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 of closed disconnected manifolds into ${{Stub}} == Introduction == <wikitex>; For notation and conventions throughout this page see [[High_codimension_embeddings|high codimension embeddings]]. `Embedded connected sum' defines a commutative group structure on $E^m(S^p\sqcup S^q)$ for $m-3\ge p,q$. See Figure 3.3. of \cite{Skopenkov2006}, \cite{Haefliger1966} \cite{Haefliger1966a}. </wikitex> == General position and the Hopf linking == <wikitex>; {{beginthm|General Position Theorem}}\label{th1} For each $n$-manifold $N$ and $m\ge2n+2$, every two embeddings $N\to\Rr^m$ are isotopic. {{endthm}} The restriction $m\ge2n+2$ in Theorem \ref{th1} is sharp for non-connected manifolds. {{beginthm|Example: the Hopf linking}}\label{hopf} For each $n$ there is an embedding $S^n\sqcup S^n\to\Rr^{2n+1}$ which is not isotopic to the standard embedding. {{endthm}} For $n=1$ the Hopf Linking is shown in Figure 2.1.a of \cite{Skopenkov2006}. For arbitrary $n$ (including $n=1$) the image of the Hopf Linking is the union of two $n$-spheres $$\partial D^{n+1}\times0\quad\text{and}\quad 0\times\partial D^{n+1}\quad\text{in}\quad\partial(D^{n+1}\times D^{n+1}).$$ Alternatively, these spheres are given by equations: $$\left\{\begin{array}{c} x_1=\dots=x_n=0\ x_{n+1}^2+\dots+x_{2n+1}^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1\end{array}\right..$$ This embedding is distinguished from the standard embedding by [[Links, i.e. embeddings of non-connected manifolds#The Zeeman construction and linking coefficient|the linking coefficient]]. </wikitex> == The Zeeman construction and linking coefficient == <wikitex>; The following table was obtained by Zeeman around 1960: $$\begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \ \#E^m(S^q\sqcup S^q) &1 &\infty &2 &2 &24 &1 &1 \end{array}$$ ====Construction of the Zeeman map $\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)$ for $p\le q$==== Take $x\in\pi_p(S^{m-q-1})$ Define embedding $\tau(x)$ on $S^q$ to be the standard embedding into $\R^m$. Take any map $\varphi:S^p\to\partial D^{m-q}$. Define embedding $\tau(x)$ on $S^p$ to be the composition $$S^p\overset{x\times i}\to\partial D^{m-q}\times S^q \subset D^{m-q}\times S^q\subset\R^m,$$ where $i:S^p\to S^q$ is the equatorial inclusion and the latter inclusion is the standard. See Figure 3.2 of \cite{Skopenkov2006}. Clearly, $\tau$ is well-defined and is a homomorphism. ====Definition of linking coefficient $\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})$ for $m\ge q+3$==== Fix orientations of $S^p$, $S^q$, $S^m$ and $D^{m-p}$. Take an embedding $f:S^p\sqcup S^q\to S^m$. Take an embedding $g:D^{m-q}\to S^m$ such that $gD^{m-q}$ intersects $fS^q$ transversally at exactly one point with positive sign (see Figure 3.1 of \cite{Skopenkov2006}). Then the restriction $h':S^{m-q-1}\to S^m-fS^q$ of $g$ to $\partial D^{m-q}$ is a homotopy equivalence. (Indeed, since $m\ge q+3$, the complement $S^m-fS^q$ is simply-connected. By Alexander duality $h'$ induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems $h'$ is a homotopy equivalence.) Let $h$ be a homotopy inverse of $h'$. Define $$\lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$$ {{beginthm|Remark}}\label{lkrem} (a) Clearly, $\lambda(f)$ is indeed independent of $g,h',h$. Clearly, $\lambda$ is a homomorphism. (b) For $m=p+q+1$ there is a simpler alternative `homological' definition. That definition works for $m=q+2$ as well. (c) Analogously one can define $\lambda_{21}(f)\in\pi_q(S^{m-p-1})$ for $m\ge p+3$. (d) This definition works for $m=q+2$ if $S^m-fS^q$ is simply-connected (or, equivalently for $q>4$, if the restriction of $f$ to $S^q$ is unknotted). (e) Clearly, $\lambda\tau=\id\pi_p(S^{m-q-1})$, even for $m=q+2$. So $\lambda$ is surjective and $\tau$ is injective. {{endthm}} ====Classification in the `metastable' range==== {{beginthm|The Haefliger-Zeeman Theorem}}\label{lkhaze} If \Rr^m$ up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings $N_1\sqcup\ldots\sqcup N_s\to S^m$ , where $N_1,\ldots, N_s$ are spheres (or even closed manifolds) and $m-3\ge\dim N_i$ for every $i$ , under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].

