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 Izv. RAN. Ser. Mat., 2021, Volume 85, Issue 3, Pages 239–260 (Mi izv8980)

Immersions of open Riemann surfaces into the Riemann sphere

F. Forstneričab

a Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
b Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

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 $\mathbb{CP}^1$. It follows in particular that this space 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 theorem for maps from Riemann surfaces to an arbitrary complex manifold, a result used in the proof of our main theorem.

Keywords: Riemann surface, holomorphic immersion, meromorphic function, $\mathrm{h}$-principle, weak homotopy equivalence.

 Funding Agency Grant Number Slovenian Research Agency J1-9104P1-0291 My research is supported by the programme P1-0291 and the grant J1-9104 from ARRS, Republic of Slovenia.

DOI: https://doi.org/10.4213/im8980

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English version:
Izvestiya: Mathematics, 2021, 85:3, 562–581

Bibliographic databases:

UDC: 517.545+517.551
MSC: 32H02, 58D10, 57R42
Revised: 16.02.2020

Citation: F. Forstnerič, “Immersions of open Riemann surfaces into the Riemann sphere”, Izv. RAN. Ser. Mat., 85:3 (2021), 239–260; Izv. Math., 85:3 (2021), 562–581

Citation in format AMSBIB
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