Publication: El teorema de Jordan-Schoenflies en el toro
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2020
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Este trabajo estudia el teorema de Jordan-Schoenflies en el toro, que clasifica por homeomorfismo ambiente las curvas de Jordan del toro según lo desconecten o no.
Se demostraría que hay dos tipos: las nulhomótopas, que desconectan; y las demás, que no. En particular, los complementos de todas las del mismo tipo son homeomorfos: a una corona circular para las que no desconectan, a un disco y un toro pinchado para las que sí. Además, se estudian los casos en los que los homeomorfismos ambientes se pueden
refinar a isotopías.
We study the Jordan-Schoenflies theorem for the torus, which classifies Jordan curves on the torus modulo ambient homeomorphism depending on whether they disconnect the torus or they do not. We show that there are two types of curves: those that are nullhomotopic, which disconnect the torus; and the rest, which do not. Besides, the complements of two curves of the same type are homeomorphic: to an annulus for those which do not disconnect, and to a disk and a punctured torus for those which do. Furthermore, we study which ambient homeomorphisms can be refined to isotopies.
We study the Jordan-Schoenflies theorem for the torus, which classifies Jordan curves on the torus modulo ambient homeomorphism depending on whether they disconnect the torus or they do not. We show that there are two types of curves: those that are nullhomotopic, which disconnect the torus; and the rest, which do not. Besides, the complements of two curves of the same type are homeomorphic: to an annulus for those which do not disconnect, and to a disk and a punctured torus for those which do. Furthermore, we study which ambient homeomorphisms can be refined to isotopies.
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