Cohomology of moduli spaces

Abstract: For many mathematical structures the collection of all objects having that structure may naturally be assembled into a space, either because the objects can be deformed or because they have symmetries: this is a moduli space. The most naive “classification” of such objects would be a description of the path-components of this moduli space, or in other words its zeroth cohomology, but to understand families of such objects one is led to investigate its higher cohomology.

It is hopeless to say anything in this generality, but topologists have developed some broad principles for studying certain kinds of moduli spaces. I will survey some of these techniques, focussing on moduli spaces of Riemann surfaces and their natural generalisation—from the point of view of differential topology—to higher dimensional smooth manifolds