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 the case of manifolds [Skopenkov2016i]. Denote by $E^m(S^{(n)})$ the set of embeddings $S^{(n)}\to S^m$ up to isotopy.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$ 1, $\S$ 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: Component-wise embedded connected sum

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$ 5] defines a commutative group structure on the set $E^m(S^{(n)})$ for $m-3\ge n_i$ [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, $\S$ 1], [Avvakumov2017, $\S$ 1.4], 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 $D^m_k \to S^m$ .

2 Examples

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

(a) 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.

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 $0\times\partial D^{q+1}$ 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:

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

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

(b) For any $p,q$ there is an embedding $S^p\sqcup S^q\to\Rr^{p+q+1}$ which is not isotopic to the standard embedding.

Analogously to (a), the image is the union of two spheres which can be described as follows:

either the spheres are $\partial D^{p+1}\times0$ and $0\times\partial D^{q+1}$ 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:

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

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

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

Figure 3: A link with prescribed linking coefficient

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

For a map $x:S^p\to S^{m-q-1}$ representing an element of $\pi_p(S^{m-q-1})$ 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 $\zeta$ is well-defined and is a homomorphism.

3 The linking coefficient

Here we define the linking coefficient and discuss its 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 $f:S^p\sqcup S^q\to S^m$ 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.

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

Then the restriction $h':S^{m-q-1}\to S^m-fS^q$ 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

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

(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 Zeeman map $\zeta$ . So $\lambda$ is surjective and $\zeta$ is injective.

(d) For $m=p+q+1$ there is a simpler alternative definition 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 $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$ . The stable suspension of the linking coefficient can be described as follows.

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)

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

For $p,q\le m-2$ define the $\alpha$ -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 $v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q}$ is the quotient map, see Figure 5.

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

The map $v^*$ is a 1--1 correspondence 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 $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$ .)

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

We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$ .

Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$ , even though in general $\lambda_{12} \neq \pm \lambda_{21}$ as we explain in Example 5.2.a,c.

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

4 Classification in the metastable range

If $1\le p\le q$ , then both $\lambda$ and $\zeta$ are isomorphisms for $m\ge\frac{3q}2+2$ in the smooth category, and for $m\ge\frac p2+q+2$ in the PL category.

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 [Haefliger1962t, Theorem in $\S$ 5], [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 in $\S$ 5]. Let $(q) = (q, \dots, q)$ be the $s$ -tuple consisting entirely of some positive integer $q$ .

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 a 1-1 correspondence for $m\ge\frac{3q}2+2$ .

5 Examples beyond the metastable range

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

First 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.

(a) There is a non-trivial embedding $S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ whose restriction to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, $\S$ 6].

In order to construct such an embedding, denote coordinates in $\Rr^{3l}$ by $(x,y,z)=(x_1,\dots,x_l,y_1,\dots,y_l,z_1,\dots,z_l)$ . The (higher-dimensional) `Borromean spheres' are 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, called Borromean link, is any embedding whose image is the union of the Borromean spheres.

(b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. More precisely, each two of the Borromean spheres span two (intersecting) $2l$ -disks disjoint from the remaining sphere.

(c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint $2l$ -disks (intersecting the remaining sphere). Moreover, we can take these $2l$ -disks so that

each one of them intersects the remaining sphere transversely by an $(l-1)$ -sphere;

the obtained two disjoint $(l-1)$ -spheres in the remaining sphere have linking number $\pm1$ , i.e. one of them spans an $l$ -disk (in the remaining sphere) itersecting the other transversely at exacly one point.

(d) The Borromean link is distinguished from the standard embedding by triple linking number of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). Take a 3-component link, i.e. an embedding $g:S^{2l-1}_1\sqcup S^{2l-1}_2\sqcup S^{2l-1}_3\to S^{3l}$ . Assume that $g$ is pairwise unlinked, i.e. every two components are contained in disjoint smooth balls. Let $D_2,D_3\subset S^{3l}$ be disjoint oriented embedded $2l$ -disks in general position to $g_1:=g|_{S^{2l-1}_1}$ , and such that $g(S^{2l-1}_i)=\partial D_i$ for $i=2,3$ . Then for $j=2,3$ the preimage $g_1^{-1}D_j$ is an oriented $(l-1)$ -submanifold of $S^{2l-1}_1$ missing $g_1^{-1}D_{5-j}$ . Let $\mu(g)$ be the linking number of $g_1^{-1}D_2$ and $g_1^{-1}D_3$ in $S^{2l-1}_1$ .

