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

## 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 $\Rr^m$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\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].

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 [Skopenkov2016i]. 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.

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):

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

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$$m-3\ge n_i$ [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, $\S$$\S$1], [Avvakumov2017, $\S$$\S$1.4], 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.

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

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

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

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

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

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see Figure 3. We have
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$\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let
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$\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$.

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

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

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

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

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

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 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}$$\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$$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)

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

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

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

The second isomorphism in this formula is 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^*$$v^*$ is a 1--1 correspondence for $m\ge q+2$$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$5.

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. 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, and 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$$\zeta$) is proved in [Haefliger1962t, Theorem in $\S$$\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$$\S$5]. Let $(q) = (q, \dots, q)$$(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

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

## 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}$$S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, $\S$$\S$6].

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:

$\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1,\end{array}\right. \qquad \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1. \end{array}\right.$
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, $\S$$\S$4], yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].)

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)$$\iota_l\in\pi_l(S^l)$ [Haefliger1962t, end of $\S$$\S$6] (I do not know a published proof of $\lambda_{21}(w)=[\iota_l,\iota_l]$$\lambda_{21}(w)=[\iota_l,\iota_l]$ except [Skopenkov2015a, the Whitehead link Lemma 2.14]). For $l=1,3,7$$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}$$S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link see [Skopenkov2015a, $\S$$\S$2.5].

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 connectivity assumptions are made on the source manifolds.

Theorem 6.1. Assume that $N_1,\ldots,N_s$$N_1,\ldots,N_s$ are closed manifolds and $\frac{m-1}2=\dim N_i\ge2$$\frac{m-1}2=\dim N_i\ge2$ for every $i$$i$. Then for an embedding $f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m$$f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m$ and every $1\le i$1\le i one defines the linking coefficient $\lambda_{ij}(f)$$\lambda_{ij}(f)$, see Remark 3.2.e. We have $\lambda_{ij}(f)\in\Z$$\lambda_{ij}(f)\in\Z$ if both $N_i$$N_i$ and $N_j$$N_j$ are orientable, and $\lambda_{ij}(f)\in\Z_2$$\lambda_{ij}(f)\in\Z_2$ otherwise. Then the collection of pairwise linking coefficients

$\displaystyle \bigoplus\limits_{1\le i

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

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

Theorem 6.2. Let $N_1$$N_1$ and $N_2$$N_2$ be closed $n$$n$-dimensional homologically $(2n-m+1)$$(2n-m+1)$-connected orientable manifolds. For an embedding $f:N_1\sqcup N_2\to\Rr^m$$f:N_1\sqcup N_2\to\Rr^m$ one can define the invariant $\alpha(f)\in\pi_{2n-m+1}^S$$\alpha(f)\in\pi_{2n-m+1}^S$ analogously to Definition 3.3. Then

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

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

Theorems 6.1 and 6.2 are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]; 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}$$[S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m}$ in Definition 3.3. Apparently Theorem 6.1 has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.

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

Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$$(S^2\times S^1)\sqcup S^3\to\Rr^6$ is isotopic to $f_{k,l,n}$$f_{k,l,n}$ for some integers $k,l,n$$k,l,n$ such that $l-n$$l-n$ is even. Two embeddings $f_{k,l,n}$$f_{k,l,n}$ and $f_{k',l',n'}$$f_{k',l',n'}$ are isotopic if and only if $k=k'$$k=k'$ and both $l-l'$$l-l'$ and $n-n'$$n-n'$ are divisible by $2k$$2k$.

The proof uses M. Skopenkov's classification of embeddings $S^3\sqcup S^3\to\Rr^6$$S^3\sqcup S^3\to\Rr^6$ (Theorem 8.1 for $m=2p=2q=6$$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 6.4. [Avvakumov2016, Corollary 1], 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'$.

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)$$(n)$ is an $s$$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:

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

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

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

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

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

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, Theorem 2.4] For $n_1,\ldots,n_s\le m-3$$n_1,\ldots,n_s\le m-3$, the homomorphism

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

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

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

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

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 [Haefliger1966a, $\S$$\S$10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of $\beta$$\beta$ see [Skopenkov2009, $\S$$\S$3], [Skopenkov2006b, $\S$$\S$5], cf. [Skopenkov2007, $\S$$\S$2] and [Crowley&Skopenkov2016, $\S$$\S$2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

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

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

(b) For any $l\not\in\{1,3,7\}$$l\not\in\{1,3,7\}$, the map

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

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$$l=2$ there is an exact sequence $\pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3)$$\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}$$\lambda_{12}$ is $\lambda_{12,PL}$$\lambda_{12,PL}$ [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$$\Sigma$ is the reduction modulo 2. By the exactness of the previous sequence, $\lambda_{21}(\ker \lambda_{12}) = 2\Z$$\lambda_{21}(\ker \lambda_{12}) = 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, Main Theorem in $\S$$\S$1] one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$$\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

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

which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number ($\S$$\S$5). This follows from [Haefliger1962t, $\S$$\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$ was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.2] contains necessary and sufficient conditions on $(n)$$(n)$ which determine when $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$ is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and $\S$$\S$1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

Theorem 9.1. There are algorithms which for integers $m,n_1,\ldots,n_s>0$$m,n_1,\ldots,n_s>0$

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

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

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

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$$s$-tuples $m-n_1,\ldots,m-n_s$$m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W :=\vee_{i=1}^s S^{m-n_i-1}$$W :=\vee_{i=1}^s S^{m-n_i-1}$. For any $j=1,\ldots,s$$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 homomorphism induced by the collapse map onto 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)$ and $\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. It can also be shown that the homotopy class of a push off of the $j$$j$th component in the complement of the link gives a well-defined map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$$\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) 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 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$$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.