Degree of analytic map between Riemann surfaces

Definition

Suppose $M,N$ are Riemann surfaces and $f:M\to N$ is an analytic map. The degree of $f$, denoted $\operatorname {deg} (f)$, is defined as follows, for any point $q\in N$:

$\operatorname {deg} (f):=\sum _{f(p)=q}v_{f}(p)$

where $v_{f}(p)$ denotes the local degree of the map $f$ in a small neighborhood of $p$. This local degree is also the same as the smallest positive $n$ such that the $n^{th}$ derivative of $f$ does not vanish, for some choice of local coordinates around $p$ and $q$.

Facts

• This definition coincides with the definition of degree in the sense of a map of differential manifolds. Indeed, if we pick $q$ to be a regular value of $f$, all the $v_{f}(p)$ are one (because the map is nondegenerate on all tangent spaces, and preserves orientation), and hence the degree is the same as the number of inverse images of the regular value.
• It also coincides with the purely topological definition of degree, because $v_{f}(p)$ equals the degree in terms of the effect of the map locally on homology classes.
• Unlike the case of topological or differential manifolds, we cannot have degree one maps between different Riemann surfaces. In fact, a degree one analytic map between Riemann surfaces must be an isomorphism. This also shows that not every homotopy class of continuous or smooth maps, has an analytic representative.