Spacetime topology (I) - Chirality and the third Stiefel-Whitney class

The standard model of particle physics poses certain limitations upon the topology of spacetime, most notably by imposing the triviality of an important family of characteristic classes, the Stiefel-Whitney classes. In this, the first of two articles, we present a physical interpretation of the first three Stiefel-Whitney classes. While the relationship of the first two to the existence of spinor fields has been known since the sixties, apparently no connection between the third class and microscopical physics seems to be known. We show that the third class is related to chirality.