High codimension links

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1 Introduction

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.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, ${{Stub}} == Introduction == <wikitex>; This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings. For a [[High_codimension_embeddings#Introduction|general introduction to embeddings]] as well as the [[High_codimension_embeddings#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\S$ 1, $\S$ 2].

A componentwise version of embedded connected sum [Skopenkov2016c, $\S$ 5] defines a commutative group structure on $E^m(S^p\sqcup S^q)$ for $m-3\ge p,q$ ; see [Skopenkov2006, Figure 3.3], [Haefliger1966], [Haefliger1966a] and [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a].

The following table was obtained by Zeeman around 1960:

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

See explanation below.

2 Examples

Recall that for each $n$ -manifold $N$ and $m\ge2n+2$ , any two embeddings $N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. The following example shows that the restriction $m\ge2n+2$ is sharp for non-connected manifolds.

For each $n$ there is an embedding $S^n\sqcup S^n\to\Rr^{2n+1}$ which is not isotopic to the standard embedding.

For $n=1$ the Hopf link is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary $n$ (including $n=1$ ) the image of the Hopf link is the union of two $n$ -spheres

$\displaystyle \partial D^{n+1}\times0\quad\text{and}\quad 0\times\partial D^{n+1}\quad\text{in}\quad\partial(D^{n+1}\times D^{n+1}).$ $\displaystyle \partial D^{n+1}\times0\quad\text{and}\quad 0\times\partial D^{n+1}\quad\text{in}\quad\partial(D^{n+1}\times D^{n+1}).$

Alternatively, these spheres are given by equations:

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

This embedding is distinguished from the standard embedding by the linking coefficient.

Analogously for each $p,q$ one constructs an embedding $S^p\sqcup S^q\to\Rr^{p+q+1}$ which is not isotopic to the standard embedding.

Take $x\in\pi_p(S^{m-q-1})$ Define embedding $\tau(x)$ on $S^q$ to be the standard embedding into $\R^m$ . Take any map $\varphi:S^p\to\partial D^{m-q}$ . Define embedding $\tau(x)$ on $S^p$ to be the composition

$\displaystyle S^p\overset{x\times i}\to\partial D^{m-q}\times S^q \subset D^{m-q}\times S^q\subset\R^m,$ $\displaystyle S^p\overset{x\times i}\to\partial D^{m-q}\times S^q \subset D^{m-q}\times S^q\subset\R^m,$

where $i:S^p\to S^q$ is the equatorial inclusion and the latter inclusion is the standard [Skopenkov2006, Figure 3.2].

Clearly, $\tau$ is well-defined and is a homomorphism.

3 Invariants

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

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

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

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

(a) Clearly, $\lambda(f)$ is indeed independent of $g,h',h$ . Clearly, $\lambda$ is a homomorphism.

(b) For $m=p+q+1$ or $m=q+2$ there are simpler alternative `homological' definitions, in which components are any closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].

(c) Analogously one can define $\lambda_{21}(f)\in\pi_q(S^{m-p-1})$ for $m\ge p+3$ .

(d) This definition works for $m=q+2$ if $S^m-fS^q$ is simply-connected (or, equivalently for $q>4$ , if the restriction of $f$ to $S^q$ is unknotted).

(e) Clearly, $\lambda\tau=\id\pi_p(S^{m-q-1})$ , even for $m=q+2$ . So $\lambda$ is surjective and $\tau$ is injective.

By Freudenthal Suspension Theorem $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$ . The stable suspension of the linking coefficient can be described alternatively as follows.

For an embedding $f:S^p\sqcup S^q\to S^m$ define a map [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 $p\le q\le m-2$ define the $\alpha$ -invariant by

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

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

We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$ by [Kervaire1959a, Lemma 5.1]. Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$ .

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

4 Classification in the `metastable' range

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

The surjectivity of $\lambda$ (=the injectivity of $\tau$ ) follows from $\lambda\tau=\id$ . The injectivity of $\lambda$ (=the surjectivity of $\tau$ ) 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 [Haefliger1966a]: the collection of pairwise linking coefficients is bijective for $2m\ge3n+4$ and $n$ -dimensional links in $\R^m$ .

5 Examples below the metastable range

Let us present an example of non-injectivity of the collection of pairwise linking coefficients.

The Borromean rings $S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ is a non-trivial embedding whose restrictions to 2-componented sublinks are trivial [Haefliger1962, 4.1], [Haefliger1962t].

