A metric approach to the topology of Bridgeland stability spaces

This thesis concerns the topology of spaces of Bridgeland stability conditions on triangulated categories. These are interesting, if difficult to compute, geometric invariants of the underlying category. Under suitable mild conditions, they are conjectured to be contractible. We study their topology using tools from the theory of metric spaces.

In particular, we study the critical points of the slicing distance from a fixed stability condition.

Using a generalisation of Morse Theory to continuous functions on a metric space, we obtain critical point criteria under which the stability space is contractible.