I want to understand relation between complex and real spheres. How to show?
$S^1(\mathbb{C}) \approx \mathbb{R} \times S^1$ $S^3(\mathbb{C}) \approx \mathbb{R} \times S^3$
$\approx$ means homotopy equivalence.
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1What is your definition of $S^1(\mathbb{C})$ and $S^3(\mathbb{C})$? – Yuchen Liu Apr 07 '13 at 03:12
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$S^n (\mathbb{C}) = { (z_0, z_1, \ldots , z_n) \in \mathbb{C}^{n+1} ; \ z_0^2 +\ldots +z_n^2=1 } $ – pjastr Apr 07 '13 at 10:05
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1I guess he know the definition, but yours is wrong, $S^n(C)={z \in \mathbb{C}^{n+1}: |z|=1}$, and yours is not even a norm. – Dominic Michaelis Apr 07 '13 at 10:11
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I don't take this as set definied by norn. For me $S^n(\mathbb{C})$ is affine quadric ddefined by the equation. – pjastr Apr 07 '13 at 10:20
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1Well but that is really not an answer to the question anyway. Your sphere is not bounded is that on purpose ? – Dominic Michaelis Apr 07 '13 at 10:28
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I don't know that. – pjastr Apr 07 '13 at 10:32
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1Well take $S^2(\mathbb{C}$ with your definition and the points $(1,n,ni)$ this makes $1^2+n^2 -n^2 =1$ neverthe less what n is – Dominic Michaelis Apr 07 '13 at 10:34
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Ok, you have right. This set may be not bounded. For me important case is for n=1,3. – pjastr Apr 07 '13 at 10:39
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1Are you Piotrekkk ? – Dominic Michaelis Apr 07 '13 at 10:39