Abstract: Recent progress in quantum gravity is underpinned by a sum over spacetimes of different topologies, providing geometric explanations for a variety of quantum properties of black holes.  To better understand the Hilbert space interpretation, I will discuss these results from the perspective of the canonical formalism and Lorentzian spacetimes. A simple model demonstrates that topology change can result in a gravitational Hilbert space with many fewer independent states than semiclassical expectations suggest. I propose an interpretation of topology changing processes as a non-perturbative extension of gravitational gauge symmetry to include discontinuous diffeomorphisms.