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)$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT--> {{Stub}} == Introduction == <wikitex>; For notation and conventions throughout this page see [[High_codimension_embeddings|high codimension embeddings]]. `Embedded connected sum' defines a commutative group structure on $E^m(S^p\sqcup S^q)$ for $m-3\ge p,q$. See Figure 3.3. of \cite{Skopenkov2006}, \cite{Haefliger1966} \cite{Haefliger1966C}. </wikitex> == General position and the Hopf linking == <wikitex>; {{beginthm|General Position Theorem}}\label{th1} For each $n$-manifold $N$ and $m\ge2n+2$, every two embeddings $N\to\Rr^m$ are isotopic. {{endthm}} The restriction $m\ge2n+2$ in Theorem \ref{th1} is sharp for non-connected manifolds. {{beginthm|Example: the Hopf linking}}\label{hopf} For each $n$ there is an embedding $S^n\sqcup S^n\to\Rr^{2n+1}$ which is not isotopic to the standard embedding. {{endthm}} For $n=1$ the Hopf Linking is shown in Figure~2.1.a of \cite{Skopenkov2006}. For arbitrary $n$ (including $n=1$) the image of the Hopf Linking is the union of two $n$-spheres: $$\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..$$ </wikitex> == The Haefliger-Zeeman classification == <wikitex>; The following table was obtained by Zeeman around 1960: $$\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}).$$ {{beginthm|Remark}}\label{lkrem} (a) Clearly, $\lambda(f)$ is indeed independent of $g,h',h$. (b) For $m=p+q+1$ there is a simpler alternative `homological' definition. That definition works for $m=q+2$ as well. (c) Analogously one can define $\lambda_{21}(f)\in\pi_q(S^{m-p-1})$ for $m\ge p+3$. (d) This definition works for $m=q+2$ if $S^m-fS^q$ is simply-connected (or, equivalently for $q>4$, if the restriction of $f$ to $S^q$ is unknotted). {{endthm}}. (e) Clearly, $\lambda\tau=\id \pi_p(S^{m-q-1})$, even for $m=q+2$. So $\lambda$ is surjective and $\tau$ is injective. ====Classification in the `metastable' range==== {{beginthm|The Haefliger-Zeeman Theorem}}\label{lkhaze} If E^m(S^p\sqcup S^q)$ for $m-3\ge p,q$$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 Haefliger-Zeeman classification

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.

If $m\ge\frac p2+q+2$$m\ge\frac p2+q+2$, then $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$$\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism.

6 References

• [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
• [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.

\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}} If $m\ge\frac p2+q+2$, then $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism. == 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 Haefliger-Zeeman classification

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.

If $m\ge\frac p2+q+2$$m\ge\frac p2+q+2$, then $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$$\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism.

6 References

• [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
• [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.