Publication: Cohomología de de Rham
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2021-07-10
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Abstract
La memoria busca introducir los conceptos y resultados básicos de cohomología para después tratar la cohomología de de Rham. Primero se desarrolla la teoría sin soporte compacto y a continuación con soporte compacto. Más adelante estudiamos lo necesario de teoría de grado y de funciones de Morse para poder enunciar y demostrar el teorema de Poincaré-Hopf para funciones de Morse. También incluiremos una aplicación topológica de los resultados sobre grado para aplicaciones diferenciables. Por último usaremos un resultado de la teoría de Morse para demostrar el teorema de Gauss-Bonnet y probamos el teorema de Reeb que caracteriza topológicamente las esferas mediante funciones de Morse.
This work seeks to introduce cohomology’s basic concepts and results, so they can be applied to de Rham cohomology. First non-compact support theory is developed, then with compact support. Afterwards we study the required theory about degree and Morse functions to be able to state and prove the Poincar´e-Hopf theorem for Morse functions. We will also include a nice topological application of degree theory. Then we show a proof of Gauss-Bonnet theorem using a result from Morse theory. Finally we prove Reeb’s theorem which gives a topological characterization of spheres through Morse functions.
This work seeks to introduce cohomology’s basic concepts and results, so they can be applied to de Rham cohomology. First non-compact support theory is developed, then with compact support. Afterwards we study the required theory about degree and Morse functions to be able to state and prove the Poincar´e-Hopf theorem for Morse functions. We will also include a nice topological application of degree theory. Then we show a proof of Gauss-Bonnet theorem using a result from Morse theory. Finally we prove Reeb’s theorem which gives a topological characterization of spheres through Morse functions.
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[2] J.M. Gamboa, J.M. Ruiz: Iniciaci´on al estudio de las Variedades Diferenciables.
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[5] F. Coltraro: Topolog´ıa Algebraica y Diferencial. Teorema de Poincar´e-Hopf.
TFG Departamento de AGT UCM, 2020. http://blogs.mat.ucm.es/jesusr/wp-content/uploads/sites/52/2020/03/franco.