Poincaré Duality for Smooth Surfaces

Abstract

The thesis at hand aims to study the de Rham cohomology of smooth surfaces and present a proof of a duality result, dating from 1895, due to H. Poincaré, namely, the Poincaré Duality Theorem. We also look into some applications of such duality involving the Euler-Poincaré characteristic and the signature of compact surfaces, and discuss its connections to the Hodge decomposition theorem. In order to do so, we develop some preliminary tools by providing an overview of basic concepts in the language of categories and functors, homological algebra and differential forms on surfaces in Euclidean spaces.