Teoria de Donaldson i aplicacions a les varietats topològiques de dimensió 4

Abstract

[en] The aim of this dissertation is to introduce the world of topological four-manifolds. The main result is Donaldson’s Theorem, which can be used to prove the existence of topological manifolds that admit no differentiable structure. First of all we will give preliminaries based on algebraic topology (cohomology mainly), following with tools of differential geometry (vector bundles, connections and the basics of gauge theories). In the most important part of this project we will check the proof of Donaldson’s Theorem and the definition of Donaldson’s invariants. We will use these results in order to see examples of manifolds that cannot have any differentiable structure as well as the construction of an exotic R 4 and various vanishing theorems.