High codimension links

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1 Introduction

For notation and conventions throughout this page see high codimension embeddings.

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

2 General position and the Hopf linking

General Position Theorem 2.1. For each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic.

The restriction $m\ge2n+2$$m\ge2n+2$ in Theorem 2.1 is sharp for non-connected manifolds.

Example: the Hopf linking 2.2. For each $n$$n$ there is an embedding $S^n\sqcup S^n\to\Rr^{2n+1}$$S^n\sqcup S^n\to\Rr^{2n+1}$ which is not isotopic to the standard embedding.

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

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

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

3 The Zeeman construction and linking coefficient

The following table was obtained by Zeeman around 1960:

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

1 Construction of the Zeeman map $\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)$$\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)$ for $p\le q$$p\le q$

Take $x\in\pi_p(S^{m-q-1})$$x\in\pi_p(S^{m-q-1})$ Define embedding $\tau(x)$$\tau(x)$ on $S^q$$S^q$ to be the standard embedding into $\R^m$$\R^m$. Take any map $\varphi:S^p\to\partial D^{m-q}$$\varphi:S^p\to\partial D^{m-q}$. Define embedding $\tau(x)$$\tau(x)$ on $S^p$$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$$i:S^p\to S^q$ is the equatorial inclusion and the latter inclusion is the standard. See Figure 3.2 of [Skopenkov2006]. Clearly, $\tau$$\tau$ is well-defined and is a homomorphism.

2 Definition of linking coefficient $\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})$$\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})$ for $m\ge q+3$$m\ge q+3$

Fix orientations of $S^p$$S^p$, $S^q$$S^q$, $S^m$$S^m$ and $D^{m-p}$$D^{m-p}$. Take an embedding $f:S^p\sqcup S^q\to S^m$$f:S^p\sqcup S^q\to S^m$. Take an embedding $g:D^{m-q}\to S^m$$g:D^{m-q}\to S^m$ such that $gD^{m-q}$$gD^{m-q}$ intersects $fS^q$$fS^q$ transversally at exactly one point with positive sign (see Figure 3.1 of [Skopenkov2006]). 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$$m\ge q+3$, the complement $S^m-fS^q$$S^m-fS^q$ is simply-connected. By Alexander duality $h'$$h'$ induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems $h'$$h'$ is a homotopy equivalence.)

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

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

Remark 3.1. (a) Clearly, $\lambda(f)$$\lambda(f)$ is indeed independent of $g,h',h$$g,h',h$. Clearly, $\lambda$$\lambda$ is a homomorphism.

(b) For $m=p+q+1$$m=p+q+1$ there is a simpler alternative `homological' definition. That definition works for $m=q+2$$m=q+2$ as well.

(c) 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$.

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

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

3 Classification in the `metastable' range

The Haefliger-Zeeman Theorem 3.2. If $1\le p\le q$$1\le p\le q$, then both $\lambda$$\lambda$ and $\tau$$\tau$ are isomorphisms for $m\ge\frac p2+q+2$$m\ge\frac p2+q+2$ and for $m\ge\frac{3q}2+2$$m\ge\frac{3q}2+2$, in the PL and DIFF cases, respectively.

The surjectivity of $\lambda$$\lambda$ (=the injectivity of $\tau$$\tau$) follows from $\lambda\tau=\id$$\lambda\tau=\id$. The injectivity of $\lambda$$\lambda$ (=the surjectivity of $\tau$$\tau$) is proved in [Haefliger1962T], [Zeeman1962] (or follows from the Haefliger-Weber Theorem 5.4 and Deleted Product Lemma 5.3.a of [Skopenkov2006]).

An analogue of this result holds for links with many components: the collection of pairwise linking coefficients is bijective for $2m\ge3n+4$$2m\ge3n+4$ and $n$$n$-dimensional links in $\R^m$$\R^m$.