(e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of $D_2,D_3$ , and of the isotopy of $g$ . The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number [Haefliger1962t, $\S$ 4], [HaefligerSteer1965], [Haefliger1966a, proof of Theorem 9.4], [Massey1968], [Massey1990, $\S$ 7] by the well-known `linking number' definition of the Whitehead invariant $\pi_{2l-1}(S^l\vee S^l)\to\Z$ [Skopenkov2020e, $\S$ 2, Sketch of a proof of (b1)]. If $g$ is pairwise unlinked, then the number $\mu$ is independent of permutation of the components, up to multiplication by $\pm1$ [HaefligerSteer1965] (this can be easily proved directly).

(f) The Borromean link is non-trivial also because joining the three components with two tubes (i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c, $\S$ 4]) yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].

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.

(a) For every positive integer $l$ there is a non-trivial embedding $w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient $\lambda_{21}(w)$ 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 the second and the third components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, $\S$ 3, $\S$ 4]; see also the 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.)

(b) The second component is null-homotopic in the complement of the first component by Example 5.1.b.

(c) For $l\ne1,3,7$ the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. More precisely, $\lambda_{12}(w)$ equals to the Whitehead square $[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$ [Haefliger1962t, end of $\S$ 6] (I do not know a written proof of this except [Skopenkov2015a, the Whitehead link Lemma 2.14] for $l$ even). For $l=1,3,7$ the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant [Skopenkov2006a].

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

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

6 Linked manifolds

In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no high-connectivity assumptions are made on the source manifolds.

Assume that $N_1,\ldots,N_s$ are closed connected manifolds and $\frac{m-1}2=\dim N_i\ge2$ for every $i$ . Then for an embedding $f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m$ and every $1\le i<j\le s$ one defines the linking coefficient $\lambda_{ij}(f)$ , see Remark 3.2.e. We have $\lambda_{ij}(f)\in\Z$ if both $N_i$ and $N_j$ are orientable, and $\lambda_{ij}(f)\in\Z_2$ otherwise. Then the collection of pairwise linking coefficients

$\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}$ $\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}$

is well-defined and is a 1-1 correspondence, where $t$ of $N_1,\ldots, N_s$ are orientable and $s-t$ are not.

We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension.

Let $N_1$ and $N_2$ be closed $n$ -dimensional homologically $(2n-m+1)$ -connected orientable manifolds. For an embedding $f:N_1\sqcup N_2\to\Rr^m$ one can define the invariant $\alpha(f)\in\pi_{2n-m+1}^S$ analogously to Definition 3.3. Then

$\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S$ $\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S$

is well-defined and is a 1-1 correspondence, provided $2m\ge3n+4$ .

Theorems 6.1 and 6.2 are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category [Ivansic&Horvatic1974]. They are also corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4] (in both categories); the calculations are analogous to the construction of the 1-1 correspondence $[S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m}$ in Definition 3.3. See [Skopenkov2000, Proposition 1.2] for the link map analogue.

Now we present an extension of Theorems 6.1 and 6.2 to a case where $m = 6$ and $n = 3$ . In particular, for the results below $m=2\dim N_i$ , $2m=3n+3$ and the manifolds $N_i$ are only $(2n-m)$ -connected. 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 $k,l,n$ such that $l-n$ is even, Avvakumov has constructed a Brunnian embedding $f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ , which appears in the next result [Avvakumov2016, $\S$ 1].

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

The proof uses M. Skopenkov's classification of embeddings $S^3\sqcup S^3\to\Rr^6$ (Theorem 8.1 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.

[Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] 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 $g'$ .

For an unpublished generalization of Theorem 6.3 and Corollary 6.4 see [Avvakumov2017].

7 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 $E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$ [Haefliger1966a, $\S\S$ 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, $\S$ 1.5].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$ -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 $s$ 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

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

Then $j$ 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

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

Informally, the homomorphism $u$ 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.

