High codimension links
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− | == | + | == The Haefliger-Zeeman classification == |
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− | + | 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}. | ||
+ | |||
+ | |||
+ | </wikitex> | ||
== Invariants == | == Invariants == |
Revision as of 12:25, 13 February 2013
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. |
Contents |
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 -manifold and , every two embeddings are isotopic.
The restriction in Theorem 2.1 is sharp for non-connected manifolds.
Example: the Hopf linking 2.2. For each there is an embedding which is not isotopic to the standard embedding.
For the Hopf Linking is shown in Figure~2.1.a of [Skopenkov2006]. For arbitrary (including ) the image of the Hopf Linking is the union of two -spheres:
3 The Haefliger-Zeeman classification
The following table was obtained by Zeeman around 1960:
Construction of the Zeeman map . Take Define embedding on to be the standard embedding into . Take any map . Define embedding on to be the composition
where is the equatorial inclusion and the latter inclusion is the standard. See Figure 3.2 of [Skopenkov2006].
4 Invariants
5 Further discussion
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.