4 Alpha-invariant

By Freudenthal Suspension Theorem $\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 alternatively as follows. For an embedding $f:S^p\sqcup S^q\to S^m$$f:S^p\sqcup S^q\to S^m$ define a map

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

See Figure 3.1 of [Skopenkov2006]. For $p\le q\le m-2$$p\le q\le m-2$ define the $\alpha$$\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 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}$$v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q}$ is the quotient map. See Figure 3.4 of [Skopenkov2006]. The map $v^*$$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)$.)

We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$ by Lemma 5.1 of [Kervaire1959L]. Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$.

Note that $\alpha$$\alpha$-invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006], \S5.

4 Classification below the metastable range

1 Higher-dimensional Borromean rings

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

Borromean rings example 4.1. The Borromean rings $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}$ is a non-trivial embedding whose restrictions to 2-componented sublinks are 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 Borromean rings are the three spheres given by the following three systems of equations:

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

See Figures 3.5 and 3.6 of [Skopenkov2006]. This embedding is distinguished from the standard embedding by the Massey invariant.

2 Higher-dimensional Whitehead link

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

Whitehead link example 4.2. The Whitehead link $w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ is a non-trivial embedding whose linking coefficient $\lambda_{12}(w)$$\lambda_{12}(w)$ is trivial.

The Whitehead link is obtained from Borromean rings by joining two components with a tube. We have $\lambda_{12}(w)=0$$\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in [Skopenkov2006a]). For $l\ne1,3,7$$l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$$\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$. (It would be interesting to find or write a published proof of this fact.) For $l=1,3,7$$l=1,3,7$ the Whitehead link is is distinguished from the standard embedding by more complicated invariants [Skopenkov2006a], [Haefliger1962T], \S3.

This example shows that the dimension restriction is sharp in Theorem 3.2.

This example seems to be discovered by Whitehead, in connection with Whitehead product. It would be interesting to find a publication where it first appeared.

3 Classification

See The Haefliger Trefoil knot. Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. Some information on this group.

The Haefliger Theorem 4.3. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

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

(c) [Haefliger1966a], Theorem 10.7, [Skopenkov2009] 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 map $\varkappa$$\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$$\varkappa$ and its right inverse $\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$$\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$ are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].

Part (b) implies that $E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\cong\pi_{2l-1}(S^l)\oplus\Z_{(l)}$$E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\cong\pi_{2l-1}(S^l)\oplus\Z_{(l)}$. This isomorphism is defined for $l\ge2$$l\ge2$, $l\ne3,7$$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$$l\ge2$, $l\ne3,7$$l\ne3,7$; the image of this map is $\{(a,b)\ :\ \Sigma a=\Sigma b\}$$\{(a,b)\ :\ \Sigma a=\Sigma b\}$ [Haefliger1962T]. Part (b) shows that $\lambda_{12}\oplus\lambda_{21}$$\lambda_{12}\oplus\lambda_{21}$ is in general not injective.

5 Further discussion

The set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ for $m\ge n_i+3$$m\ge n_i+3$ has been described in terms of exact sequences involving homotopy groups of spheres [Haefliger1966], [Haefliger1966a], cf. [Levine 1965], [Habegger1986].

See [Skopenkov2009], [Crowley&Ferry&Skopenkov2011].

6 References

• [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)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962T] Template:Haefliger1962T
• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$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
• [Kervaire1959L] Template:Kervaire1959L
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [Levine 1965] Template:Levine 1965
• [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
• [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

