Prove that $$\{(a,b,c) \in \mathbb{C}^3|a^5+b^3+c^2=0,|a|^2+|b|^2+|c|^2=1\}$$ is a 3 dimensional compact smooth (real) manifold and calculate its fundamental group $\pi_1$.
I wonder if there's a way to check without invoking(and with brute force) regular value theorem.And I don't how to find the fundamental group.