for $l=2$$l=2$ and general remark are intended for specialists.

For $l=2$$l=2$ there is an exact sequence $\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$$\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$, where $\lambda_{12}$$\lambda_{12}$ is $\lambda_{12,PL}$$\lambda_{12,PL}$ not $\lambda_{12,D}$$\lambda_{12,D}$ [Haefliger1966a, Corollary 10.3]. We have $\pi_3(S^2)\cong\Zz$$\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$$\pi_4(S^3)\cong\Zz_2$ and $\Sigma$$\Sigma$ is the reduction modulo 2. By the exactness, $\text{im}\lambda_{21}=2\Z$$\text{im}\lambda_{21}=2\Z$. By (a) $\ker\lambda_{12}\cong\Z$$\ker\lambda_{12}\cong\Z$. Hence $\lambda_{21}|_{\ker\lambda_{12}}$$\lambda_{21}|_{\ker\lambda_{12}}$ is injective. We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$$\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of $\beta$$\beta$ [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5] and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$$\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ [Koschorke&Sanderson1977] one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$$\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$. Then (b) would follow.

(c) For any $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, $\S$$\S$6].

(d) For any $l\ge4$$l\ge4$, $l\ne7$$l\ne7$ we have an isomorphism

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

which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number ($\S$$\S$8). This follows from [Haefliger1962t, $\S$$\S$6]. We conjecture that this result holds also for $l=2,3,7$$l=2,3,7$.

9 Classification in codimension at least 3

In this section we assume that $(n)$$(n)$ is an $s$$s$-tuple such that $m\ge n_i+3$$m\ge n_i+3$. For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$$E^m_{PL}(S^{(n)})\otimes\Qq$ is obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.2] one are necessary and sufficient conditions on $(n)$$(n)$ when $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers $m,n_1,\ldots,n_s>0$$m,n_1,\ldots,n_s>0$

(a) calculate

Tex syntax error

${\rm rk}E^m_{PL}(S^{(n)})$.

(b) find out whether $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite.

Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups

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

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. In the above sequence the $s$$s$-tuples $m-n_1,\ldots,m-n_s$$m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$$W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$. For any $j=1,\ldots,s$$j=1,\ldots,s$ and positive integer $k$$k$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$$p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$$j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$$(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$$\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$.

Analogous to Definition 3.1 there is a canonical homotopy equivalence $C_f \sim W$$C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$$S^{(n)} \to S^m$ has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement. One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$$\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$$\lambda_j$ is a generalisation of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$$\lambda:= \oplus_{j=1}^s \lambda_j$.

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

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

Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

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

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

In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving the Haefliger Theorem 8.1 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

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• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
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• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Massey1968] W. S. Massey, Higher order linking numbers, Proc. Conf. on Algebraic Topology, Univ. Illinois, Chicago Circle, Chicago, Ill., (1968) pp. 174--205. MR0254832 (40 #8039), see also MR1625365 (99e:57016) massey.pdf.
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• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
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\le p\le q$, then both $\lambda$ and $\zeta$ are isomorphisms for $m\ge\frac{3q}2+2$ in the smooth category. (PL) If \Rr^m. Known cases are embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$$S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$$m-3\ge n_i$, embeddings $N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$$N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$ for $m\ge 2n_i+1$$m\ge 2n_i+1$ for every $i$$i$, and embeddings of some disconnected 3-manifolds in $S^6$$S^6$. For a related classification of knotted tori see [Skopenkov2016k].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3].

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

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

Figure 1: Embedded connected sum

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{(n)})$$E^m(S^{(n)})$ for $m-3\ge n_i$$m-3\ge n_i$ [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see Figure 1.

The standard embedding $S^k\to D^m$$S^k\to D^m$ is defined by $(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$$(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$. Fix $s$$s$ pairwise disjoint $m$$m$-discs $D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$$D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$. The standard embedding $S^{(n)}\to S^m$$S^{(n)}\to S^m$ is defined by taking the union of the compositions of the standard embeddings $S^{n_k} \to D^m_k$$S^{n_k} \to D^m_k$ with the fixed inclusions $D^m_k \to S^m$$D^m_k \to S^m$.

2 Examples

Recall that for any $q$$q$-manifold $N$$N$ and $m\ge2q+2$$m\ge2q+2$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction $m\ge2q+2$$m\ge2q+2$ is sharp for non-connected manifolds.

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

Figure 2: The Hopf link (a) and the trivial link (b)

For $q=1$$q=1$ the Hopf link is shown in Figure 2. For all $q$$q$ the image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows:

• either the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$;

• or they are given as the sets of points in $\Rr^{2q+1}$$\Rr^{2q+1}$ satisfying the following equations:

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

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

Analogously for any $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 which can be described as follows:

• either the spheres are $\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})\cong S^{p+q+1}$$\partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}$.

• or they are given as the points in $\Rr^{p+q+1}$$\Rr^{p+q+1}$ satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see $\S$$\S$3).

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient $\pm 1$$\pm 1$

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

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

Tex syntax error

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

see Figure 3. We have

Tex syntax error

$\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let

Tex syntax error

$\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$.

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$$g:D^{m-q}\to S^m$ such that $g(D^{m-q})$$g(D^{m-q})$ intersects $f(S^q)$$f(S^q)$ transversely at exactly one point with positive sign; see Figure 4.

Figure 4: The disc $gD^{m-q}$$gD^{m-q}$ and Gauss map $\widetilde f$$\widetilde f$

Then the restriction $h':S^{m-q-1}\to S^m-fS^q$$h':S^{m-q-1}\to S^m-fS^q$ of $g$$g$ to $\partial D^{m-q}$$\partial D^{m-q}$ is a homotopy equivalence.

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$$\lambda[f]$ is well-defined, i.e. is independent of the choices of $g,h',h$$g,h',h$ and of the representative $f$$f$ of $[f]$$[f]$. One can check that $\lambda$$\lambda$ is a homomorphism.

(b) 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.

(c) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$$\lambda\zeta=\id\pi_p(S^{m-q-1})$ for the Zeman map $\zeta$$\zeta$. So $\lambda$$\lambda$ is surjective and $\zeta$$\zeta$ is injective.

(d) For $m=p+q+1$$m=p+q+1$ there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].

Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism $\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 stable suspension 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 the Gauss map (see Figure 4)

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

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 the suspension isomorphism. 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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

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)$.)

One can easily check that $\alpha$$\alpha$ is well-defined and for $p,q\le m-3$$p,q\le m-3$ is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$$p,q\le m-3$.

Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$$\lambda_{12} \neq \pm \lambda_{21}$ as we explain in $\S$$\S$8.

Note that the $\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, each of the same dimension [Haefliger1962t, Theorem at the end of $\S$$\S$5], [Haefliger1966a]. Let $(q) = (q, \dots, q)$$(q) = (q, \dots, q)$ be the $s$$s$-tuple consisting entirely of some positive integer $q$$q$.

Theorem 4.2. The collection of pairwise linking coefficients

$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$ $$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$$

is bijective for $m\ge\frac{3q}2+2$$m\ge\frac{3q}2+2$.

