Algebraic Topology and Obstruction Theory

Abstract

This thesis aims to serve as an excursion into the theories of homology, cohomology and characteristic classes, conventional tools used to study topological spaces and vector bundles with origins in Algebraic Topology. Important definitions and results like the Excision theorem, Mayer-Vietoris sequences, the Universal Coefficient Theorem, Künneth formulae and Poincaré Duality are presented. The Stiefel-Whitney class and the Euler class are introduced and their properties discussed. Meanwhile, the obstructions they pose to the existence of sections and orientability are also mentioned. The thesis concludes with an application of the tools acquired to the computation of the equivariant cohomology and Stiefel-Whitney classes of a chosen space under a particular group action.