Draw the schemes of the Riemann surfaces and compute the monodromy groups for:
($a$) $ \sqrt{\sqrt{z^2 +1}-2} $
($b$)$ \sqrt{2-z^{\frac{1}{3}}} $
How do I calculate the monordromy groups for these functions? What is the procedure? I'm even more confused about the Riemann surfaces for these functions. Can anyone provide some guidance?
Start with $f(z)=\sqrt{2-\sqrt{z}}$ analytic at $z=10$.
For each branch $f_j(z)=a_j \sqrt{2-b_j \sqrt{z}}$ (where $a_j=\pm 1,b_j=\pm 1$)
analytic at $z=10$ there is a closed-loop $\gamma_j:10\to 10$ such that when continuing $f$ analytically along $\gamma_j$ it will become $f_j$.
Your problem is to find this closed-loop, then to find what each $f_i$ becomes when continued analytically along it.
The analytic continuation along $\gamma_j$ is permuting the $f_i$.
The monodromy group is the group generated by those permutations, each permutation being represented by (an homotopy class of) closed-loop $10\to 10$, the composition of two permutations being represented by the concatenation of the two underlying closed-loops.