4-manifolds in 7-space
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− | + | These two embeddings $\tau^1,\tau^2:S^1\times S^3\to\Rr^7$ are defined \cite{Skopenkov2006} as compositions $S^1\times S^3\overset{p_2\times t^i}\to\rightarrow S^3\times S^3\subset\Rr^7$, where $i=1,2$, $p_2$ is the projection onto the second factor, $\subset$ is | |
− | These two embeddings $\tau^1,\tau^2:S^1\times S^3\to\Rr^7$ are defined as | + | the standard inclusion and maps $t^i:S^1\times S^3\to S^3$ are defined below. We shall see that $t^i|_{S^1\times y}$ are embeddings for each $y\in S^3$, hence $\tau^1$ and $\tau^2$ are embeddings. |
− | compositions | + | |
− | $S^1\times S^3\overset{p_2\times t^i}\to\rightarrow S^3\times S^3\subset\Rr^7$, | + | |
− | where $i=1,2$, $p_2$ is the projection onto the second factor, $\subset$ is | + | |
− | the standard inclusion and maps $t^i:S^1\times S^3\to S^3$ are defined below. | + | |
− | We shall see that $t^i|_{S^1\times y}$ are embeddings for | + | |
− | each $y\in S^3$, hence $\tau^1$ and $\tau^2$ are embeddings. | + | |
− | Define $t^1(s,y):=sy$, where $S^3$ is identified with the set of | + | Define $t^1(s,y):=sy$, where $S^3$ is identified with the set of unit length quaternions and $S^1\subset S^3$ with the set of unit |
− | unit length quaternions and $S^1\subset S^3$ with the set of unit | + | |
length complex numbers. | length complex numbers. | ||
− | Define $t^2(e^{i\theta},y):= | + | Define $t^2(e^{i\theta},y):=\eta(y)\cos\theta+\sin\theta$, where $S^2$ is identified with the 2-sphere formed by unit |
− | + | ||
length quaternions of the form $ai+bj+ck$. | length quaternions of the form $ai+bj+ck$. | ||
Revision as of 12:07, 7 March 2010
This page has been accepted for publication in the Bulletin of the Manifold Atlas. |
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
These two embeddings are defined [Skopenkov2006] 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 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:
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.
This page has not been refereed. The information given here might be incomplete or provisional. |