【Seminar】 De Rham cohomology for non-Hausdorff manifolds

Speaker:Mr. David O'Connell (OIST sttudent)

Title: De Rham cohomology for non-Hausdorff manifolds

Abstract: It is standard practice to impose the Hausdorff property within the definition of a manifold, and this is for good reason: Hausdorffness gives us access to partitions of unity subordinate to any open cover, and these in turn can be used to construct various objects of geometric interest. In this talk we will boldly take the opposite approach, and see what happens when we relax the Hausdorff property in the definition of a manifold. A priori, it may seem that such spaces are too difficult to study without arbitrarily-existent partitions of unity.  However, through a series of tricks it is possible to avoid this problem and construct various things like smooth structures, bundles, differential forms and integrals. We will then use these tricks to describe de Rham cohomology (both compactly-supported and otherwise) and prove non-Hausdorff versions of de Rham’s theorem and Poincaré duality. Finally, we finish with some ongoing ideas regarding the Čech-de Rham equivalence for non-Hausdorff manifolds.