Chern classes via differential forms - Tezôto, Ivan Tagliaferro de Oliveira (2022)
The objective of this dissertation is to present, through differential topology, some of the mathematical foundations to construct the Chern classes on complex vector bundles π: E → M, where M is a differentiable manifold. In this work we cover some preliminary topics of multilinear algebra, general topology, homological algebra and category theory in order to present the necessary background to develop the concepts here present. Next, we discuss the theory of differentiable manifolds needed, such as basic definitions, tangent space, differentiability, orientation and boundary. From the notion of manifolds, we introduce differential forms and their main properties, which allows us to work with integration on differentiable manifolds in a simplified way due to the algebraic properties that the graded space Ω*(M) possesses. Using the theory of differential forms we construct a cohomology theory, called de Rhams Cohomology, which is defined from the vector spaces of differential forms. The cohomology groups are essential in this work, because from them we have the basis to present several of the important results in the thesis such as the Poincaré duality, the Künneth formula and the Leray-Hirsch theorem. Also, they are important for the definition of Euler classes on real vector bundles of rank 2 and, consequently, the definition of the first Chern class on complex line bundles. We then give an overview of the general construction of Chern classes and give some of its properties. Finally, it is important to emphasize the importance of the topological concept of vector bundles in the work, both real and complex, in view of its relevance to define the desired classes.