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Immersions of open Riemann surfaces into the Riemann sphere

Franc Forstneric

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.

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 . We show in particular that this space has path components, where . 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.

Paper Structure

This paper contains 5 sections, 9 theorems, 47 equations.

Table of Contents

  1. The main result
  2. Topological preliminaries
  3. A parametric approximation theorem for immersions from discs to $\mathbb{CP}^1$
  4. Parametric Mergelyan theorem for manifold valued maps
  5. Proof of Theorem \ref{['th:WHE']}

Key Result

Theorem 1.1

For every open Riemann surface, $M$, the map $\Phi$eq:Phi from the space of holomorphic immersions $M\to\mathbb{CP}^1$ to the space of formal immersions satisfies the parametric h-principle, and hence is a weak homotopy equivalence.

Theorems & Definitions (19)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Example 1.4
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 3.1
  • proof
  • ...and 9 more