On projective connections and their potential application to four-dimensional topology

The understanding of shapes is developed at an early age with the stretching and compressing of malleable substances and the sorting of spheres and cubes. However, a major difference between the former and latter is that the former welcomes deformations that are continuous. Hence, the game of Play-Doh fosters a mindset that is interested in the existence of holes/singularities and connectedness, but completely erases any concerns over the notion of distance. The study of the properties of a shape that are preserved under continuous transformations, but where no tearing is allowed is called topology.

It is remarkable that waves originating in the study of the fundamental forces of nature have a say in this. Instead of considering waves that can propagate endlessly and that are infinite in extent, one can imagine waves that are both temporally and spatially constrained. We call these waves, instantons and they share the transient nature and smallness of humans, and the lack of a preferred perspective. The feature which enables instantons to shift between many mathematically equivalent perspectives is called gauge invariance.

This thesis is an account of past methods that were used to probe the topology of 4-dimensional shapes called 4-manifolds. We propose the study of a different kind of gauge in-variance called projective invariance, which is rooted in the study of geodesics, where one elevates the idea of parallelism from a flat world to a curved world, where parallelism is not as clear.