High codimension links
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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:
![\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..](/images/math/1/8/b/18b1d7e000cc51571b57f92b03227c8f.png)
3 The Haefliger-Zeeman classification
The following table was obtained by Zeeman around 1960:
![\displaystyle \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}](/images/math/b/2/2/b22cac6dff110756c80daca3082b8178.png)
Construction of the Zeeman map
![\displaystyle \tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q).](/images/math/3/4/1/341d57ce5a10ef3bf8d754904e18a278.png)
Take
Define embedding
on
to be the standard embedding into
.
Take any map
.
Define embedding
on
to be the composition
![\displaystyle S^p\overset{x\times i}\to\partial D^{m-q}\times S^q \subset D^{m-q}\times S^q\subset\R^m,](/images/math/a/2/9/a29aef5baaba94d8741863f8a2bbe06c.png)
where is the equatorial inclusion and the latter inclusion is the standard.
See Figure 3.2 of [Skopenkov2006].
Construction of the linking coefficient
![\displaystyle \lambda:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.](/images/math/b/9/f/b9f3d259cf48b2448a3de8b6444e93e0.png)
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
![\displaystyle \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}).](/images/math/0/5/c/05c1bffa3c231c4e8f9df10db3009149.png)
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.