Denote coordinates in $\Rr^{3l}$ by $(x,y,z)=(x_1,\dots,x_l,y_1,\dots,y_l,z_1,\dots,z_l)$ . The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6] are the three spheres given by the following three systems of equations:

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

This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because the connected sum of the three components yields a non-trivial knot [Haefliger1962]).

Let us present an example of non-injectivity of the linking coefficient.

The Whitehead link $w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ is a non-trivial embedding whose linking coefficient $\lambda_{12}(w)$ is trivial.

The (higher-dimensional) Whitehead link is obtained from Borromean rings by joining two components with a tube.

We have $\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in [Skopenkov2006a]). For $l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$ . This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18]. For $l=1,3,7$ the Whitehead link is is distinguished from the standard embedding by more complicated invariants [Skopenkov2006a], [Haefliger1962t, $\S$ 3].

This example shows that the dimension restriction is sharp in Theorem 4.1.

This example seems to be discovered by Whitehead, in connection with Whitehead product.

Cf. the Haefliger Trefoil knot [Skopenkov2016t].

6 Classification below the metastable range

Let $C_q^{m-q}:=E^m_D(S^q)$ . For some information on this group see [Skopenkov2006, $\S$ 3.3].

(a) [Haefliger1966a] If $p,q\le m-3$ , then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$

(b) [Haefliger1966a, Theorem 10.7], [Skopenkov2009] If $p\le q\le m-3$ and $3m\ge2p+2q+6$ , then there is a map $\varkappa$ for which the following map is an isomorphism

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

The map $\varkappa$ and its right inverse $\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$ are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].

Part (b) implies that

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

This isomorphism is defined for $l\ge2$ , $l\ne3,7$ by map

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

This map is injective for $l\ge2$ , $l\ne3,7$ ; the image of this map is $\{(a,b)\ :\ \Sigma a=\Sigma b\}$ [Haefliger1962t]. Thus part (b) shows that $\lambda_{12}\oplus\lambda_{21}$ is in general not injective.

The set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ for $m\ge n_i+3$ has been described in terms of exact sequences involving homotopy groups of spheres [Haefliger1966], [Haefliger1966a], cf. [Levine1965], [Habegger1986].

See [Skopenkov2009], [Crowley&Ferry&Skopenkov2011], [Avvakumov2016], [Skopenkov2015a, $\S$ 2.5], [Skopenkov2016k].

7 References

[Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
[Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
[Habegger1986] N. Habegger, Knots and links in codimension greater than 2, Topology, 25:3 (1986) 253--260.
[Haefliger1962] A. Haefliger, Knotted $(4k-1)$ -spheres in $6k$ -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
[Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

[Haefliger1966] A. Haefliger, Differential embeddings of $S^n$ in $S^{n+q}$ for $q>2$ , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
[Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
[Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
[Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
[Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
[Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
[Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
[Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
[Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
[Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

[Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
[Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
[Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
[Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
[Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
[Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069

