Immersions of open Riemann surfaces into the Riemann sphere

TL;DR

This paper characterizes the space of holomorphic immersions from open Riemann surfaces into the Riemann sphere, revealing its homotopy type and component structure, and extends approximation theorems to complex manifolds.

Contribution

It establishes the homotopy equivalence of immersion spaces to continuous maps and proves a parametric Mergelyan approximation theorem for Riemann surface maps.

Findings

The space of immersions has 2^k path components, where k is the rank of the first homology group.

Homotopy equivalence between immersion spaces and continuous map spaces.

Extension of Mergelyan approximation theorem to complex manifolds.

Abstract

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 $\mathbb{CP}^1$. We show in particular that this space has $2^k$ path components, where $H_1(M,\mathbb Z)={\mathbb Z}^k$. We also prove a parametric version of Mergelyan approximation theorem for maps from Riemann surfaces into any complex manifold, a result used in the proof of our main theorem.