ModuLow · Moduli spaces in low dimensions

The topological properties of low dimensional moduli spaces play a fundamental role across algebra, geometry, and topology. My research programme will build homotopical tools for moduli spaces of 3- and 4-manifolds, and moduli spaces of links in 3-manifolds, with my main applications being to manifold symmetries and configurations. The upgrading of discrete techniques for ‘an object’ to space-level techniques on ‘the moduli space of all such objects’ is a central theme.Moduli spaces of manifolds, or, equivalently, classifying spaces of manifold diffeomorphism groups, are foundational objects which classify smooth manifold bundles. The diffeomorphism group of a manifold—the group of symmetries—is a topological group which ought to be studied through a homotopical lens. My recent joint work solved a conjecture of Kontsevich on the homotopy type of moduli spaces of 3-manifolds. This breakthrough lays the groundwork for some of my goals: I will use it to compute rational cohomology rings of salient 3-manifold moduli spaces, yielding characteristic classes of 3-manifold bundles. Furthermore, I will classify the existence of sections for natural maps between 3-manifold and 4-manifold moduli spaces.The topological properties of moduli spaces of unparametrised links in 3-manifolds will play a central role: these can be viewed as configuration spaces of 1-manifolds in 3-manifolds. These spaces are yet to be thoroughly understood, and it is a fundamental problem to describe their homotopy type. I will prove a finiteness theorem, and develop a framework to compute motion groups of configurations of links in 3-manifolds. An underlying theme is homological stability, and I will show families of these spaces satisfy (higher) homological stability.One of my main aims is to combine my work on all three moduli spaces, by introducing an innovative method to represent families of 4-manifold diffeomorphisms via motions of Kirby diagrams (decorated links) in a 3-manifold.