The Oka principle for holomorphic Legendrian curves in $$\mathbb {C}^{2n+1}$$ C 2 n + 1
| Autor: | Franc Forstneric, Finnur Larusson |
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| Rok vydání: | 2017 |
| Předmět: |
Homotopy group
General Mathematics Riemann surface Homotopy 010102 general mathematics Structure (category theory) Holomorphic function Type (model theory) Space (mathematics) 01 natural sciences Combinatorics symbols.namesake 0103 physical sciences symbols 010307 mathematical physics 0101 mathematics Mathematics |
| Zdroj: | Mathematische Zeitschrift. 288:643-663 |
| ISSN: | 1432-1823 0025-5874 |
| DOI: | 10.1007/s00209-017-1904-1 |
| Popis: | Let M be a connected open Riemann surface. We prove that the space \(\mathscr {L}(M,\mathbb {C}^{2n+1})\) of all holomorphic Legendrian immersions of M to \(\mathbb {C}^{2n+1}\), \(n\ge 1\), endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space \(\mathscr {C}(M,\mathbb {S}^{4n-1})\) of continuous maps from M to the sphere \(\mathbb {S}^{4n-1}\). If M has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of \(\mathscr {L}(M,\mathbb {C}^{2n+1})\) in terms of the homotopy groups of \(\mathbb {S}^{4n-1}\). It follows that \(\mathscr {L}(M,\mathbb {C}^{2n+1})\) is \((4n-3)\)-connected. |
| Databáze: | OpenAIRE |
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