Could someone please explain the concept of branch points to me? I have tried searching online and had a read of the textbook Visual Complex Analysis by T. Needham, but I am still not very clear how they work.
An excerpt I found online from Introduction to Complex Analysis by H. Priestly says that
$a$ is a branch point for [$w(z)$] if, for all sufficiently small $r>0$, it is not possible to choose $f(z)\in[w(z)]$ so that $f$ is a continuous function on $\gamma(a;r)^*$.
Firstly, I couldn't find what $\gamma(a;r)^*$ is … I presume it is an open ball around the point $a$ with radius $r$?
Secondly, I just don't understand what it is saying. Why is there no continuous function? How when asked to find branch points would I know which points in $\mathbb C$ have this property?
Needham's book basically says a branch point is one which if we circle it once we don't get back to the same point… but I still don't get it!
Then I read something about branch cuts and Riemann spheres which really don't help to clarify anything at all!
Thank you for your time.
[Added] For example if I have a map of the form $f(z) =[(z-a)(z-b)…(z-n)]^{1\over m}$ how may branch points are there?
Best Answer
Given a non-constant morphism of Riemann surfaces $f:X\to Y$, a critical point of $f$ is a point $x\in X$ such that equivalently :
$\bullet$ For every neighbourhood $x\in U\subset X$ the restriction $f|U:U\to Y$ is not injective.
$\bullet$ The differential $df(x)=0$
$\bullet$ In local cooordinates at $x$ and $f(x)$, $f$ can be written as $z\mapsto z^n $ with $n\geq 2$
The critical values of $f$ are the $y\in Y$ which can be written $y=f(x)$ for some critical $x\in X$.
The terminology "branchpoint" is unfortunately ambiguous: some authors (e.g. Griffiths and Forster) use it for critical point and others (e.g. Miranda) for critical values.
For example, given distinct $a_1, a_2,...,a_n\in \mathbb C$, if you consider the Riemann surface $X$ of the "function"$\sqrt {(z-a_1)(z-a_2)...(z-a_n)}$, you will obtain a morphism $f:X\to \mathbb P^1(\mathbb C)$ whose critical values in $\mathbb C$ are $a_1,a_2,..., a_n$ .
Moreover $\infty\in \mathbb P^1(\mathbb C) $ will also be a critical value precisely when $n$ is odd.
In this simple but basic example each critical value has exactly one critical point mapping to it.
Although the above might look a bit abstract, be very wary of the "concrete" approach of this kind of problems by cut and paste techniques with pictures of sheets crossing themselves.
It has elicited some very harsh words from Serge Lang (for example) in his book here , Chapter XI, ยง1: "It should be emphasised that the picture is totally and irretrievably misleading".
I recommend Forster's Riemann Surfaces for a completely rigorous and definitive treatment.