[Haefliger1966a, Theorem 2.4] For $n_1,\ldots,n_s\le m-3$ , 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 $E^m_D(S^q)$ see [Skopenkov2016s], [Skopenkov2006, $\S$ 3.3].

8 Classification beyond the metastable range

Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] If $p\le q\le m-3$ and $3m\ge2p+2q+6$ , then there is a homomorphism $\beta$ 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 $\beta$ and its right inverse $\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$ are constructed in [Haefliger1966a, $\S$ 10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of $\beta$ see [Skopenkov2009, $\S$ 3], [Skopenkov2006b, $\S$ 5], cf. [Skopenkov2007, $\S$ 2] and [Crowley&Skopenkov2016, $\S$ 2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

(a) Theorem 8.1 implies that for any $l\ge2$ 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 $l\not\in\{1,3,7\}$ , 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 $\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$ .

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

For $l=2$ there is an exact sequence $\pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3)$ , where $\lambda_{12}$ is $\lambda_{12,PL}$ [Haefliger1966a, Corollary 10.3]. 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 of the previous sequence, $\lambda_{21}(\ker \lambda_{12}) = 2\Z$ . By (a) $\ker \lambda_{12} \cong \Z$ . Hence $\lambda_{21}|_{\ker\lambda_{12}}$ is injective. We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of $\beta$ [Skopenkov2009, $\S$ 3], [Skopenkov2006b, $\S$ 5] and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ [Koschorke&Sanderson1977, Main Theorem in $\S$ 1] 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 [Haefliger1962t, $\S$ 6].

For any $l>2$ there an isomorphism

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

which is the sum of 3 pairwise invariants of Remark 8.2.a, and the triple linking number ( $\S$ 5).

This follows from [Haefliger1966a, Theorem 9.4], see also [Haefliger1962t, $\S$ 6].

9 Classification in codimension at least 3

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 $E^m_{PL}(S^{(n)})\otimes\Qq$ was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, $\S$ 1.2] contains necessary and sufficient conditions on $(n)$ which determine when $E^m_{PL}(S^{(n)})$ is finite. These results are elementary, but the precise formulations are technical. Hence, we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and $\S$ 1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

There are algorithms which for integers $m,n_1,\ldots,n_s>0$

(a) calculate

Tex syntax error

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

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

Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] 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. We first note that in the sequence above the $s$ -tuples $m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W :=\vee_{i=1}^s S^{m-n_i-1}$ . For 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 homomorphism induced by the collapse map onto to the $j$ -component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$ and $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})\subset\pi_{m-1}(W)$ .

It can be shown that each component of a link $f:S^{(n)} \to S^m$ has a non-zero normal vector field. So each component of a link can be pushed off along such a vector field into the complement $C_f$ . Analogously to Definition 3.1 there is a canonical homotopy equivalence $C_f \sim W$ . It can also be shown that the homotopy class in $C_f$ of the push off of the $j$ th component 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 generalization of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$ .

Taking the Whitehead product with the class of the inclusion of $S^{m-n_j-1}$ into $W$ 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 [Haefliger1966a, 1.5].

