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 the classification of embeddings $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$$m-3\ge n_i$. For the classification of embeddings in 6-space of some disconnected 3-manifolds see [Avvakumov2016], [Avvakumov2017].

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):

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

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

A component-wise version of embedded connected sum [Skopenkov2016c, $\S$$\S$5] defines a commutative group structure on the set $E^m(S^{(n)})$$E^m(S^{(n)})$ for $m-3\ge n_i$$m-3\ge n_i$ [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].

We define the standard embedding $S^{(n)}\to S^m$$S^{(n)}\to S^m$ as follows. The standard embedding

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${\rm i}:S^k\to D^m$ is defined by

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${\rm i}(x_1, \ldots, x_k) = (x_1, \dots, x_k, 0, \ldots, 0)$. Fix $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$. The standard embedding $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$$S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ is defined as the composition

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$g\circ({\rm i}_1\sqcup\ldots\sqcup{\rm i}_s)$ of the union of the standard embeddings with $g$$g$.

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$$q=1$ the Hopf link is shown e.g. in [Skopenkov2006, Figure 2.1.a]. For all $q$$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 (The Zeeman map). We define a map

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

Denote by $i_{m,p}:S^p\to S^m$$i_{m,p}:S^p\to S^m$ the equatorial inclusion. For a map $x:S^p\to S^{m-q-1}$$x:S^p\to S^{m-q-1}$ representing an element of $\pi_p(S^{m-q-1})$$\pi_p(S^{m-q-1})$ let

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

where $i$$i$ is the natural `standard embedding' defined in [Skopenkov2015a, $\S$$\S$2.1], see [Skopenkov2006, Figure 3.2]. We have $\overline\zeta_x(S^p)\cap i_{m,q}(S^q)\subset i(S^{m-q-1}\times S^p)\cap i(0_{m-q}\times S^q) =\emptyset$$\overline\zeta_x(S^p)\cap i_{m,q}(S^q)\subset i(S^{m-q-1}\times S^p)\cap i(0_{m-q}\times S^q) =\emptyset$. Let $\zeta[x]:=[\overline\zeta_x\sqcup i_{m,q}]$$\zeta[x]:=[\overline\zeta_x\sqcup i_{m,q}]$.

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

3 The linking coefficient

Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.

Definition 3.1 (The linking coefficient). We define a map

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

Take an embedding $f:S^p\sqcup S^q\to S^m$$f:S^p\sqcup S^q\to S^m$ representing an element $[f]\in E^m(S^p\sqcup S^q)$$[f]\in E^m(S^p\sqcup S^q)$. Take an embedding $g:D^{m-q}\to S^m$$g:D^{m-q}\to S^m$ such that $g(D^{m-q})$$g(D^{m-q})$ intersects $f(S^q)$$f(S^q)$ transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1]. Then the restriction $h':S^{m-q-1}\to S^m-fS^q$$h':S^{m-q-1}\to S^m-fS^q$ of $g$$g$ to $\partial D^{m-q}$$\partial D^{m-q}$ is a homotopy equivalence.

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$$\lambda[f]$ is well-defined, i.e. is independent of the choises of $g,h',h$$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})$$\lambda\zeta=\id\pi_p(S^{m-q-1})$ (see Definition 2.2; this also holds for $m=q+2$$m=q+2$, see (d) and (e) below). 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].

Let $\pi_k^S := \pi_{2k+2}(S^{k+2})$$\pi_k^S := \pi_{2k+2}(S^{k+2})$ be the $k$$k$-th stable homotopy group of spheres. 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 a map [Skopenkov2006, Figure 3.1]

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

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

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

The second isomorphism in this formula is 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}$$v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q}$ is the quotient map. See [Skopenkov2006, Figure 3.4]. The map $v^*$$v^*$ is an isomorphism for $m\ge q+2$$m\ge q+2$. (For $m\ge q+3$$m\ge q+3$ this follows by general position and for $m=q+2$$m=q+2$ by the cofibration Barratt-Puppe exact sequence of pair $(S^p\times S^q,S^p\vee S^q)$$(S^p\times S^q,S^p\vee S^q)$ and by the existence of a retraction $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$$\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.)

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

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$.

Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$$\lambda_{12} \neq \pm \lambda_{21}$ as we explain in $\S$$\S$7.

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$.

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' [Skopenkov2006, Figures 3.5 and 3.6] are the three spheres given by the following three systems of equations:

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

The required embedding is any embedding whose image consists of 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.

Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, $\S$$\S$3, $\S$$\S$4], cf. Wikipedia article on Whitehead link.

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)=[\iota_l,\iota_l]\ne0$$\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$. This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14]. For $l=1,3,7$$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.

6 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 6.1. [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)$$E^m_D(S^q)$ see [Skopenkov2006, $\S$$\S$3.3]. (See also foundational paper [Haefliger1966] on $E^m_D(S^q)$$E^m_D(S^q)$ where this information is less complete and harder to find, and the foundational paper [Levine1965] on different but related group.)

7 Classification beyond the metastable range

The Haefliger Theorem 7.1. [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 7.2. (a) The Haefliger Theorem 7.1 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$$\S$7). This follows from [Haefliger1962t, $\S$$\S$6]. We conjecture that this result holds also for $l=2,3,7$$l=2,3,7$.

8 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$.

Definition 8.1 (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 integer $k>0$$k>0$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$$p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphisms induced by the projection to the $j$$j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$$(n-1):=(n_1 -1, \ldots, n_s-1)$. Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$$\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$. Analogous to Definition 3.1 there is a canonical homotopy equivalence $C_f \sim W$$C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$$S^{(n)} \to S^m$ has a normal bundle with a section and so each component of a link can be pushed off along such a section into the complement. One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined a map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$$\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$$\lambda_j$ is a generalization of linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$$\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 sketched in [Haefliger1966a, 1.5].

Theorem 8.2. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

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

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

In [Crowley&Ferry&Skopenkov2011, $\S$$\S$1.2, $\S$$\S$1.3] one can find necessary and sufficient conditions on $p$$p$ and $q$$q$ when $E^m_{PL}(S^p\sqcup S^q)$$E^m_{PL}(S^p\sqcup S^q)$ is finite, as well as an effective procedure for computing the rank of the group $E^m_{PL}(S^{(n)})$$E^m_{PL}(S^{(n)})$.

For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Avvakumov2017], [Skopenkov2015a, $\S$$\S$2.5], [Skopenkov2016k].

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