\le p\le q$, then both $\lambda$ and $\tau$ are isomorphisms for $m\ge\frac p2+q+2$ and for $m\ge\frac{3q}2+2$, in the PL and DIFF cases, respectively. {{endthm}} The surjectivity of $\lambda$ (=the injectivity of $\tau$) follows from $\lambda\tau=\id$. The injectivity of $\lambda$ (=the surjectivity of $\tau$) is proved in \cite{Haefliger1962T}, \cite{Zeeman1962} (or follows from the Haefliger-Weber Theorem 5.4 and Deleted Product Lemma 5.3.a of \cite{Skopenkov2006}). An analogue of this result holds for links with many components: ''the collection of pairwise linking coefficients is bijective for m\ge3n+4$ and $n$-dimensional links in $\R^m$''. ====Alpha-invariant==== By Freudenthal Suspension Theorem $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$. The stable suspension of the linking coefficient can be described alternatively as follows. For an embedding $f:S^p\sqcup S^q\to S^m$ define a map $$\widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.$$ See Figure 3.1 of \cite{Skopenkov2006}. For $p\le q\le m-2$ define the $\alpha$-invariant by $$\alpha(f)=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.$$ The second isomorphism in this formula is given by the Freudenthal Suspension Theorem. The map $v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q}$ is the quotient map. See Figure 3.4 of \cite{Skopenkov2006}. The map $v^*$ is an isomorphism for $m\ge q+2$. (For $m\ge q+3$ this follows by general position and for $m=q+2$ by the cofibration Barratt-Puppe exact sequence of pair $(S^p\times S^q,S^p\vee S^q)$ and by the existence of a retraction $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.) We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$ by Lemma 5.1 of \cite{Kervaire1959L}. Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$. Note that $\alpha$-invariant can be defined in more general situations \cite{Koschorke1988}, \cite{Skopenkov2006}, \S5. == Classification below the metastable range == ; ====Higher-dimensional Borromean rings==== Let us present an example of ''non-injectivity of the collection of pairwise linking coefficients''. {{beginthm|Borromean rings example}}\label{belmetbor} The Borromean rings $S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ is a non-trivial embedding whose restrictions to 2-componented sublinks are trivial \cite{Haefliger1962}, 4.1, \cite{Haefliger1962T}. {{endthm}} Denote coordinates in $\Rr^{3l}$ by $(x,y,z)=(x_1,\dots,x_l,y_1,\dots,y_l,z_1,\dots,z_l)$. The ''Borromean rings'' are the three spheres given by the following three systems of equations: $$\left\{\begin{array}{c} x=0\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\ |x|^2+2|y|^2=1 \end{array}\right..$$ See Figures 3.5 and 3.6 of \cite{Skopenkov2006}. This embedding is distinguished from the standard embedding by ''the Massey invariant''. ====Higher-dimensional Whitehead link==== Let us present an example of ''non-injectivity of the linking coefficient''. {{beginthm|Whitehead link example}}\label{belmetwhi} The Whitehead link $w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ is a non-trivial embedding whose linking coefficient $\lambda_{12}(w)$ is trivial. {{endthm}} The Whitehead link is obtained from Borromean rings by joining two components with a tube. We have $\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in \cite{Skopenkov2006a}). For $l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$. (It would be interesting to find or write a published proof of this fact.) For $l=1,3,7$ the Whitehead link is is distinguished from the standard embedding by more complicated invariants \cite{Skopenkov2006a}, \cite{Haefliger1962T}, \S3. This example shows that the dimension restriction is sharp in Theorem \ref{lkhaze}. This example seems to be discovered by Whitehead, in connection with Whitehead product. It would be interesting to find a publication where it first appeared. ====Classification==== See [[3-manifolds in 6-space#Examples#The Haefliger trefoil knot |The Haefliger Trefoil knot]]. Let $C_q^{m-q}:=E^m_D(S^q)$. [[Knots, i.e. embeddings of spheres|Some information on this group]]. {{beginthm|The Haefliger Theorem}}\label{belmethae} (a) \cite{Haefliger1966a} If $p,q\le m-3$, then $$E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$$ (c) \cite{Haefliger1966a}, Theorem 10.7, \cite{Skopenkov2009} If $p\le q\le m-3$ and m\ge2p+2q+6$, then there is a map $\varkappa$ for which the following map is an isomorphism $$\lambda_{12}\oplus\varkappa:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).$$ {{endthm}} The map $\varkappa$ and its right inverse $\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$ are constructed in \cite{Haefliger1966} and \cite{Haefliger1966a}, cf. \cite{Skopenkov2009}. Part (b) implies that $E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\cong\pi_{2l-1}(S^l)\oplus\Z_{(l)}$. This isomorphism is defined for $l\ge2$, $l\ne3,7$ by map $$\lambda_{12}\oplus\lambda_{21}:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\pi_{2l-1}(S^l).$$ This map is injective for $l\ge2$, $l\ne3,7$; the image of this map is $\{(a,b)\ :\ \Sigma a=\Sigma b\}$ \cite{Haefliger1962T}. Part (b) shows that $\lambda_{12}\oplus\lambda_{21}$ is in general not injective. == Further discussion == ; The set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ for $m\ge n_i+3$ has been described in terms of exact sequences involving homotopy groups of spheres \cite{Haefliger1966}, \cite{Haefliger1966a}, cf. \cite{Levine 1965}, \cite{Habegger1986}. See \cite{Skopenkov2009}, \cite{Crowley&Ferry&Skopenkov2011}. == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]E^m(S^p\sqcup S^q) for $m-3\ge p,q$$m-3\ge p,q$. See Figure 3.3. of [Skopenkov2006], [Haefliger1966] [Haefliger1966a].

