Cohomologia de varietats i càlcul d’índexs

Abstract

[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.