Assume that $N_1,\ldots N_s$$N_1,\ldots N_s$ are closed manifolds, $m=2\dim N_i+1>2\dim N_k+1$$m=2\dim N_i+1>2\dim N_k+1$ for every $1\le i\le t<k\le s$$1\le i\le t<k\le s$ and $N_1,\ldots N_t$$N_1,\ldots N_t$ are orientable. Then for an embedding $N_1,\ldots N_s$$N_1,\ldots N_s$ and every $1\le i<j\le t$$1\le i<j\le t$ one can define the linking coefficient $\lambda_{ij}(f)\in\Z$$\lambda_{ij}(f)\in\Z$, see Remark 3.2.e.

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.

Example 5.1 (Borromean rings). 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' are the three spheres given by the following three systems of equations:

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

Figure 6: The Borromean rings

See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.

An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, $\S$$\S$4.5 `Triple linking modulo 2' and $\S$$\S$4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.

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

Figure 7: The Borromean rings and Whitehead link for $l = 1$$l = 1$

Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, $\S$$\S$3, $\S$$\S$4]; see also the Wikipedia article on the Whitehead link. (For $l=1$$l=1$ the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

We have $\lambda_{12}(w)=0$$\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]). For $l\ne1,3,7$$l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)$$\lambda_{21}(w)$ equal to the Whitehead square $[\iota_l,\iota_l]\ne0$$[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$$\iota_l\in\pi_l(S^l)$. This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14]. For $l=1,3,7$$l=1,3,7$ the Whitehead link 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.

For some results on links $S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link [Skopenkov2015a, $\S$$\S$2.5].

6 Linked 3-manifolds in 6-space

An embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$$k,m,n$ such that $m-n$$m-n$ is even, one can explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ so that the following theorem holds.

Theorem 6.1. [Avvakumov2016] Any Brunnian embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is isotopic to fk,m,n for some integers k,m,n such that $m-n$$m-n$ is even. Two embeddings fk,m,n and fk′,m′,n′ are isotopic if and only if $k=k′$$k=k′$ and both $m-m′$$m-m′$ and $n-n′$$n-n′$ are divisible by $2k$$2k$.

The proof uses classification of embeddings $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ (the Haefliger Theorem 8.1 for $m=2p=2q=6$$m=2p=2q=6$). The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ is not trivial.

Corollary 6.2. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $g,g':S^3\sqcup S^3\to\Rr^6$$g,g':S^3\sqcup S^3\to\Rr^6$ such that the componentwise embedded connected sum $f\#g$$f\#g$ is isotopic to $f\#g'$$f\#g'$ but $g$$g$ is not isotopic to $g'$$g'$.

See an unpublished generalization in [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define $E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$$E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$ to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding $S^{n_i} \to S^m$$S^{n_i} \to S^m$. We remark that $E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$$E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$ [Haefliger1966a, $\S\S$$\S\S$ 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.5].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$$s$-tuple of the isotopy classes of its components:

$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$ $$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$$

Take $s$$s$ pairwise disjoint $m$$m$-discs in $S^m$$S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$$g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$. Define

$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$ $$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$$

Then $j$$j$ is a right inverse of the restriction homomorphism $r$$r$, i.e. $r\circ j=\mathrm{id}$$r\circ j=\mathrm{id}$. The unknotting homomorphism $u$$u$ is defined to be the homomorphism

$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$ $$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$$

Informally, the homomorphism $u$$u$ is obtained by taking embedded connected sums of components with knots $h_i:S^{n_i}\to S^m$$h_i:S^{n_i}\to S^m$ representing the elements of $E^m_D(S^{n_i})$$E^m_D(S^{n_i})$ inverse to the components, whose images $h_i(S^{n_i})$$h_i(S^{n_i})$ are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For $n_1,\ldots,n_s\le m-3$$n_1,\ldots,n_s\le m-3$, the homomorphism

$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$ $$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$$

is an isomorphism.

For some information on the groups $E^m_D(S^q)$$E^m_D(S^q)$ see [Skopenkov2016s], [Skopenkov2006, $\S$$\S$3.3].

8 Classification beyond the metastable range

The Haefliger Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of $\beta$$\beta$ see [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5], cf. [Skopenkov2007, $\S$$\S$2] and [Crowley&Skopenkov2016, $\S$$\S$2.3].

Remark 8.2. (a) The Haefliger Theorem 8.1 implies that for any $l\ge2$$l\ge2$ we have an isomorphism

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

(b) For any $l\not\in\{1,3,7\}$$l\not\in\{1,3,7\}$, 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+(-1)^lb)=0\}$$\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$.

For $l\ge4$$l\ge4$ see [Haefliger1962t, $\S$$\S$6]. The following proof for $l=2$$l=2$ and general remark are intended for specialists.

For $l=2$$l=2$ there is an exact sequence $\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$$\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$, where $\lambda_{12}$$\lambda_{12}$ is $\lambda_{12,PL}$$\lambda_{12,PL}$ not $\lambda_{12,D}$$\lambda_{12,D}$ [Haefliger1966a, Corollary 10.3]. We have $\pi_3(S^2)\cong\Zz$$\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$$\pi_4(S^3)\cong\Zz_2$ and $\Sigma$$\Sigma$ is the reduction modulo 2. By the exactness, $\text{im}\lambda_{21}=2\Z$$\text{im}\lambda_{21}=2\Z$. By (a) $\ker\lambda_{12}\cong\Z$$\ker\lambda_{12}\cong\Z$. Hence $\lambda_{21}|_{\ker\lambda_{12}}$$\lambda_{21}|_{\ker\lambda_{12}}$ is injective. We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$$\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of $\beta$$\beta$ [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5] and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$$\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ [Koschorke&Sanderson1977] one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$$\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$. Then (b) would follow.

(c) For any $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, $\S$$\S$6].

(d) For any $l\ge4$$l\ge4$, $l\ne7$$l\ne7$ we have an isomorphism

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

which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number ($\S$$\S$8). This follows from [Haefliger1962t, $\S$$\S$6]. We conjecture that this result holds also for $l=2,3,7$$l=2,3,7$.

9 Classification in codimension at least 3

In this section we assume that $(n)$$(n)$ is an $s$$s$-tuple such that $m\ge n_i+3$$m\ge n_i+3$. For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$$E^m_{PL}(S^{(n)})\otimes\Qq$ is obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.2] one are necessary and sufficient conditions on $(n)$$(n)$ when $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers $m,n_1,\ldots,n_s>0$$m,n_1,\ldots,n_s>0$

(a) calculate

Tex syntax error

${\rm rk}E^m_{PL}(S^{(n)})$.

(b) find out whether $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite.

Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups

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

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. In the above sequence the $s$$s$-tuples $m-n_1,\ldots,m-n_s$$m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$$W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$. For any $j=1,\ldots,s$$j=1,\ldots,s$ and positive integer $k$$k$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$$p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$$j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$$(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$$\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$.

Analogous to Definition 3.1 there is a canonical homotopy equivalence $C_f \sim W$$C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$$S^{(n)} \to S^m$ has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement. One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$$\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$$\lambda_j$ is a generalisation of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$$\lambda:= \oplus_{j=1}^s \lambda_j$.

