The Topology, Geometry and Physics of Non-Hausdorff Manifolds

Abstract

Okinawa Institute of Science and Technology Graduate UniversityDoctor of PhilosophyIn this thesis we investigate various mathematical and physical problems surrounding the theory of non-Hausdorff manifolds. We start by introducing a topological theory of nonHausdorff manifolds from first principles. We then pass our discussion into a smooth category by defining various structures of geometric interest on non-Hausdorff manifolds, all whilst circumventing the technical issues surrounding partitions of unity. We complete our mathematical contribution by describing de Rham cohomology for non-Hausdorff manifolds, ultimately proving a generalized version of de Rham’s Theorem. For the physical contribution, we focus our attention on certain two-dimensional non-Hausdorff manifolds that may be interpreted as a type of topology-changing spacetime. We define a gravitational action for 2d non-Hausdorff spacetimes, and determine the angular conventions required to suppress their contribution within a Lorentzian-signature path integral for gravity that sums over topologies.doctoral thesi