High codimension links
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where $i:S^p\to S^q$ is the equatorial inclusion and the latter inclusion is the standard. | 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}. | See Figure 3.2 of \cite{Skopenkov2006}. | ||
+ | |||
+ | '' Construction of the linking coefficient'' | ||
+ | $$\lambda:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.$$ | ||
+ | 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 of $g$ to $\partial D^{m-q}$ is an orientation preserving homotopy equivalence $S^{m-q-1}\to S^m-fS^q$. | ||
+ | Let $h$ be a homotopy inverse. | ||
+ | Define | ||
+ | $$\lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$$ | ||
+ | |||
+ | Clearly, $\lambda_{12}(f)$ is indeed independent of $g,h$. | ||
+ | The isomorphism of homotopy groups induced by $h$ does not depend on $g$. | ||
+ | |||
+ | Analogously we may define $\lambda_{21}(f)\in\pi_q(S^{m-p-1})$ for $m\ge p+3$. | ||
+ | The definition works for $m=q+2$ if the restriction of $f$ to $S^q$ | ||
+ | is PL unknotted (this is always so for $m\ge q+3$ by Theorem \wi5.a). | ||
+ | For $m=p+q+1$ there is a simpler alternative definition. | ||
+ | |||
Revision as of 12:44, 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].
Construction of the linking coefficient
Fix orientations of , , and . Take an embedding . Take an embedding such that intersects transversally at exactly one point with positive sign (see Figure 3.1 of [Skopenkov2006]). Then the restriction of to is an orientation preserving homotopy equivalence . Let be a homotopy inverse. Define
Clearly, is indeed independent of . The isomorphism of homotopy groups induced by does not depend on .
Analogously we may define for . The definition works for if the restriction of to is PL unknotted (this is always so for by Theorem \wi5.a). For there is a simpler alternative definition.
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.