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

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

Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

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

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

In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving the Haefliger Theorem 8.1 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

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\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, each of the same dimension \cite[Theorem at the end of $\S]{Haefliger1962t}, \cite{Haefliger1966a}. Let $(q) = (q, \dots, q)$ be the $s$-tuple consisting entirely of some positive integer $q$. {{beginthm|Theorem}}\label{t:lkmany} The collection of pairwise linking coefficients $$\bigoplus\limits_{1\le i ==Examples beyond the metastable range== ;\label{s:ebmr} We present an example of the ''non-injectivity of the collection of pairwise linking coefficients'', which shows that the dimension restriction is sharp in Theorem \ref{t:lkmany}. {{beginthm|Example|(Borromean rings)}}\label{belmetbor} The embedding defined below is a non-trivial embedding $S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ whose restrictions to each 2-component sublink is trivial \cite[4.1]{Haefliger1962}, \cite{Haefliger1962t}. Denote coordinates in $\Rr^{3l}$ by $(x,y,z)=(x_1,\dots,x_l,y_1,\dots,y_l,z_1,\dots,z_l)$. The (higher-dimensional) [[Wikipedia:Borromean_rings|`Borromean rings']] are the three spheres given by the following three systems of equations: $$\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.$$ [[Image:Borromean.jpg|thumb|400px|Figure 6: The Borromean rings]] See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings. {{endthm}} An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known ''triple linking number'' called the ''Massey number'' \cite{Massey1968}, for an elementary definition see \cite[$\S.5 `Triple linking modulo 2' and $\S.6 `Massey-Milnor and Sato-Levine numbers']{Skopenkov2017}. (Also, this embedding is not isotopic to the standard embedding 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|Example|(Whitehead link)}}\label{belmetwhi} For every positive integer $l$ 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. [[Image:Borromean-to-Whitehead.jpg|thumb|400px|Figure 7: The Borromean rings and Whitehead link for $l = 1$]] Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the [[Embeddings_just_below_the_stable_range:_classification#Embedded_connected_sum|embedded connected sum]] of the components \cite[$\S, $\S]{Skopenkov2016c}; see also the [[Wikipedia:Whitehead_link|Wikipedia article on the Whitehead link]]. (For $l=1$ the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7. {{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 e.g. in \cite{Skopenkov2006a}). For $l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)$ equal to the Whitehead square $[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$. This fact should be well-known, but I do not know a published proof except \cite[the Whitehead link Lemma 2.14]{Skopenkov2015a}. For $l=1,3,7$ the Whitehead link 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. For some results on links $S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link \cite[$\S.5]{Skopenkov2015a}. == Linked 3-manifolds in 6-space == ; An embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$ is ''Brunnian'' if its restriction to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$ such that $m-n$ is even, one can explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ so that the following theorem holds. {{beginthm|Theorem}}\label{t:avv} \cite{Avvakumov2016} Any Brunnian embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$ is isotopic to fk,m,n for some integers k,m,n such that $m-n$ is even. Two embeddings fk,m,n and fk′,m′,n′ are isotopic if and only if $k=k′$ and both $m-m′$ and $n-n′$ are divisible by k$. {{endthm}} The proof uses classification of embeddings $S^3\sqcup S^3\to\Rr^6$ (the Haefliger Theorem \ref{belmethae} for $m=2p=2q=6$). The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $S^3\sqcup S^3\to\Rr^6$ is not trivial. {{beginthm|Corollary}}\label{t:avv} \cite{Avvakumov2016}, cf. \cite[Corollary 3.5.b]{Skopenkov2016t} There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $g,g':S^3\sqcup S^3\to\Rr^6$ such that the componentwise embedded connected sum $f\#g$ is isotopic to $f\#g'$ but $g$ is not isotopic to $g'$. {{endthm}} See an unpublished generalization in \cite{Avvakumov2017}. == Reduction to the case with unknotted components == ; In this section we assume that $(n)$ is an $s$-tuple such that $m\ge n_i+3$. Define $E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$ to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding $S^{n_i} \to S^m$. We remark that $E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$ \cite[$\S\S$ 2.4, 2.6 and 9.3]{Haefliger1966a}, \cite[$\S\Rr^m. Known cases are 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$ for $m-3\ge n_i$$m-3\ge n_i$, embeddings $N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$$N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$ for $m\ge 2n_i+1$$m\ge 2n_i+1$ for every $i$$i$, and embeddings of some disconnected 3-manifolds in $S^6$$S^6$. For a related classification of knotted tori see [Skopenkov2016k].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3].

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

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

Figure 1: Embedded connected sum

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{(n)})$$E^m(S^{(n)})$ for $m-3\ge n_i$$m-3\ge n_i$ [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see Figure 1.

The standard embedding $S^k\to D^m$$S^k\to D^m$ is defined by $(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$$(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$. Fix $s$$s$ pairwise disjoint $m$$m$-discs $D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$$D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$. The standard embedding $S^{(n)}\to S^m$$S^{(n)}\to S^m$ is defined by taking the union of the compositions of the standard embeddings $S^{n_k} \to D^m_k$$S^{n_k} \to D^m_k$ with the fixed inclusions $D^m_k \to S^m$$D^m_k \to S^m$.

2 Examples

Recall that for any $q$$q$-manifold $N$$N$ and $m\ge2q+2$$m\ge2q+2$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction $m\ge2q+2$$m\ge2q+2$ is sharp for non-connected manifolds.

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

Figure 2: The Hopf link (a) and the trivial link (b)

For $q=1$$q=1$ the Hopf link is shown in Figure 2. For all $q$$q$ the image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows:

• either the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$;

• or they are given as the sets of points in $\Rr^{2q+1}$$\Rr^{2q+1}$ satisfying the following equations:

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

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

Analogously for any $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 which can be described as follows:

• either the spheres are $\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})\cong S^{p+q+1}$$\partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}$.

• or they are given as the points in $\Rr^{p+q+1}$$\Rr^{p+q+1}$ satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see $\S$$\S$3).

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient $\pm 1$$\pm 1$

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

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

Tex syntax error

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

see Figure 3. We have

Tex syntax error

$\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let

Tex syntax error

$\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$.

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$$g:D^{m-q}\to S^m$ such that $g(D^{m-q})$$g(D^{m-q})$ intersects $f(S^q)$$f(S^q)$ transversely at exactly one point with positive sign; see Figure 4.

Figure 4: The disc $gD^{m-q}$$gD^{m-q}$ and Gauss map $\widetilde f$$\widetilde f$

Then the restriction $h':S^{m-q-1}\to S^m-fS^q$$h':S^{m-q-1}\to S^m-fS^q$ of $g$$g$ to $\partial D^{m-q}$$\partial D^{m-q}$ is a homotopy equivalence.

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$$\lambda[f]$ is well-defined, i.e. is independent of the choices of $g,h',h$$g,h',h$ and of the representative $f$$f$ of $[f]$$[f]$. One can check that $\lambda$$\lambda$ is a homomorphism.

(b) 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.

(c) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$$\lambda\zeta=\id\pi_p(S^{m-q-1})$ for the Zeman map $\zeta$$\zeta$. So $\lambda$$\lambda$ is surjective and $\zeta$$\zeta$ is injective.

(d) For $m=p+q+1$$m=p+q+1$ there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].

Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism $\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 stable suspension 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 the Gauss map (see Figure 4)

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

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 the suspension isomorphism. 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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

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)$.)

One can easily check that $\alpha$$\alpha$ is well-defined and for $p,q\le m-3$$p,q\le m-3$ is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$$p,q\le m-3$.

Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$$\lambda_{12} \neq \pm \lambda_{21}$ as we explain in $\S$$\S$8.

