How do you describe a Riemann surface? I am looking at $f(z)=(z-1)^{1/3}.$ I have found the branch points and know that there needs to be a branch cut touching each of the branch points, but these branch cuts can be taken in many different ways.
I do not understand how to formally "describe" the Riemann surface or define the branch cuts. Do I visually  draw my branch cuts and this interpretation of the Riemann surface, or is there a way to formally write it up?
I would also appreciate any references you can recommend on the subject.
 A: I doubt whether you'll like this, but here goes.
Your Riemann surface is a triple branched cover of the Riemann sphere $\Bbb C_\infty$
with three branch points $1$, $\omega$, $\omega^2$. Deleting these gives
a triple covering of the thrice punctured Riemann sphere $X=\Bbb C_\infty-\{1,\omega,\omega^2\}$. Pick a base point $a\in X$. Coverings of $X$ correspond to subgroups
of the fundamental group $G=\pi_1(X,a)$. A three-fold covering will correspond to
an index $3$ subgroup $H$ of $G$.
How can we describe $H$? For $n\ge2$ a the fundamental group of an $n$-punctured
sphere is free on $n-1$ generators. Here we can take two generators for $G$,
informally these can be loops based at $a$ "encircling" two of the branch points.
Call these $g_1$ and $g_2$. How can we define $H$? As $H$ has index $3$, $H$ is a point
stabiliser of the action of $G$ on the cosets of $G$. So for a suitable homomorphism
$\phi:G\to S_3$, with transitive image, $H$ is the inverse image of a point stabiliser
in $S_3$.
We have $g_1$ and $g_2$ generating $G$. What do $\phi(g_1)$ and $\phi(g_2)$ represent?
Well imagine travelling around the loop corresponding to $g_1$ on the Riemann surface.
You may not end up on the sheet you started with. Traversing $g_1$ then induces
a permutation of the sheets of the Riemann surface in a neighbourhood of $a$. This is the
permutation corresponding to $\phi(g_1)$.
An accessible book on Riemann surfaces is that by Jones and Singerman.
