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This article is cited in 1 scientific paper (total in 1 paper)
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.
Received: 14.10.2019 Revised: 16.02.2020
Citation:
F. Forstnerič, “Immersions of open Riemann surfaces into the Riemann sphere”, Izv. Math., 85:3 (2021), 562–581
Linking options:
https://www.mathnet.ru/eng/im8980https://doi.org/10.1070/IM8980 https://www.mathnet.ru/eng/im/v85/i3/p239
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Abstract page: | 268 | Russian version PDF: | 51 | English version PDF: | 64 | Russian version HTML: | 108 | References: | 39 | First page: | 10 |
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