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I am reading the book geometry and topology by Bredon and I got quite confused with example 7.8, which says the following:

Consider $V = \{(z_1,z_2,z_3) \in \mathbb{C}^3-\{0\}|z_1^3+ z_2^3 + z_3^3 =0 \}$. Note that $0$ is a regular value of $(z_1,z_2,z_3) \mapsto z_1^3 + z_2^2 + z_3^2$ of $\mathbb{C}^3 - \{0\} \rightarrow \mathbb{C}$, so that $V$ is a 4-manifold. Let $S = \mathbb{S}^5 = \{(z_1,z_2,z_3) \in \mathbb{C}^3 | |z_1|^2 + |z_2|^2 + |z_3|^2 = 1\}$. Then, we claim that $V \pitchfork S$ and hence that $V \cap S$ is a 3-manifold.

My questions are: why is $V$ a 4-manifold? How can we make sure that $0$ is a regular value for the map proposed? Finally, would $S$ be a 3-manifold? Thanks for your help!

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    What have you tried? For example, what happened when you tried to apply the definition of a regular value in order to see whether $0$ is a regular value for the map proposed? – 2017-02-13
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    Also, do you know the implicit function theorem? And do you know about transversality of manifolds? – 2017-02-13

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