3-manifolds in 6-space
(→The Hopf construction of an embedding $\Rr P^3\to S^5$) |
(→The Hopf construction of an embedding $\Rr P^3\to S^5$) |
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Represent $\Rr P^3=\{(x,y)\in\Cc^2\ |\ |x|^2+|y|^2=1\}/\pm1.$ Define | Represent $\Rr P^3=\{(x,y)\in\Cc^2\ |\ |x|^2+|y|^2=1\}/\pm1.$ Define | ||
− | $$Ho:\Rr P^3\to S^5\subset\Cc^3\quad\text{by}\quad | + | $$Ho:\Rr P^3\to S^5\subset\Cc^3\quad\text{by}\quad Ho[(x,y)]=(x^2,2xy,y^2).$$ |
It is easy to check that $Ho$ is an embedding. (The image of this embedding is given by the equations $b^2=4ac$, $|a|^2+|b|^2+|c|^2=1$.) | It is easy to check that $Ho$ is an embedding. (The image of this embedding is given by the equations $b^2=4ac$, $|a|^2+|b|^2+|c|^2=1$.) | ||
Revision as of 16:27, 14 February 2010
This page has been accepted for publication in the Bulletin of the Manifold Atlas. |
For notation and conventions see high codimension embeddings.
Contents |
1 The Haefliger trefoil knot
Let us construct a smooth embedding (which is a generator of ) [Haefliger1962], 4.1. A miraculous property of this embedding is that it is not isotopic to the standard embedding, but is isotopic to the standard embedding.
Denote coordinates in by . The higher-dimensional trefoil knot is obtained by joining with two tubes the higher-dimensional , i.e. the three spheres given by the following three systems of equations:
See Figures 3.5 and 3.6 of [Skopenkov2006].
2 The Hopf construction of an embedding
Represent Define
It is easy to check that is an embedding. (The image of this embedding is given by the equations , .)
It would be interesting to obtain an explicit construction of an embedding which is not isotopic to the composition of the Hopf embedding with the standard inclusion . (Such an embedding is unique up to PL isotopy by classification just below the stable range.)
3 Algebraic embeddings from the theory of integrable systems
Some 3-manifolds appear in the theory of integrable systems together with their embeddings into (given by a system of algebraic equations) [BolsinovFomenko2004], Chapter 14. E.g. the following system of equations corresponds to the Euler integrability case:
where and are variables while and are constants. This defines embeddings of , or into .
4 References
- [BolsinovFomenko2004] Template:BolsinovFomenko2004
- [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. |