High codimension links
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\#E^m(S^q\sqcup S^q) &1 &\infty &2 &2 &24 &1 &1 | \#E^m(S^q\sqcup S^q) &1 &\infty &2 &2 &24 &1 &1 | ||
\end{array}$$ | \end{array}$$ | ||
+ | |||
====Construction of the Zeeman map $\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q).$==== | ====Construction of the Zeeman map $\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q).$==== | ||
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$$\lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$$ | $$\lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$$ | ||
− | {{beginthm|Remark}}\label{ | + | {{beginthm|Remark}}\label{lkrem} |
(a) Clearly, $\lambda(f)$ is indeed independent of $g,h',h$. | (a) Clearly, $\lambda(f)$ is indeed independent of $g,h',h$. | ||
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(d) This definition works for $m=q+2$ if $S^m-fS^q$ is simply-connected | (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). | (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 $1\le p\le q$, 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}} | {{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. | ||
Revision as of 13:10, 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 Definition of 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
Remark 3.1. (a) Clearly, is indeed independent of .
(b) For there is a simpler alternative `homological' definition. That definition works for as well.
(c) Analogously one can define for .
(d) This definition works for if is simply-connected (or, equivalently for , if the restriction of to is unknotted).
(e) Clearly, , even for . So is surjective and is injective.
3 Classification in the `metastable' range
The Haefliger-Zeeman Theorem 3.2. If , then both and are isomorphisms for and for , in the PL and DIFF cases respectively.
If , then is an isomorphism.
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.