Immersions of open Riemann surfaces into the Riemann sphere. (English. Russian original) Zbl 1470.32037
Izv. Math. 85, No. 3, 562-581 (2021); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 85, No. 3, 239-260 (2021).
Summary: 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.
MSC:
| 32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |
| 58D10 | Spaces of embeddings and immersions |
| 57R42 | Immersions in differential topology |
Keywords:
Riemann surface; holomorphic immersion; meromorphic function; h-principle; weak homotopy equivalenceReferences:
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