4-manifolds in 7-space
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Contents |
1 Introduction
For notation and conventions see high codimension embeddings.
2 Examples
There is The Hudson torus
.
Analogously to the case for an orientable 4-manifold
, an embedding
and a class
one can construct an embedding
. However, this embedding is no longer well-defined.
We have for the Whitney invariant (which is defined analogously to The Whitney invariant for
.
2.1 An embedding of CP2 into R7
We follow [Boechat&Haefliger1970], p. 164. It suffices to construct an embedding such that the boundary 3-sphere is standardly embedded into
. Recall that
is the mapping cylinder of the Hopf map
. Recall that
. Define
, where
. In other words, the segment joining
and
is mapped onto the arc in
joining
to
.
2.2 The Lambrechts torus and the Hudson torus
[Skopenkov2006].
These two embeddings are defined as
compositions
,
where
,
is the projection onto the second factor,
is
the standard inclusion and maps
are defined below.
We shall see that
are embeddings for
each
, hence
and
are embeddings.
Define , where
is identified with the set of
unit length quaternions and
with the set of unit
length complex numbers.
Define , where
is the Hopf map and
is identified with the 2-sphere formed by unit
length quaternions of the form
.
Note that is PL isotopic to
The Hudson torus
.
Take the Hopf fibration . Take the standard embeding
. Its complement has the homotopy type of
. Then
. This is the construction of Lambrechts motivated by the following property:
![\displaystyle S^7-im\tau^1\simeq \eta^{-1}(S^2)\cong S^2\times S^3\not\simeq S^2\vee S^3\vee S^5\simeq S^7-im f_0,](/images/math/1/b/b/1bbdafdc2296d2623edc8f6b06241099.png)
where is the standard embedding.
2.3 The Haefliger torus
This is a PL embedding which is (locally flat but) not PL isotopic to a smooth embedding [Haefliger1962], [Boechat&Haefliger1970], p.165, [Boechat1971], 6.2. Take the Haefliger trefoil knot
.
Extend this knot to a conical embedding
. By [Haefliger1962], the trefoil knot also extends to a smooth embedding
(see [Skopenkov2006], Figure 3.7.a). These two extensions together form the Haefliger torus
(see [Skopenkov2006], 3.7.b).
3 The Boechat-Haefliger invariant
4 Classification
5 References
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension
dans
, (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
- [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension
dans
, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
- [Haefliger1962] A. Haefliger, Knotted
-spheres in
-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [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.
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