2 General position and the Hopf linking

General Position Theorem 2.1. For each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic.

The restriction $m\ge2n+2$$m\ge2n+2$ in Theorem 2.1 is sharp for non-connected manifolds.

Example: the Hopf linking 2.2. For each $n$$n$ there is an embedding $S^n\sqcup S^n\to\Rr^{2n+1}$$S^n\sqcup S^n\to\Rr^{2n+1}$ which is not isotopic to the standard embedding.

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

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

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

3 The Zeeman construction and linking coefficient

The following table was obtained by Zeeman around 1960:

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

1 Construction of the Zeeman map $\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)$$\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)$ for $p\le q$$p\le q$

Take $x\in\pi_p(S^{m-q-1})$$x\in\pi_p(S^{m-q-1})$ Define embedding $\tau(x)$$\tau(x)$ on $S^q$$S^q$ to be the standard embedding into $\R^m$$\R^m$. Take any map $\varphi:S^p\to\partial D^{m-q}$$\varphi:S^p\to\partial D^{m-q}$. Define embedding $\tau(x)$$\tau(x)$ on $S^p$$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$$i:S^p\to S^q$ is the equatorial inclusion and the latter inclusion is the standard. See Figure 3.2 of [Skopenkov2006]. Clearly, $\tau$$\tau$ is well-defined and is a homomorphism.

2 Definition of linking coefficient $\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})$$\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})$ for $m\ge q+3$$m\ge q+3$

Fix orientations of $S^p$$S^p$, $S^q$$S^q$, $S^m$$S^m$ and $D^{m-p}$$D^{m-p}$. Take an embedding $f:S^p\sqcup S^q\to S^m$$f:S^p\sqcup S^q\to S^m$. Take an embedding $g:D^{m-q}\to S^m$$g:D^{m-q}\to S^m$ such that $gD^{m-q}$$gD^{m-q}$ intersects $fS^q$$fS^q$ transversally at exactly one point with positive sign (see Figure 3.1 of [Skopenkov2006]). 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$$m\ge q+3$, the complement $S^m-fS^q$$S^m-fS^q$ is simply-connected. By Alexander duality $h'$$h'$ induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems $h'$$h'$ is a homotopy equivalence.)

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

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

Remark 3.1. (a) Clearly, $\lambda(f)$$\lambda(f)$ is indeed independent of $g,h',h$$g,h',h$. Clearly, $\lambda$$\lambda$ is a homomorphism.

(b) For $m=p+q+1$$m=p+q+1$ there is a simpler alternative `homological' definition. That definition works for $m=q+2$$m=q+2$ as well.

(c) 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$.

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

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

3 Classification in the `metastable' range

The Haefliger-Zeeman Theorem 3.2. If $1\le p\le q$$1\le p\le q$, then both $\lambda$$\lambda$ and $\tau$$\tau$ are isomorphisms for $m\ge\frac p2+q+2$$m\ge\frac p2+q+2$ and for $m\ge\frac{3q}2+2$$m\ge\frac{3q}2+2$, in the PL and DIFF cases, respectively.

