Publication: Cohomología de de Rham y grado de Brouwer-Kronecker
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2019-07
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En este trabajo se estudian algunos apartados de la cohomolog´ıa de de Rham. Veremos la definici´on de la derivada de Lie y del producto interior y se tratará la integral de cohomología, comprobando que es un isomorfismo. También se probará el lema de Poincaré, conectando la cohomología con la homotopía. Por último, se presentará el grado de Brouwer-Kronecker y se utilizarán todas estas herramientas para demostrar algunos teoremas de Brouwer y el teorema de la invarianza del dominio, extraer algunas conclusiones acerca de las esferas Sm y culminar con el teorema de Hopf.
In this work we study some aspects of the de Rham cohomology. We will introduce the Lie derivative and inner product and we will explore the integration of cohomology classes, showing that it is an isomorphism. We will also prove Poincaré’s lemma, linking the cohomology to results on homotopy. In the end, we will present the Brouwer-Kronecker degree and use the previous tools to prove some of Brouwer’s theorems and the invariance of domain theorem, to achieve some conclusions about the spheres Sm and to culminate with Hopf’s theorem.
In this work we study some aspects of the de Rham cohomology. We will introduce the Lie derivative and inner product and we will explore the integration of cohomology classes, showing that it is an isomorphism. We will also prove Poincaré’s lemma, linking the cohomology to results on homotopy. In the end, we will present the Brouwer-Kronecker degree and use the previous tools to prove some of Brouwer’s theorems and the invariance of domain theorem, to achieve some conclusions about the spheres Sm and to culminate with Hopf’s theorem.
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