Homologia singular e orientação em variedades - Cremasco, Victorio Amadeo (2025)
This work explores the concepts of orientation and orientability in topological manifolds. To begin, we start with a preliminary study of the elementary theory of modulii, in order to develop the algebraic language necessary for what follows. Next, we proceed with the study of singular homology theory, whose most useful outcomes will turn out to be the exact sequence of homology, the excision theorem, and the Mayer-Vietoris sequence. Following this, we discuss topological manifolds and covering spaces as preparation for the definition and construction of the concept of orientation itself, as well as another major associated structure known as the R-orientation sheaf of a manifold. This latter structure will provide us with a wealth of orientability criteria, which shall be applied to obtain numerous examples after the final section of the work, which discusses cellular homology and will allow us to compute the singular complex of various spaces presented throughout this monograph.