High codimension links

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1 Introduction

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

2 General position and the Hopf linking

General Position Theorem 2.1. For each $n$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT--> {{Stub}} == Introduction == <wikitex>; For notation and conventions throughout this page see [[High_codimension_embeddings|high codimension embeddings]]. </wikitex> == General position and the Hopf linking == <wikitex>; {{beginthm|General Position Theorem}}\label{th1} For each $n$-manifold $N$ and $m\ge2n+2$, every two embeddings $N\to\Rr^m$ are isotopic. {{endthm}} The restriction $m\ge2n+2$ in Theorem \ref{th1} is sharp for non-connected manifolds. {{beginthm|Example: the Hopf linking}}\label{hopf} For each $n$ there is an embedding $S^n\sqcup S^n\to\Rr^{2n+1}$ which is not isotopic to the standard embedding. {{endthm}} For $n=1$ the Hopf Linking is shown in Figure~2.1.a of \cite{Skopenkov2006}. For arbitrary $n$ (including $n=1$) the image of the Hopf Linking is the union of two $n$-spheres: $$\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..$$ </wikitex> == Classification/Characterization == <wikitex>; </wikitex> == Invariants == <wikitex>; </wikitex> == Further discussion == <wikitex>; </wikitex> <!-- == Acknowledgments == == Footnotes == <references/> --> == References == {{#RefList:}} <!-- == External links == * The Wikipedia page about [[Wikipedia:Page_name|link text]]. --> <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]n$-manifold $N$$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:

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

6 References

• [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.