Exotic Smoothness, Branched Covering Spaces, Andquantum Gravity

This thesis addresses the role of topology and geometry in quantum gravity. A major topic will be how inequivalent differentiable structures (exotic smoothness) can play a physically significant role in both semiclassical gravity and loop quantum gravity. We will discuss the result of including these structures into a physical theory, and describe some approaches to fully account for them. We will also be able to use our construction to study the topology of loop quantum gravity. In our framework, topology change will be a natural part of the theory. The approaches discussed in this thesis will be inspired by novel mathematical results, applied to established physical models. It is hoped that the methods described herein will lead to a greater understanding of the deep connection between geometry and physics, particularly as it relates to the geometrical nature of the gravitational field.