1

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.

Focus
  • 754
  • As we are looking for a real manifold, the conditions should be interpreted as three real polynomials in real and imaginary parts, the first two polynomials being homogenuous, the last describing the unit sphere. With this abstraction, it should be easier (than with concrete polynomials) to see that the set is a compact manifold of dimension $\ge 3$. -- An idea for the fundamental group: Maybe see how you can glue suitable parts together? – Hagen von Eitzen Apr 07 '18 at 09:47

0 Answers0