Casacuberta, CarlesPicazo Archilla, Àngel2019-09-192019-09-192019-01-18https://hdl.handle.net/2445/140505Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Carles Casacuberta[en] The index calculation that we study is a special case of the Atiyah–Singer theorem, which relates a topological index with an analytical index. By using cohomology in manifolds we state the de Rham theorem, define orientations, and relate the signature of a manifold with characteristic classes by means of a theorem due to Hirzebruch. In the main part of the work, we prove that the signature of a Riemannian manifold is equal to the index of a differential operator closely related with the Hodge Laplacian.36 p.application/pdfcatcc-by-nc-nd (c) Àngel Picazo Archilla, 2019http://creativecommons.org/licenses/by-nc-nd/3.0/es/HomologiaTreballs de fi de grauVarietats de RiemannLaplaciàOperadors diferencialsHomologyBachelor's thesesRiemannian manifoldsLaplacian operatorDifferential operatorsTopologia diferencialDifferential topologyinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/openAccess