Obstruction Theory for Spin Structures

The aim of this thesis is to study the mathematical structures involved with these three problems: the existence of spin structures on a manifold, the reduction of a Spin(3,1)-structure to a SU(2)-structure and the induction of the Dirac equation on submanifolds. To formally state these problems, we recall the main de?nitions and results on G-structures, singular and cellular cohomology, obstruction theory and characteristic classes. For spin structures on a manifold M, we carry out a direct construction which shows that M is a spin mainfold if a characteristic class of a certain vector bundle E_{r;s} (dependent on the signature) is trivial; this class is then identi?fied as the second Stiefel-Whitney class of the speci?fied vector bundle. The reduction of a Spin(3,1)-structure to a SU(2)-structure is shown to be equivalent to another problem, namely the existence of global sections in a particular bundle having Spin(3,1)=SU(2) as fi?ber. We also show, through a more physical argument, that the reduction exists if the third Stiefel-Whitney class of E_{r;s} is trivial.