Resumen

In this work we present an introduction to the theory of principal bundles with singularities, i.e. principal bundles which reduce to a closed subgroup of the structure group outside of a closed subset of the base space. In the first part we give the definition of these new structures, and we define morphisms between them. Then, we prove that these structures and their morphisms form a category which contains the bundles induced by transversal maps, and we use the obstruction theory to construct characteristic class of principal bundles with singularities. In the second part we present a version of Gauss-Bonnet-Hopf-Poincaré Formula for locally trivial fiber bundles over 2-dimensional manifolds and we prove that this result generalizes the classical Gauss-Bonnet theorem. Then, we define branched sections of locally trivial fiber bundles, index of a singular point of a branched section, and give examples of its calculation, in particular for branched sections defined by binary differential equations. We also define a resolution of singularities of a branched section, and prove an analog of Gauss-Bonnet-Hopf-Poincaré formula for the branched sections admitting a resolution in case the manifold M has dimension two