High codimension links

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

This page has not been refereed. The information given here might be incomplete or provisional.

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)$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT--> {{Stub}} == Introduction == <wikitex>; For notation and conventions throughout this page see [[High_codimension_embeddings|high codimension embeddings]]. `Embedded connected sum' defines a commutative group structure on $E^m(S^p\sqcup S^q)$ for $m-3\ge p,q$. See Figure 3.3. of \cite{Skopenkov2006}, \cite{Haefliger1966} \cite{Haefliger1966C}. </wikitex> == General position and the Hopf linking == <wikitex>; {{beginthm|General Position Theorem}}\label{th1} For each $n$-manifold $N$ and $m\ge2n+2$, every two embeddings $N\to\Rr^m$ are isotopic. {{endthm}} The restriction $m\ge2n+2$ in Theorem \ref{th1} is sharp for non-connected manifolds. {{beginthm|Example: the Hopf linking}}\label{hopf} For each $n$ there is an embedding $S^n\sqcup S^n\to\Rr^{2n+1}$ which is not isotopic to the standard embedding. {{endthm}} For $n=1$ the Hopf Linking is shown in Figure~2.1.a of \cite{Skopenkov2006}. For arbitrary $n$ (including $n=1$) the image of the Hopf Linking is the union of two $n$-spheres: $$\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..$$ </wikitex> == The Zeeman construction and linking coefficient == <wikitex>; The following table was obtained by Zeeman around 1960: $$\begin{array}{c|c|c|c|c|c|c|c} m &2q+2 &2q+1 &2q &2q-1 &2q-2 &2q-3 &2q-4 \ \#E^m(S^q\sqcup S^q) &1 &\infty &2 &2 &24 &1 &1 \end{array}$$ ====Construction of the Zeeman map $\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)$==== Take $x\in\pi_p(S^{m-q-1})$ Define embedding $\tau(x)$ on $S^q$ to be the standard embedding into $\R^m$. Take any map $\varphi:S^p\to\partial D^{m-q}$. Define embedding $\tau(x)$ on $S^p$ to be the composition $$S^p\overset{x\times i}\to\partial D^{m-q}\times S^q \subset D^{m-q}\times S^q\subset\R^m,$$ where $i:S^p\to S^q$ is the equatorial inclusion and the latter inclusion is the standard. See Figure 3.2 of \cite{Skopenkov2006}. ====Definition of linking coefficient $\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})$ for $m\ge q+3$==== Fix orientations of $S^p$, $S^q$, $S^m$ and $D^{m-p}$. Take an embedding $f:S^p\sqcup S^q\to S^m$. Take an embedding $g:D^{m-q}\to S^m$ such that $gD^{m-q}$ intersects $fS^q$ transversally at exactly one point with positive sign (see Figure 3.1 of \cite{Skopenkov2006}). Then the restriction $h':S^{m-q-1}\to S^m-fS^q$ of $g$ to $\partial D^{m-q}$ is a homotopy equivalence. (Indeed, since $m\ge q+3$, the complement $S^m-fS^q$ is simply-connected. By Alexander duality $h'$ induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems $h'$ is a homotopy equivalence.) Let $h$ be a homotopy inverse of $h'$. Define $$\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}).$$ {{beginthm|Remark}}\label{lkrem} (a) Clearly, $\lambda(f)$ is indeed independent of $g,h',h$. (b) For $m=p+q+1$ there is a simpler alternative `homological' definition. That definition works for $m=q+2$ as well. (c) Analogously one can define $\lambda_{21}(f)\in\pi_q(S^{m-p-1})$ for $m\ge p+3$. (d) This definition works for $m=q+2$ if $S^m-fS^q$ is simply-connected (or, equivalently for $q>4$, if the restriction of $f$ to $S^q$ is unknotted). (e) Clearly, $\lambda\tau=\id \pi_p(S^{m-q-1})$, even for $m=q+2$. So $\lambda$ is surjective and $\tau$ is injective. {{endthm}} ====Classification in the `metastable' range==== {{beginthm|The Haefliger-Zeeman Theorem}}\label{lkhaze} If E^m(S^p\sqcup S^q)$ for $m-3\ge p,q$$m-3\ge p,q$. See Figure 3.3. of [Skopenkov2006], [Haefliger1966] [Haefliger1966C].

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:

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

3 The Zeeman construction and linking coefficient

The following table was obtained by Zeeman around 1960:

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

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)$

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].

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$.

(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]).

4 $\alpha$$\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

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

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 [Kervaire1959].

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

6 References

• [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
• [Haefliger1966C] Template:Haefliger1966C
• [Kervaire1959] M. A. Kervaire, A note on obstructions and characteristic classes, Amer. J. Math. 81 (1959), 773–784. MR0107863 (21 #6585) Zbl 0124.16302
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [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.
• [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}). ====$\alpha$-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 $$\alpha(f)=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\to\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.$$ 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{Kervaire1959}. Note that $\alpha$-invariant can be defined in more general situations \cite{Koschorke1988}, \cite{Skopenkov2006}. == Invariants == ; == Further discussion == ; == References == {{#RefList:}} [[Category: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] [Haefliger1966C].

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:

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

3 The Zeeman construction and linking coefficient

The following table was obtained by Zeeman around 1960:

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

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)$

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].

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$.

(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]).

4 $\alpha$$\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

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

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 [Kervaire1959].

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

6 References

• [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
• [Haefliger1966C] Template:Haefliger1966C
• [Kervaire1959] M. A. Kervaire, A note on obstructions and characteristic classes, Amer. J. Math. 81 (1959), 773–784. MR0107863 (21 #6585) Zbl 0124.16302
• [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
• [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.
• [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