Note that the $\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, each of the same dimension [Haefliger1962t, Theorem at the end of $\S$$\S$5], [Haefliger1966a]. Let $(q) = (q, \dots, q)$$(q) = (q, \dots, q)$ be the $s$$s$-tuple consisting entirely of some positive integer $q$$q$.

Theorem 4.2. The collection of pairwise linking coefficients

$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$ $$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$$

is bijective for $m\ge\frac{3q}2+2$$m\ge\frac{3q}2+2$.

Assume that $N_1,\ldots N_s$$N_1,\ldots N_s$ are closed manifolds, $m=2\dim N_i+1>2\dim N_k+1$$m=2\dim N_i+1>2\dim N_k+1$ for every $1\le i\le t<k\le s$$1\le i\le t<k\le s$ and $N_1,\ldots N_t$$N_1,\ldots N_t$ are orientable. Then for an embedding $N_1,\ldots N_s$$N_1,\ldots N_s$ and every $1\le i<j\le t$$1\le i<j\le t$ one can define the linking coefficient $\lambda_{ij}(f)\in\Z$$\lambda_{ij}(f)\in\Z$, see Remark 3.2.e.

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.

Example 5.1 (Borromean rings). 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' are the three spheres given by the following three systems of equations:

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

Figure 6: The Borromean rings

See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.

An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, $\S$$\S$4.5 `Triple linking modulo 2' and $\S$$\S$4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.

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

Figure 7: The Borromean rings and Whitehead link for $l = 1$$l = 1$

Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, $\S$$\S$3, $\S$$\S$4]; see also the Wikipedia article on the Whitehead link. (For $l=1$$l=1$ the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

We have $\lambda_{12}(w)=0$$\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]). For $l\ne1,3,7$$l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)$$\lambda_{21}(w)$ equal to the Whitehead square $[\iota_l,\iota_l]\ne0$$[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$$\iota_l\in\pi_l(S^l)$. This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14]. For $l=1,3,7$$l=1,3,7$ the Whitehead link 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.

For some results on links $S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link [Skopenkov2015a, $\S$$\S$2.5].

6 Linked 3-manifolds in 6-space

An embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$$k,m,n$ such that $m-n$$m-n$ is even, one can explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ so that the following theorem holds.

Theorem 6.1. [Avvakumov2016] Any Brunnian embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is isotopic to fk,m,n for some integers k,m,n such that $m-n$$m-n$ is even. Two embeddings fk,m,n and fk′,m′,n′ are isotopic if and only if $k=k′$$k=k′$ and both $m-m′$$m-m′$ and $n-n′$$n-n′$ are divisible by $2k$$2k$.

The proof uses classification of embeddings $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ (the Haefliger Theorem 8.1 for $m=2p=2q=6$$m=2p=2q=6$). The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ is not trivial.

Corollary 6.2. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $g,g':S^3\sqcup S^3\to\Rr^6$$g,g':S^3\sqcup S^3\to\Rr^6$ such that the componentwise embedded connected sum $f\#g$$f\#g$ is isotopic to $f\#g'$$f\#g'$ but $g$$g$ is not isotopic to $g'$$g'$.

See an unpublished generalization in [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define $E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$$E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$ to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding $S^{n_i} \to S^m$$S^{n_i} \to S^m$. We remark that $E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$$E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$ [Haefliger1966a, $\S\S$$\S\S$ 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.5].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$$s$-tuple of the isotopy classes of its components:

$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$ $$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$$

Take $s$$s$ pairwise disjoint $m$$m$-discs in $S^m$$S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$$g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$. Define

$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$ $$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$$

Then $j$$j$ is a right inverse of the restriction homomorphism $r$$r$, i.e. $r\circ j=\mathrm{id}$$r\circ j=\mathrm{id}$. The unknotting homomorphism $u$$u$ is defined to be the homomorphism

$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$ $$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$$

Informally, the homomorphism $u$$u$ is obtained by taking embedded connected sums of components with knots $h_i:S^{n_i}\to S^m$$h_i:S^{n_i}\to S^m$ representing the elements of $E^m_D(S^{n_i})$$E^m_D(S^{n_i})$ inverse to the components, whose images $h_i(S^{n_i})$$h_i(S^{n_i})$ are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For $n_1,\ldots,n_s\le m-3$$n_1,\ldots,n_s\le m-3$, the homomorphism

$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$ $$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$$

is an isomorphism.

For some information on the groups $E^m_D(S^q)$$E^m_D(S^q)$ see [Skopenkov2016s], [Skopenkov2006, $\S$$\S$3.3].

8 Classification beyond the metastable range

The Haefliger Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of $\beta$$\beta$ see [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5], cf. [Skopenkov2007, $\S$$\S$2] and [Crowley&Skopenkov2016, $\S$$\S$2.3].

Remark 8.2. (a) The Haefliger Theorem 8.1 implies that for any $l\ge2$$l\ge2$ we have an isomorphism

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

(b) For any $l\not\in\{1,3,7\}$$l\not\in\{1,3,7\}$, 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+(-1)^lb)=0\}$$\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$.

For $l\ge4$$l\ge4$ see [Haefliger1962t, $\S$$\S$6]. The following proof for $l=2$$l=2$ and general remark are intended for specialists.

For $l=2$$l=2$ there is an exact sequence $\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$$\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$, where $\lambda_{12}$$\lambda_{12}$ is $\lambda_{12,PL}$$\lambda_{12,PL}$ not $\lambda_{12,D}$$\lambda_{12,D}$ [Haefliger1966a, Corollary 10.3]. We have $\pi_3(S^2)\cong\Zz$$\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$$\pi_4(S^3)\cong\Zz_2$ and $\Sigma$$\Sigma$ is the reduction modulo 2. By the exactness, $\text{im}\lambda_{21}=2\Z$$\text{im}\lambda_{21}=2\Z$. By (a) $\ker\lambda_{12}\cong\Z$$\ker\lambda_{12}\cong\Z$. Hence $\lambda_{21}|_{\ker\lambda_{12}}$$\lambda_{21}|_{\ker\lambda_{12}}$ is injective. We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$$\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of $\beta$$\beta$ [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5] and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$$\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ [Koschorke&Sanderson1977] one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$$\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$. Then (b) would follow.

(c) For any $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, $\S$$\S$6].

(d) For any $l\ge4$$l\ge4$, $l\ne7$$l\ne7$ we have an isomorphism

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

which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number ($\S$$\S$8). This follows from [Haefliger1962t, $\S$$\S$6]. We conjecture that this result holds also for $l=2,3,7$$l=2,3,7$.

9 Classification in codimension at least 3

In this section we assume that $(n)$$(n)$ is an $s$$s$-tuple such that $m\ge n_i+3$$m\ge n_i+3$. For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$$E^m_{PL}(S^{(n)})\otimes\Qq$ is obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.2] one are necessary and sufficient conditions on $(n)$$(n)$ when $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers $m,n_1,\ldots,n_s>0$$m,n_1,\ldots,n_s>0$

(a) calculate

Tex syntax error

${\rm rk}E^m_{PL}(S^{(n)})$.

(b) find out whether $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite.

Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups

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

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. In the above sequence the $s$$s$-tuples $m-n_1,\ldots,m-n_s$$m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$$W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$. For any $j=1,\ldots,s$$j=1,\ldots,s$ and positive integer $k$$k$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$$p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$$j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$$(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$$\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$.