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

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

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

In general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for $3m\ge2p+2q+7$ (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [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 $\tau$ are isomorphisms for $m\ge\frac p2+q+2$ and for $m\ge\frac{3q}2+2$, in the PL and DIFF cases, respectively. {{endthm}} The surjectivity of $\lambda$ (=the injectivity of $\tau$) follows from $\lambda\tau=\id$. The injectivity of $\lambda$ (=the surjectivity of $\tau$) is proved in \cite{Haefliger1962T}, \cite{Zeeman1962} (or follows from the Haefliger-Weber Theorem 5.4 and Deleted Product Lemma 5.3.a of \cite{Skopenkov2006}). An analogue of this result holds for links with many components: ''the collection of pairwise linking coefficients is bijective for m\ge3n+4$ and $n$-dimensional links in $\R^m$''. ====Alpha-invariant==== By Freudenthal Suspension Theorem $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$. The stable suspension of the linking coefficient can be described alternatively as follows. For an embedding $f:S^p\sqcup S^q\to S^m$ define a map $$\widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.$$ See Figure 3.1 of \cite{Skopenkov2006}. For $p\le q\le m-2$ define the $\alpha$-invariant by $$\alpha(f)=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.$$ The second isomorphism in this formula is given by the Freudenthal Suspension Theorem. The map $v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q}$ is the quotient map. See Figure 3.4 of \cite{Skopenkov2006}. The map $v^*$ is an isomorphism for $m\ge q+2$. (For $m\ge q+3$ this follows by general position and for $m=q+2$ by the cofibration Barratt-Puppe exact sequence of pair $(S^p\times S^q,S^p\vee S^q)$ and by the existence of a retraction $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.) We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$ by Lemma 5.1 of \cite{Kervaire1959L}. Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$. Note that $\alpha$-invariant can be defined in more general situations \cite{Koschorke1988}, \cite{Skopenkov2006}, \S5. == Classification below the metastable range == ; ====Higher-dimensional Borromean rings==== Let us present an example of ''non-injectivity of the collection of pairwise linking coefficients''. {{beginthm|Borromean rings example}}\label{belmetbor} The Borromean rings $S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ is a non-trivial embedding whose restrictions to 2-componented sublinks are trivial \cite{Haefliger1962}, 4.1, \cite{Haefliger1962T}. {{endthm}} 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 ''Borromean rings'' are the three spheres given by the following three systems of equations: $$\left\{\begin{array}{c} x=0\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\ |x|^2+2|y|^2=1 \end{array}\right..$$ See Figures 3.5 and 3.6 of \cite{Skopenkov2006}. This embedding is distinguished from the standard embedding by ''the Massey invariant''. ====Higher-dimensional Whitehead link==== Let us present an example of ''non-injectivity of the linking coefficient''. {{beginthm|Whitehead link example}}\label{belmetwhi} The Whitehead link $w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ is a non-trivial embedding whose linking coefficient $\lambda_{12}(w)$ is trivial. {{endthm}} The Whitehead link is obtained from Borromean rings by joining two components with a tube. We have $\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in \cite{Skopenkov2006a}). For $l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$. (It would be interesting to find or write a published proof of this fact.) For $l=1,3,7$ the Whitehead link is is distinguished from the standard embedding by more complicated invariants \cite{Skopenkov2006a}, \cite{Haefliger1962T}, \S3. This example shows that the dimension restriction is sharp in Theorem \ref{lkhaze}. This example seems to be discovered by Whitehead, in connection with Whitehead product. It would be interesting to find a publication where it first appeared. ====Classification==== See [[3-manifolds in 6-space#Examples#The Haefliger trefoil knot |The Haefliger Trefoil knot]]. Let $C_q^{m-q}:=E^m_D(S^q)$. [[Knots, i.e. embeddings of spheres|Some information on this group]]. {{beginthm|The Haefliger Theorem}}\label{belmethae} (a) \cite{Haefliger1966a} If $p,q\le m-3$, then $$E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$ (b) \cite{Haefliger1966a}, Theorem 10.7, \cite{Skopenkov2009} If $p\le q\le m-3$ and m\ge2p+2q+6$, then there is a map $\varkappa$ for which the following map is an isomorphism $$\lambda_{12}\oplus\varkappa:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).$$ {{endthm}} The map $\varkappa$ and its right inverse $\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$ are constructed in \cite{Haefliger1966} and \cite{Haefliger1966a}, cf. \cite{Skopenkov2009}. Part (b) implies that $$E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\cong\pi_{2l-1}(S^l)\oplus\Z_{(l)}.$$ This isomorphism is defined for $l\ge2$, $l\ne3,7$ by map $$\lambda_{12}\oplus\lambda_{21}:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\pi_{2l-1}(S^l).$$ This map is injective for $l\ge2$, $l\ne3,7$; the image of this map is $\{(a,b)\ :\ \Sigma a=\Sigma b\}$ \cite{Haefliger1962T}. Thus part (b) shows that $\lambda_{12}\oplus\lambda_{21}$ is in general not injective. == Further discussion == ; The set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ for $m\ge n_i+3$ has been described in terms of exact sequences involving homotopy groups of spheres \cite{Haefliger1966}, \cite{Haefliger1966a}, cf. \cite{Levine 1965}, \cite{Habegger1986}. See \cite{Skopenkov2009}, \cite{Crowley&Ferry&Skopenkov2011}. == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\Rr^m up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings $N_1\sqcup\ldots\sqcup N_s\to S^m$ $N_1\sqcup\ldots\sqcup N_s\to S^m$ , where $N_1,\ldots, N_s$ $N_1,\ldots, N_s$ are spheres (or even closed manifolds) and $m-3\ge\dim N_i$ $m-3\ge\dim N_i$ for every $i$ $i$ , under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].