URegina Topology & Geometry Seminar: Francis Bischoff

Can you embed the real projective plane in three-dimensional space? Motivated by this and similar questions, I will introduce the correspondence between real line bundles and hypersurfaces, the first Stiefel-Whitney class, and Poincare duality. In particular, I will explain why every hypersurface of a smooth manifold is the regular level set of a 'twisted function' in a canonical way and will show you why this implies that it is a regular level set of an ordinary function if and only if it intersects every loop in an even number of points.