The surjectivity of $\lambda$$\lambda$ (=the injectivity of $\tau$$\tau$) follows from $\lambda\tau=\id$$\lambda\tau=\id$. The injectivity of $\lambda$$\lambda$ (=the surjectivity of $\tau$$\tau$) is proved in [Haefliger1962T], [Zeeman1962] (or follows from the Haefliger-Weber Theorem 5.4 and Deleted Product Lemma 5.3.a of [Skopenkov2006]).

An analogue of this result holds for links with many components: the collection of pairwise linking coefficients is bijective for $2m\ge3n+4$$2m\ge3n+4$ and $n$$n$-dimensional links in $\R^m$$\R^m$.

4 Alpha-invariant

By Freudenthal Suspension Theorem $\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 alternatively as follows. For an embedding $f:S^p\sqcup S^q\to S^m$$f:S^p\sqcup S^q\to S^m$ define a map

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

See Figure 3.1 of [Skopenkov2006]. For $p\le q\le m-2$$p\le q\le m-2$ define the $\alpha$$\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 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}$$v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q}$ is the quotient map. See Figure 3.4 of [Skopenkov2006]. The map $v^*$$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)$.)

We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$$\alpha=\pm\Sigma^{\infty}\lambda_{12}$ by Lemma 5.1 of [Kervaire1959L]. Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$$\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$.

Note that $\alpha$$\alpha$-invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006], \S5.

4 Classification below the metastable range

1 Higher-dimensional Borromean rings

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

Borromean rings example 4.1. The Borromean rings $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}$ is a non-trivial embedding whose restrictions to 2-componented sublinks are 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 Borromean rings are the three spheres given by the following three systems of equations:

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

See Figures 3.5 and 3.6 of [Skopenkov2006]. This embedding is distinguished from the standard embedding by the Massey invariant.

2 Higher-dimensional Whitehead link

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

Whitehead link example 4.2. The Whitehead link $w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$$w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ is a non-trivial embedding whose linking coefficient $\lambda_{12}(w)$$\lambda_{12}(w)$ is trivial.

The Whitehead link is obtained from Borromean rings by joining two components with a tube. We have $\lambda_{12}(w)=0$$\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in [Skopenkov2006a]). For $l\ne1,3,7$$l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$$\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$. (It would be interesting to find or write a published proof of this fact.) For $l=1,3,7$$l=1,3,7$ the Whitehead link is is distinguished from the standard embedding by more complicated invariants [Skopenkov2006a], [Haefliger1962T], \S3.

This example shows that the dimension restriction is sharp in Theorem 3.2.

This example seems to be discovered by Whitehead, in connection with Whitehead product. It would be interesting to find a publication where it first appeared.

3 Classification

See The Haefliger Trefoil knot. Let $C_q^{m-q}:=E^m_D(S^q)$$C_q^{m-q}:=E^m_D(S^q)$. Some information on this group.

The Haefliger Theorem 4.3. (a) [Haefliger1966a] If $p,q\le m-3$$p,q\le m-3$, then

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

(c) [Haefliger1966a], Theorem 10.7, [Skopenkov2009] 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 map $\varkappa$$\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$$\varkappa$ and its right inverse $\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$$\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$ are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].

Part (b) implies that $E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\cong\pi_{2l-1}(S^l)\oplus\Z_{(l)}$$E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\cong\pi_{2l-1}(S^l)\oplus\Z_{(l)}$. This isomorphism is defined for $l\ge2$$l\ge2$, $l\ne3,7$$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$$l\ge2$, $l\ne3,7$$l\ne3,7$; the image of this map is $\{(a,b)\ :\ \Sigma a=\Sigma b\}$$\{(a,b)\ :\ \Sigma a=\Sigma b\}$ [Haefliger1962T]. Part (b) shows that $\lambda_{12}\oplus\lambda_{21}$$\lambda_{12}\oplus\lambda_{21}$ is in general not injective.

5 Further discussion

The set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$$E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ for $m\ge n_i+3$$m\ge n_i+3$ has been described in terms of exact sequences involving homotopy groups of spheres [Haefliger1966], [Haefliger1966a], cf. [Levine 1965], [Habegger1986].

See [Skopenkov2009], [Crowley&Ferry&Skopenkov2011].

6 References

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