High codimension links

Example 2.1 (The Hopf Link). (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:

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

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:

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

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

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

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

Tex syntax error

Tex syntax error

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 $f:S^p\sqcup S^q\to S^m$ 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 $gD^{m-q}$ and Gauss map $\widetilde f$

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

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$ 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].

Definition 3.3 (The $\alpha$-invariant). We define a map $E^m(S^p\sqcup S^q)\to\pi^S_{p+q+1-m}$ for $p,q\le m-2$. Take an embedding $f:S^p\sqcup S^q\to S^m$ representing an element $[f]\in E^m(S^p\sqcup S^q)$. Define the Gauss map (see Figure 4)

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

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

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

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 $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.)

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

The Haefliger-Zeeman Theorem 4.1. 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.

Theorem 4.2. The collection of pairwise linking coefficients

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

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

Example 5.1 (Borromean rings). (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:

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

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].

Example 5.2 (Whitehead link). (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 $l = 1$

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].

Theorem 6.1. 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

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

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

Theorem 6.2. 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

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

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

Theorem 6.3. [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 $2k$.

Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ 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 $g'$.

Theorem 7.1. [Haefliger1966a, Theorem 2.4] For $n_1,\ldots,n_s\le m-3$, the homomorphism

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

is an isomorphism.

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

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

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

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

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

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

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].

Theorem 8.3. For any $l>2$ there an isomorphism

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

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

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

Tex syntax error

(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

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

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.

Example 2.1 (The Hopf Link). (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:

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

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:

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

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

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

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

Tex syntax error

Tex syntax error

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 $f:S^p\sqcup S^q\to S^m$ 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 $gD^{m-q}$ and Gauss map $\widetilde f$

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

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$ 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].

Definition 3.3 (The $\alpha$-invariant). We define a map $E^m(S^p\sqcup S^q)\to\pi^S_{p+q+1-m}$ for $p,q\le m-2$. Take an embedding $f:S^p\sqcup S^q\to S^m$ representing an element $[f]\in E^m(S^p\sqcup S^q)$. Define the Gauss map (see Figure 4)

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

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

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

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 $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.)

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

The Haefliger-Zeeman Theorem 4.1. 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.

Theorem 4.2. The collection of pairwise linking coefficients

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

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

Example 5.1 (Borromean rings). (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:

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

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].

Example 5.2 (Whitehead link). (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 $l = 1$

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].

Theorem 6.1. 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

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

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

Theorem 6.2. 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

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

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

Theorem 6.3. [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 $2k$.

Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ 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 $g'$.

Theorem 7.1. [Haefliger1966a, Theorem 2.4] For $n_1,\ldots,n_s\le m-3$, the homomorphism

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

is an isomorphism.

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

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

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

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

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

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

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].

Theorem 8.3. For any $l>2$ there an isomorphism

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

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

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

Tex syntax error

(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

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

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.

Example 2.1 (The Hopf Link). (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:

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

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:

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

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

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

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

Tex syntax error

Tex syntax error

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 $f:S^p\sqcup S^q\to S^m$ 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 $gD^{m-q}$ and Gauss map $\widetilde f$

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

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

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

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

Remark 3.2. (a) Clearly, $\lambda[f]$ 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].

Definition 3.3 (The $\alpha$-invariant). We define a map $E^m(S^p\sqcup S^q)\to\pi^S_{p+q+1-m}$ for $p,q\le m-2$. Take an embedding $f:S^p\sqcup S^q\to S^m$ representing an element $[f]\in E^m(S^p\sqcup S^q)$. Define the Gauss map (see Figure 4)

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

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

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

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 $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.)

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

The Haefliger-Zeeman Theorem 4.1. 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.

Theorem 4.2. The collection of pairwise linking coefficients

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

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

Example 5.1 (Borromean rings). (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:

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

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].

Example 5.2 (Whitehead link). (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 $l = 1$

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].

Theorem 6.1. 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

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

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

Theorem 6.2. 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

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

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

Theorem 6.3. [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 $2k$.

Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ 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 $g'$.