Describe the Riemann surface associated with $w=(z^2-1)^2$

Question

I've been asked to describe the Riemann surface associated with the function $w= (z^2-1)^2$. I found the branches to be $\pm 1$, but I am confused about proceeding.

I know if the function were just $w= (z^2-1)$, the corresponding Riemann surface would be 2-sheeted, but for my questions I am confused about whether the surface would be 2-sheeted or 4-sheeted, and how to tell in the future.

Thanks for any help!

Answer 1

I guess you have copied Exercise 2 from Section 3.4.3 of Ahlfors. Some other books use a different definition of Riemann surface, which would render your question meaningless. (Hence some comments that you have already received.)

Let's use Ahlfors' definition here. We want to divide $\mathbb{C}$ into non-overlapping regions so that $f$, when restricted to each region, be a 1-1 mapping.

Let's use a concrete example. Imagine a simple closed curve $z(t)$ that moves from 2, 2i, -2, -2i, and back to 2. You will get $f(z)=9$ four times. This means you need four sheets.

In fact, we can take the four quadrants of the complex plane to be your four fundamental regions.

Below is some modification to my original answer following Maxim's comment. (I went back to the textbooks that I used before. Ahlfors is actually a lot less clear on this than e.g. Needham. So if you are using Ahlfors, please get another book to get a different perspective.)

For the branch points, it is easier to look at $z=\sqrt{1+\sqrt{w}}$.

$w$ needs to go around a small circle around 0 twice for $z$ to go around either +1 or -1, depending on which branch you take for the outer square root, once. Therefore $w=0$ is a branch point of order 1.

Similarly, if we take the negative inner square root, $w$ needs to go around a small circle around 1 twice for $z$ to go around 0 once. Therefore $w=1$ is also a branch point of order 1. (The other two copies of $w$, corresponding to $z=\pm\sqrt{2}$, are not branching points.)

Finally, $w$ needs to go around a big enough circle, e.g. $|w| = 10$, four times, for $z$ to return to its original value. Therefore $\infty$ is a branching point of order 3.