Analogous to Definition 3.1 there is a canonical homotopy equivalence $C_f \sim W$$C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$$S^{(n)} \to S^m$ has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement. One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$$\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$$\lambda_j$ is a generalisation of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$$\lambda:= \oplus_{j=1}^s \lambda_j$.

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

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

Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

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

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

In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving the Haefliger Theorem 8.1 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

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.5]{Crowley&Ferry&Skopenkov2011}. Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$-tuple of the isotopy classes of its components: $$ r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$$ Take $s$ pairwise disjoint $m$-discs in $S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$. Define $$j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$$ Then $j$ is a right inverse of the restriction homomorphism $r$, i.e. $r\circ j=\mathrm{id}$. The unknotting homomorphism $u$ is defined to be the homomorphism $$ u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$$ Informally, the homomorphism $u$ is obtained by taking embedded connected sums of components with knots $h_i:S^{n_i}\to S^m$ representing the elements of $E^m_D(S^{n_i})$ inverse to the components, whose images $h_i(S^{n_i})$ are small and are close to the components. {{beginthm|Theorem}}\label{dpl} \cite{Haefliger1966a} For $n_1,\ldots,n_s\le m-3$, the homomorphism $$u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$$ is an isomorphism. {{endthm}} For [[Knots,_i.e._embeddings_of_spheres|some information on the groups $E^m_D(S^q)$]] see \cite{Skopenkov2016s}, \cite[$\S.3]{Skopenkov2006}. ==Classification beyond the metastable range== ;\label{s:ebmr} {{beginthm|The Haefliger Theorem}}\label{belmethae} \cite[Theorem 10.7]{Haefliger1966a}, \cite[Theorem 1.1]{Skopenkov2009}, \cite[Theorem 1.1]{Skopenkov2006b} 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}. For alternative geometric (and presumably equivalent) definitons of $\beta$ see \cite[$\S]{Skopenkov2009}, \cite[$\S]{Skopenkov2006b}, cf. \cite[$\S]{Skopenkov2007} and \cite[$\S.3]{Crowley&Skopenkov2016}. {{beginthm|Remark}}\label{r:bel} (a) The Haefliger Theorem \ref{belmethae} implies that for any $l\ge2$ we have an isomorphism $$\lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{\varepsilon(l)}.$$ (b) For any $l\not\in\{1,3,7\}$, 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 $\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$. For $l\ge4$ see \cite[$\S]{Haefliger1962t}. The following proof for $l=2$ and general remark are intended for specialists. For $l=2$ there is an exact sequence $\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$, where $\lambda_{12}$ is $\lambda_{12,PL}$ not $\lambda_{12,D}$ \cite[Corollary 10.3]{Haefliger1966a}. We have $\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$ and $\Sigma$ is the reduction modulo 2. By the exactness, $\text{im}\lambda_{21}=2\Z$. By (a) $\ker\lambda_{12}\cong\Z$. Hence $\lambda_{21}|_{\ker\lambda_{12}}$ is injective. We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by \cite[Proposition 10.2]{Haefliger1966a}. So the formula of (b) follows. Analogously to \cite[Theorem 3.5]{Skopenkov2009} using geometric definitons of $\beta$ \cite[$\S]{Skopenkov2009}, \cite[$\S]{Skopenkov2006b} and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ \cite{Koschorke&Sanderson1977} one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$. Then (b) would follow. (c) For any $l\in\{1,3,7\}$ the map $\lambda_{12}\oplus\lambda_{21}$ in (b) above is not injective \cite[$\S]{Haefliger1962t}. (d) For any $l\ge4$, $l\ne7$ we have an isomorphism $$E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{\varepsilon(l)}^4$$ which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number ($\S$\ref{s:ebmr}). This follows from \cite[$\S]{Haefliger1962t}. We conjecture that this result holds also for $l=2,3,7$. {{endthm}} == Classification in codimension at least 3 == ;\label{s:cl3} In this section we assume that $(n)$ is an $s$-tuple such that $m\ge n_i+3$. For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$ is obtained in \cite[Theorem 1.9]{Crowley&Ferry&Skopenkov2011}. Corollaries \cite[$\S\Rr^m. Known cases are embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$$S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$$m-3\ge n_i$, embeddings $N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$$N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$ for $m\ge 2n_i+1$$m\ge 2n_i+1$ for every $i$$i$, and embeddings of some disconnected 3-manifolds in $S^6$$S^6$. For a related classification of knotted tori see [Skopenkov2016k].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3].

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

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

Figure 1: Embedded connected sum

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{(n)})$$E^m(S^{(n)})$ for $m-3\ge n_i$$m-3\ge n_i$ [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see Figure 1.

The standard embedding $S^k\to D^m$$S^k\to D^m$ is defined by $(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$$(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$. Fix $s$$s$ pairwise disjoint $m$$m$-discs $D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$$D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$. The standard embedding $S^{(n)}\to S^m$$S^{(n)}\to S^m$ is defined by taking the union of the compositions of the standard embeddings $S^{n_k} \to D^m_k$$S^{n_k} \to D^m_k$ with the fixed inclusions $D^m_k \to S^m$$D^m_k \to S^m$.

2 Examples

Recall that for any $q$$q$-manifold $N$$N$ and $m\ge2q+2$$m\ge2q+2$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction $m\ge2q+2$$m\ge2q+2$ is sharp for non-connected manifolds.

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

Figure 2: The Hopf link (a) and the trivial link (b)

For $q=1$$q=1$ the Hopf link is shown in Figure 2. For all $q$$q$ the image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows:

• either the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$;

• or they are given as the sets of points in $\Rr^{2q+1}$$\Rr^{2q+1}$ satisfying the following equations:

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

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

Analogously for any $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 which can be described as follows:

• either the spheres are $\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})\cong S^{p+q+1}$$\partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}$.

• or they are given as the points in $\Rr^{p+q+1}$$\Rr^{p+q+1}$ satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see $\S$$\S$3).

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient $\pm 1$$\pm 1$

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

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

Tex syntax error

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

see Figure 3. We have

Tex syntax error

$\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let

Tex syntax error

$\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$.

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$$g:D^{m-q}\to S^m$ such that $g(D^{m-q})$$g(D^{m-q})$ intersects $f(S^q)$$f(S^q)$ transversely at exactly one point with positive sign; see Figure 4.

Figure 4: The disc $gD^{m-q}$$gD^{m-q}$ and Gauss map $\widetilde f$$\widetilde f$

Then the restriction $h':S^{m-q-1}\to S^m-fS^q$$h':S^{m-q-1}\to S^m-fS^q$ of $g$$g$ to $\partial D^{m-q}$$\partial D^{m-q}$ is a homotopy equivalence.

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$$\lambda[f]$ is well-defined, i.e. is independent of the choices of $g,h',h$$g,h',h$ and of the representative $f$$f$ of $[f]$$[f]$. One can check that $\lambda$$\lambda$ is a homomorphism.

(b) 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.

(c) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$$\lambda\zeta=\id\pi_p(S^{m-q-1})$ for the Zeman map $\zeta$$\zeta$. So $\lambda$$\lambda$ is surjective and $\zeta$$\zeta$ is injective.

(d) For $m=p+q+1$$m=p+q+1$ there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].

Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism $\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 stable suspension 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 the Gauss map (see Figure 4)

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

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 the suspension isomorphism. 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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

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)$.)

One can easily check that $\alpha$$\alpha$ is well-defined and for $p,q\le m-3$$p,q\le m-3$ is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$$p,q\le m-3$.

Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$$\lambda_{12} \neq \pm \lambda_{21}$ as we explain in $\S$$\S$8.

Note that the $\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, each of the same dimension [Haefliger1962t, Theorem at the end of $\S$$\S$5], [Haefliger1966a]. Let $(q) = (q, \dots, q)$$(q) = (q, \dots, q)$ be the $s$$s$-tuple consisting entirely of some positive integer $q$$q$.

Theorem 4.2. The collection of pairwise linking coefficients

$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$ $$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$$

is bijective for $m\ge\frac{3q}2+2$$m\ge\frac{3q}2+2$.

Assume that $N_1,\ldots N_s$$N_1,\ldots N_s$ are closed manifolds, $m=2\dim N_i+1>2\dim N_k+1$$m=2\dim N_i+1>2\dim N_k+1$ for every $1\le i\le t<k\le s$$1\le i\le t<k\le s$ and $N_1,\ldots N_t$$N_1,\ldots N_t$ are orientable. Then for an embedding $N_1,\ldots N_s$$N_1,\ldots N_s$ and every $1\le i<j\le t$$1\le i<j\le t$ one can define the linking coefficient $\lambda_{ij}(f)\in\Z$$\lambda_{ij}(f)\in\Z$, see Remark 3.2.e.

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.

Example 5.1 (Borromean rings). 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' are the three spheres given by the following three systems of equations:

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

Figure 6: The Borromean rings

See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.

An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, $\S$$\S$4.5 `Triple linking modulo 2' and $\S$$\S$4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.

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

Figure 7: The Borromean rings and Whitehead link for $l = 1$$l = 1$

Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, $\S$$\S$3, $\S$$\S$4]; see also the Wikipedia article on the Whitehead link. (For $l=1$$l=1$ the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

We have $\lambda_{12}(w)=0$$\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]). For $l\ne1,3,7$$l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)$$\lambda_{21}(w)$ equal to the Whitehead square $[\iota_l,\iota_l]\ne0$$[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$$\iota_l\in\pi_l(S^l)$. This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14]. For $l=1,3,7$$l=1,3,7$ the Whitehead link 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.

For some results on links $S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link [Skopenkov2015a, $\S$$\S$2.5].

6 Linked 3-manifolds in 6-space

An embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$$k,m,n$ such that $m-n$$m-n$ is even, one can explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ so that the following theorem holds.

Theorem 6.1. [Avvakumov2016] Any Brunnian embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is isotopic to fk,m,n for some integers k,m,n such that $m-n$$m-n$ is even. Two embeddings fk,m,n and fk′,m′,n′ are isotopic if and only if $k=k′$$k=k′$ and both $m-m′$$m-m′$ and $n-n′$$n-n′$ are divisible by $2k$$2k$.

The proof uses classification of embeddings $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ (the Haefliger Theorem 8.1 for $m=2p=2q=6$$m=2p=2q=6$). The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ is not trivial.

Corollary 6.2. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $g,g':S^3\sqcup S^3\to\Rr^6$$g,g':S^3\sqcup S^3\to\Rr^6$ such that the componentwise embedded connected sum $f\#g$$f\#g$ is isotopic to $f\#g'$$f\#g'$ but $g$$g$ is not isotopic to $g'$$g'$.

See an unpublished generalization in [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define $E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$$E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$ to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding $S^{n_i} \to S^m$$S^{n_i} \to S^m$. We remark that $E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$$E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$ [Haefliger1966a, $\S\S$$\S\S$ 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.5].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$$s$-tuple of the isotopy classes of its components:

$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$ $$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$$

Take $s$$s$ pairwise disjoint $m$$m$-discs in $S^m$$S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$$g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$. Define

$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$ $$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$$

Then $j$$j$ is a right inverse of the restriction homomorphism $r$$r$, i.e. $r\circ j=\mathrm{id}$$r\circ j=\mathrm{id}$. The unknotting homomorphism $u$$u$ is defined to be the homomorphism

$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$ $$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$$

Informally, the homomorphism $u$$u$ is obtained by taking embedded connected sums of components with knots $h_i:S^{n_i}\to S^m$$h_i:S^{n_i}\to S^m$ representing the elements of $E^m_D(S^{n_i})$$E^m_D(S^{n_i})$ inverse to the components, whose images $h_i(S^{n_i})$$h_i(S^{n_i})$ are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For $n_1,\ldots,n_s\le m-3$$n_1,\ldots,n_s\le m-3$, the homomorphism

$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$ $$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$$

is an isomorphism.

For some information on the groups $E^m_D(S^q)$$E^m_D(S^q)$ see [Skopenkov2016s], [Skopenkov2006, $\S$$\S$3.3].

8 Classification beyond the metastable range

The Haefliger Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of $\beta$$\beta$ see [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5], cf. [Skopenkov2007, $\S$$\S$2] and [Crowley&Skopenkov2016, $\S$$\S$2.3].

Remark 8.2. (a) The Haefliger Theorem 8.1 implies that for any $l\ge2$$l\ge2$ we have an isomorphism

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

(b) For any $l\not\in\{1,3,7\}$$l\not\in\{1,3,7\}$, 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+(-1)^lb)=0\}$$\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$.

For $l\ge4$$l\ge4$ see [Haefliger1962t, $\S$$\S$6]. The following proof for $l=2$$l=2$ and general remark are intended for specialists.

For $l=2$$l=2$ there is an exact sequence $\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$$\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$, where $\lambda_{12}$$\lambda_{12}$ is $\lambda_{12,PL}$$\lambda_{12,PL}$ not $\lambda_{12,D}$$\lambda_{12,D}$ [Haefliger1966a, Corollary 10.3]. We have $\pi_3(S^2)\cong\Zz$$\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$$\pi_4(S^3)\cong\Zz_2$ and $\Sigma$$\Sigma$ is the reduction modulo 2. By the exactness, $\text{im}\lambda_{21}=2\Z$$\text{im}\lambda_{21}=2\Z$. By (a) $\ker\lambda_{12}\cong\Z$$\ker\lambda_{12}\cong\Z$. Hence $\lambda_{21}|_{\ker\lambda_{12}}$$\lambda_{21}|_{\ker\lambda_{12}}$ is injective. We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$$\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of $\beta$$\beta$ [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5] and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$$\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ [Koschorke&Sanderson1977] one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$$\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$. Then (b) would follow.

(c) For any $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, $\S$$\S$6].

(d) For any $l\ge4$$l\ge4$, $l\ne7$$l\ne7$ we have an isomorphism

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

which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number ($\S$$\S$8). This follows from [Haefliger1962t, $\S$$\S$6]. We conjecture that this result holds also for $l=2,3,7$$l=2,3,7$.

9 Classification in codimension at least 3

In this section we assume that $(n)$$(n)$ is an $s$$s$-tuple such that $m\ge n_i+3$$m\ge n_i+3$. For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$$E^m_{PL}(S^{(n)})\otimes\Qq$ is obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.2] one are necessary and sufficient conditions on $(n)$$(n)$ when $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers $m,n_1,\ldots,n_s>0$$m,n_1,\ldots,n_s>0$

(a) calculate

Tex syntax error

${\rm rk}E^m_{PL}(S^{(n)})$.

(b) find out whether $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite.

Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups

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

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. In the above sequence the $s$$s$-tuples $m-n_1,\ldots,m-n_s$$m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$$W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$. For any $j=1,\ldots,s$$j=1,\ldots,s$ and positive integer $k$$k$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$$p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$$j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$$(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$$\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$.

Analogous to Definition 3.1 there is a canonical homotopy equivalence $C_f \sim W$$C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$$S^{(n)} \to S^m$ has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement. One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$$\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$$\lambda_j$ is a generalisation of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$$\lambda:= \oplus_{j=1}^s \lambda_j$.

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

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

Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

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

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

In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving the Haefliger Theorem 8.1 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

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.2]{Crowley&Ferry&Skopenkov2011} one are necessary and sufficient conditions on $(n)$ when $E^m_{PL}(S^{(n)})$ is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition \ref{d:hclink} and Theorem \ref{thm:hclink}. {{beginthm|Theorem}}\label{t:cfs} \cite{Crowley&Ferry&Skopenkov2011} There are algorithms which for integers $m,n_1,\ldots,n_s>0$ (a) calculate ${\rm rk}E^m_{PL}(S^{(n)})$. (b) find out whether $E^m_{PL}(S^{(n)})$ is finite. {{endthm}} {{beginthm|Definition|(The Haefliger link sequence)}}\label{d:hclink} In \cite{Haefliger1966a} Haefliger defined a long exact sequence of abelian groups $$ \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.$$ We briefly define the groups and homomorphisms in this sequence, refering to \cite[1.4]{Haefliger1966a} for details. In the above sequence the $s$-tuples $m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$. For any $j=1,\ldots,s$ and positive integer $k$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$. Analogous to Definition \ref{dl} there is a canonical homotopy equivalence $C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$ has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement. One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$ is a generalisation of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$. Taking the Whitehead product with the class of the identity in $\pi_{n_j}(S^{n_j})$ defines a homomorphism $w_j \colon \ker p_{n_j,j}\to \Pi_{m-2}^{(n-1)}$. Define $w:=\oplus_{j=1}^s w_j$. The definition of the homomorphism $\mu$ is given in \cite[1.5]{Haefliger1966a}. {{endthm}} {{beginthm|Theorem}}\label{thm:hclink} (a) \cite[Theorem 1.3]{Haefliger1966a} The Haefliger link sequence is exact. (b) \cite{Crowley&Ferry&Skopenkov2011} The map $\lambda\otimes\Qq \colon E^m_{PL}(S^{(n)})\otimes\Qq \to \ker(w\otimes\Qq)$ is an isomorphism. {{endthm}} Part (b) follows because $\mu \otimes \Qq = 0$ \cite[Lemma 1.3]{Crowley&Ferry&Skopenkov2011}, and so the Haefliger link sequence after tensoring with the rational numbers $\Qq$ splits into short exact sequences. In general, the computation of the objects of Theorem \ref{thm:hclink} is difficult. For (a) no computations are known except for those giving (b) and those giving the Haefliger Theorem \ref{belmethae} (note that Theorem \ref{belmethae} has a direct proof easier than proof of Theorem \ref{thm:hclink}.a). For (b) the computations constitute most of the non-trivial paper \cite{Crowley&Ferry&Skopenkov2011}. Hence Theorem \ref{thm:hclink} is not a readily calculable classification in general. == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\Rr^m. Known cases are embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$$S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$$m-3\ge n_i$, embeddings $N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$$N^{n_1}\sqcup\ldots\sqcup N^{n_s}\to S^m$ for $m\ge 2n_i+1$$m\ge 2n_i+1$ for every $i$$i$, and embeddings of some disconnected 3-manifolds in $S^6$$S^6$. For a related classification of knotted tori see [Skopenkov2016k].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3].

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

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

Figure 1: Embedded connected sum

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{(n)})$$E^m(S^{(n)})$ for $m-3\ge n_i$$m-3\ge n_i$ [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see Figure 1.

The standard embedding $S^k\to D^m$$S^k\to D^m$ is defined by $(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$$(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$. Fix $s$$s$ pairwise disjoint $m$$m$-discs $D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$$D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$. The standard embedding $S^{(n)}\to S^m$$S^{(n)}\to S^m$ is defined by taking the union of the compositions of the standard embeddings $S^{n_k} \to D^m_k$$S^{n_k} \to D^m_k$ with the fixed inclusions $D^m_k \to S^m$$D^m_k \to S^m$.

2 Examples

Recall that for any $q$$q$-manifold $N$$N$ and $m\ge2q+2$$m\ge2q+2$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction $m\ge2q+2$$m\ge2q+2$ is sharp for non-connected manifolds.

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

Figure 2: The Hopf link (a) and the trivial link (b)

For $q=1$$q=1$ the Hopf link is shown in Figure 2. For all $q$$q$ the image of the Hopf link is the union of two $q$$q$-spheres which can be described as follows:

• either the spheres are $\partial D^{q+1}\times0$$\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$$0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$$\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$;

• or they are given as the sets of points in $\Rr^{2q+1}$$\Rr^{2q+1}$ satisfying the following equations:

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

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

Analogously for any $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 which can be described as follows:

• either the spheres are $\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})\cong S^{p+q+1}$$\partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}$.

• or they are given as the points in $\Rr^{p+q+1}$$\Rr^{p+q+1}$ satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see $\S$$\S$3).

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient $\pm 1$$\pm 1$

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

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

Tex syntax error

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

see Figure 3. We have

Tex syntax error

$\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let

Tex syntax error

$\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$.

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$$g:D^{m-q}\to S^m$ such that $g(D^{m-q})$$g(D^{m-q})$ intersects $f(S^q)$$f(S^q)$ transversely at exactly one point with positive sign; see Figure 4.

Figure 4: The disc $gD^{m-q}$$gD^{m-q}$ and Gauss map $\widetilde f$$\widetilde f$

Then the restriction $h':S^{m-q-1}\to S^m-fS^q$$h':S^{m-q-1}\to S^m-fS^q$ of $g$$g$ to $\partial D^{m-q}$$\partial D^{m-q}$ is a homotopy equivalence.

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$$\lambda[f]$ is well-defined, i.e. is independent of the choices of $g,h',h$$g,h',h$ and of the representative $f$$f$ of $[f]$$[f]$. One can check that $\lambda$$\lambda$ is a homomorphism.

(b) 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.

(c) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$$\lambda\zeta=\id\pi_p(S^{m-q-1})$ for the Zeman map $\zeta$$\zeta$. So $\lambda$$\lambda$ is surjective and $\zeta$$\zeta$ is injective.

(d) For $m=p+q+1$$m=p+q+1$ there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].

Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism $\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 stable suspension 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 the Gauss map (see Figure 4)

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

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 the suspension isomorphism. 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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

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)$.)

One can easily check that $\alpha$$\alpha$ is well-defined and for $p,q\le m-3$$p,q\le m-3$ is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$$p,q\le m-3$.

Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$$\lambda_{12} \neq \pm \lambda_{21}$ as we explain in $\S$$\S$8.

Note that the $\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, each of the same dimension [Haefliger1962t, Theorem at the end of $\S$$\S$5], [Haefliger1966a]. Let $(q) = (q, \dots, q)$$(q) = (q, \dots, q)$ be the $s$$s$-tuple consisting entirely of some positive integer $q$$q$.

Theorem 4.2. The collection of pairwise linking coefficients

$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$ $$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$$

is bijective for $m\ge\frac{3q}2+2$$m\ge\frac{3q}2+2$.

