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I am having difficulty answering the following question

a) for $u\neq 0$, $C_u$ is smooth submanifold

b) $f: \mathbb C\to C_0; z\mapsto (z^3,z^2)$ is surjective

c) $C_0$ is a topological manifold

a) define $F:\mathbb C^2\to \mathbb C$ by $F(z_1,z_2)=z_1^2-z_2^3$ this has constant rank of 1 over $\mathbb C^2\backslash \{(0,0)\}$. Thus, $u$ is a regular value. So, $F^{-1}(u)$ is a smooth submanifold.

b) for any $(a,b)\in C_0$ then find $z=a^{1/3}$ (maybe take the principal complex root) then $z=b^{1/2}$ since $z^6=a^2=b^3$. Then $f(z)=(z^3,z^2)=(a,b).$

c) I do not know. EDIT the regular value test fails (thanks to Ted Shifrin remark)

salmauuu
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  • How do you know it’s not a smooth submanifold? – Ted Shifrin Aug 22 '22 at 04:13
  • @TedShifrin Ah I see even though $(0,0)$ is not a regular point and so $0$ is not regular value of $F$. This does not mean it is not a smooth submanifold. Just the regular value test is not applicable. Thanks I am editing the question (I can not upvote :( – salmauuu Aug 22 '22 at 05:45
  • @Sumanta Would you please write it as answer so that I can accept it? – salmauuu Aug 22 '22 at 11:44
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    Welcome to Math.SE! <> The mapping $f$ is a bijection with continuous inverse, isn't it...? – Andrew D. Hwang Aug 22 '22 at 15:18
  • @salmauuu, what I have written was wrong; see Andrew's comment; $f$ is a bijection. Thus, you can give topological manifold structure on $C_0$. See this page https://math.stackexchange.com/questions/3398697/manifold-structure-such-that-a-bijection-of-sets-becomes-a-diffeomorphism-of-man – Sumanta Aug 22 '22 at 18:32

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