Immersions of open Riemann surfaces into the Riemann sphere

In this paper we show that the space of holomorphic immersions from any given open Riemann surface $M$into the Riemann sphere $\mathbb{CP}^1$ is weakly homotopy equivalent to the space of continuous maps from$M$ to the complement of the zero section in the tangent bundle of $\mathbb{CP}^1$. It follows in particular that thisspace has $2^k$ path components, where $k$ is the number of generators of the first homology group$H_1(M,\mathbb{Z})=\mathbb{Z}^k$. We also prove a parametric version of the Mergelyan approximation theoremfor maps from Riemann surfaces to an arbitrary complex manifold, a result used in the proof of our main theorem.

Riemann surface, holomorphic immersion, meromorphic function, -principle, weak homotopy equivalence