Assume that $N_1,\ldots N_s$$N_1,\ldots N_s$ are closed manifolds, $m=2\dim N_i+1>2\dim N_k+1$$m=2\dim N_i+1>2\dim N_k+1$ for every $1\le i\le t<k\le s$$1\le i\le t<k\le s$ and $N_1,\ldots N_t$$N_1,\ldots N_t$ are orientable. Then for an embedding $N_1,\ldots N_s$$N_1,\ldots N_s$ and every $1\le i<j\le t$$1\le i<j\le t$ one can define the linking coefficient $\lambda_{ij}(f)\in\Z$$\lambda_{ij}(f)\in\Z$, see Remark 3.2.e.

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.

Example 5.1 (Borromean rings). 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' are the three spheres given by the following three systems of equations:

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

Figure 6: The Borromean rings

See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.

An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, $\S$$\S$4.5 `Triple linking modulo 2' and $\S$$\S$4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.

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

Figure 7: The Borromean rings and Whitehead link for $l = 1$$l = 1$

Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, $\S$$\S$3, $\S$$\S$4]; see also the Wikipedia article on the Whitehead link. (For $l=1$$l=1$ the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

We have $\lambda_{12}(w)=0$$\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]). For $l\ne1,3,7$$l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)$$\lambda_{21}(w)$ equal to the Whitehead square $[\iota_l,\iota_l]\ne0$$[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$$\iota_l\in\pi_l(S^l)$. This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14]. For $l=1,3,7$$l=1,3,7$ the Whitehead link 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.

For some results on links $S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link [Skopenkov2015a, $\S$$\S$2.5].

6 Linked 3-manifolds in 6-space

An embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$$k,m,n$ such that $m-n$$m-n$ is even, one can explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f_{k,m,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ so that the following theorem holds.

Theorem 6.1. [Avvakumov2016] Any Brunnian embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is isotopic to fk,m,n for some integers k,m,n such that $m-n$$m-n$ is even. Two embeddings fk,m,n and fk′,m′,n′ are isotopic if and only if $k=k′$$k=k′$ and both $m-m′$$m-m′$ and $n-n′$$n-n′$ are divisible by $2k$$2k$.

The proof uses classification of embeddings $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ (the Haefliger Theorem 8.1 for $m=2p=2q=6$$m=2p=2q=6$). The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ is not trivial.

Corollary 6.2. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$$f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $g,g':S^3\sqcup S^3\to\Rr^6$$g,g':S^3\sqcup S^3\to\Rr^6$ such that the componentwise embedded connected sum $f\#g$$f\#g$ is isotopic to $f\#g'$$f\#g'$ but $g$$g$ is not isotopic to $g'$$g'$.

See an unpublished generalization in [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define $E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$$E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$ to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding $S^{n_i} \to S^m$$S^{n_i} \to S^m$. We remark that $E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$$E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$ [Haefliger1966a, $\S\S$$\S\S$ 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.5].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$$s$-tuple of the isotopy classes of its components:

$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$ $$\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$$

Take $s$$s$ pairwise disjoint $m$$m$-discs in $S^m$$S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$$g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$. Define

$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$ $$\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$$

Then $j$$j$ is a right inverse of the restriction homomorphism $r$$r$, i.e. $r\circ j=\mathrm{id}$$r\circ j=\mathrm{id}$. The unknotting homomorphism $u$$u$ is defined to be the homomorphism

$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$ $$\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$$

Informally, the homomorphism $u$$u$ is obtained by taking embedded connected sums of components with knots $h_i:S^{n_i}\to S^m$$h_i:S^{n_i}\to S^m$ representing the elements of $E^m_D(S^{n_i})$$E^m_D(S^{n_i})$ inverse to the components, whose images $h_i(S^{n_i})$$h_i(S^{n_i})$ are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For $n_1,\ldots,n_s\le m-3$$n_1,\ldots,n_s\le m-3$, the homomorphism

$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$ $$\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$$

is an isomorphism.

For some information on the groups $E^m_D(S^q)$$E^m_D(S^q)$ see [Skopenkov2016s], [Skopenkov2006, $\S$$\S$3.3].

8 Classification beyond the metastable range

The Haefliger Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of $\beta$$\beta$ see [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5], cf. [Skopenkov2007, $\S$$\S$2] and [Crowley&Skopenkov2016, $\S$$\S$2.3].

Remark 8.2. (a) The Haefliger Theorem 8.1 implies that for any $l\ge2$$l\ge2$ we have an isomorphism

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

(b) For any $l\not\in\{1,3,7\}$$l\not\in\{1,3,7\}$, 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+(-1)^lb)=0\}$$\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$.

For $l\ge4$$l\ge4$ see [Haefliger1962t, $\S$$\S$6]. The following proof for $l=2$$l=2$ and general remark are intended for specialists.

For $l=2$$l=2$ there is an exact sequence $\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$$\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$, where $\lambda_{12}$$\lambda_{12}$ is $\lambda_{12,PL}$$\lambda_{12,PL}$ not $\lambda_{12,D}$$\lambda_{12,D}$ [Haefliger1966a, Corollary 10.3]. We have $\pi_3(S^2)\cong\Zz$$\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$$\pi_4(S^3)\cong\Zz_2$ and $\Sigma$$\Sigma$ is the reduction modulo 2. By the exactness, $\text{im}\lambda_{21}=2\Z$$\text{im}\lambda_{21}=2\Z$. By (a) $\ker\lambda_{12}\cong\Z$$\ker\lambda_{12}\cong\Z$. Hence $\lambda_{21}|_{\ker\lambda_{12}}$$\lambda_{21}|_{\ker\lambda_{12}}$ is injective. We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$$\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of $\beta$$\beta$ [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5] and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$$\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ [Koschorke&Sanderson1977] one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$$\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$. Then (b) would follow.

(c) For any $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, $\S$$\S$6].

(d) For any $l\ge4$$l\ge4$, $l\ne7$$l\ne7$ we have an isomorphism

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

which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number ($\S$$\S$8). This follows from [Haefliger1962t, $\S$$\S$6]. We conjecture that this result holds also for $l=2,3,7$$l=2,3,7$.

9 Classification in codimension at least 3

In this section we assume that $(n)$$(n)$ is an $s$$s$-tuple such that $m\ge n_i+3$$m\ge n_i+3$. For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$$E^m_{PL}(S^{(n)})\otimes\Qq$ is obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.2] one are necessary and sufficient conditions on $(n)$$(n)$ when $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers $m,n_1,\ldots,n_s>0$$m,n_1,\ldots,n_s>0$

(a) calculate

Tex syntax error

${\rm rk}E^m_{PL}(S^{(n)})$.

(b) find out whether $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite.

Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups

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

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. In the above sequence the $s$$s$-tuples $m-n_1,\ldots,m-n_s$$m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$$W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$. For any $j=1,\ldots,s$$j=1,\ldots,s$ and positive integer $k$$k$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$$p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$$j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$$(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$$\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$.

Analogous to Definition 3.1 there is a canonical homotopy equivalence $C_f \sim W$$C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$$S^{(n)} \to S^m$ has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement. One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$$\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$$\lambda_j$ is a generalisation of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$$\lambda:= \oplus_{j=1}^s \lambda_j$.

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

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

Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

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

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

In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving the Haefliger Theorem 8.1 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

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