, $\S]{Skopenkov2016c}. A componentwise version of [[High_codimension_embeddings#Embedded_connected_sum|embedded connected sum]] \cite[$\S]{Skopenkov2016c} defines a commutative group structure on $E^m(S^p\sqcup S^q)$ for $m-3\ge p,q$; see \cite[Figure 3.3]{Skopenkov2006}, \cite{Haefliger1966}, \cite{Haefliger1966a} and \cite[Group Structure Lemma 2.2 and Remark 2.3.a]{Skopenkov2015}. The following table was obtained by Zeeman around 1960: $$\begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \ |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}$$ See explanation below. == Examples == ; [[High_codimension_embeddings:_classification#Unknotting_theorems|Recall that]] for each $n$-manifold $N$ and $m\ge2n+2$, any two embeddings $N\to\Rr^m$ are isotopic \cite[General Position Theorem 3.1]{Skopenkov2016c}. The following example shows that the restriction $m\ge2n+2$ is sharp for non-connected manifolds. {{beginthm|Example|(The Hopf Link)}}\label{hopf} For each $n$ there is an embedding $S^n\sqcup S^n\to\Rr^{2n+1}$ which is not isotopic to the standard embedding. {{endthm}} For $n=1$ the Hopf link is shown in \cite[Figure 2.1.a]{Skopenkov2006}. For arbitrary $n$ (including $n=1$) the image of the Hopf link is the union of two $n$-spheres $$\partial D^{n+1}\times0\quad\text{and}\quad 0\times\partial D^{n+1}\quad\text{in}\quad\partial(D^{n+1}\times D^{n+1}).$$ Alternatively, these spheres are given by equations: $$\left\{\begin{array}{c} x_1=\dots=x_n=0\ x_{n+1}^2+\dots+x_{2n+1}^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1\end{array}\right..$$ This embedding is distinguished from the standard embedding by [[#Invariants|the linking coefficient]]. Analogously for each $p,q$ one constructs an embedding $S^p\sqcup S^q\to\Rr^{p+q+1}$ which is not isotopic to the standard embedding. {{beginthm|Definition|of the Zeeman map $\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)$ for $p\le q$}}\label{dz} Take $x\in\pi_p(S^{m-q-1})$ Define embedding $\tau(x)$ on $S^q$ to be the standard embedding into $\R^m$. Take any map $\varphi:S^p\to\partial D^{m-q}$. Define embedding $\tau(x)$ on $S^p$ to be the composition $$S^p\overset{x\times i}\to\partial D^{m-q}\times S^q \subset D^{m-q}\times S^q\subset\R^m,$$ where $i:S^p\to S^q$ is the equatorial inclusion and the latter inclusion is the standard \cite[Figure 3.2]{Skopenkov2006}. {{endthm}} Clearly, $\tau$ is well-defined and is a homomorphism. == Invariants == ; {{beginthm|Definition|of linking coefficient $\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})$ for $m\ge q+3$}}\label{dl} Fix orientations of $S^p$, $S^q$, $S^m$ and $D^{m-p}$. Take an embedding $f:S^p\sqcup S^q\to S^m$. Take an embedding $g:D^{m-q}\to S^m$ such that $gD^{m-q}$ intersects $fS^q$ transversally at exactly one point with positive sign \cite[Figure 3.1]{Skopenkov2006}. Then the restriction $h':S^{m-q-1}\to S^m-fS^q$ of $g$ to $\partial D^{m-q}$ is a homotopy equivalence. (Indeed, since $m\ge q+3$, the complement $S^m-fS^q$ is simply-connected. By Alexander duality $h'$ induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems $h'$ is a homotopy equivalence.) Let $h$ be a homotopy inverse of $h'$. Define $$\lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$$ {{endthm}} {{beginthm|Remark}}\label{lkrem} (a) Clearly, $\lambda(f)$ is indeed independent of $g,h',h$. Clearly, $\lambda$ is a homomorphism. (b) For $m=p+q+1$ or $m=q+2$ there are simpler alternative `homological' definitions, in which components are any closed orientable manifolds, cf. [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|Remark 5.3.f]] of \cite{Skopenkov2016e}. (c) Analogously one can define $\lambda_{21}(f)\in\pi_q(S^{m-p-1})$ for $m\ge p+3$. (d) This definition works for $m=q+2$ if $S^m-fS^q$ is simply-connected (or, equivalently for $q>4$, if the restriction of $f$ to $S^q$ is unknotted). (e) Clearly, $\lambda\tau=\id\pi_p(S^{m-q-1})$, even for $m=q+2$. So $\lambda$ is surjective and $\tau$ is injective. {{endthm}} By Freudenthal Suspension Theorem $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$. The stable suspension of the linking coefficient can be described alternatively as follows. {{beginthm|Definition|of the $\alpha$-invariant $E^m(S^p\sqcup S^q)\to\pi^S_{p+q+1-m}$}} For an embedding $f:S^p\sqcup S^q\to S^m$ define a map \cite[Figure 3.1]{Skopenkov2006} $$\widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.$$ For $p\le q\le m-2$ define the $\alpha$-invariant by $$\alpha(f)=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.$$ The second isomorphism in this formula is given by the Freudenthal Suspension Theorem. The map $v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q}$ is the quotient map. See \cite[Figure 3.4]{Skopenkov2006}. The map $v^*$ is an isomorphism for $m\ge q+2$. (For $m\ge q+3$ this follows by general position and for $m=q+2$ by the cofibration Barratt-Puppe exact sequence of pair $(S^p\times S^q,S^p\vee S^q)$ and by the existence of a retraction $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.) {{endthm}} We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$ by \cite[Lemma 5.1]{Kervaire1959a}. Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$. Note that $\alpha$-invariant can be defined in more general situations \cite{Koschorke1988}, \cite[$\S]{Skopenkov2006}. ==Classification in the `metastable' range== ; {{beginthm|The Haefliger-Zeeman Theorem}}\label{lkhaze} If \S1, $\S$ $\S$ 2].