High codimension links
Askopenkov (Talk | contribs) (→The Haefliger-Zeeman classification) |
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− | + | ====Construction of the Zeeman map $\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q).$==== | |
− | + | ||
Take $x\in\pi_p(S^{m-q-1})$ | Take $x\in\pi_p(S^{m-q-1})$ | ||
Define embedding $\tau(x)$ on $S^q$ to be the standard embedding into $\R^m$. | Define embedding $\tau(x)$ on $S^q$ to be the standard embedding into $\R^m$. | ||
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See Figure 3.2 of \cite{Skopenkov2006}. | See Figure 3.2 of \cite{Skopenkov2006}. | ||
− | + | ====Construction of the linking coefficient $\lambda:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})$ for $m\ge q+3.$==== | |
− | + | ||
Fix orientations of $S^p$, $S^q$, $S^m$ and $D^{m-p}$. | 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 $f:S^p\sqcup S^q\to S^m$. | ||
Take an embedding $g:D^{m-q}\to S^m$ such that $gD^{m-q}$ intersects $fS^q$ | 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}). | 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 | + | Then the restriction $h':S^{m-q-1}\to S^m-fS^q$ of $g$ to $\partial D^{m-q}$ is a homotopy equivalence. |
− | Let $h$ be a homotopy inverse. | + | |
+ | (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 | Define | ||
$$\lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$$ | $$\lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$$ |
Revision as of 12:53, 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:
1 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].
2 Construction of the linking coefficient for
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 a homotopy equivalence.
(Indeed, since , the complement is simply-connected. By Alexander duality induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . 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.