Immersions of open Riemann surfaces into the Riemann sphere

Abstract

In this paper we show that the space of holomorphic immersions from any given open Riemann surface into the Riemann sphere is weakly homotopy equivalent to the space of continuous maps from to the complement of the zero section in the tangent bundle of . It follows in particular that this space has path components, where is the number of generators of the first homology group . We also prove a parametric version of the Mergelyan approximation theorem for maps from Riemann surfaces to an arbitrary complex manifold, a result used in the proof of our main theorem.