林剑锋 教授，清华大学

The moduli space of a smooth manifold X is defined to be the classifying space of its diffeomorphism group. Understanding the cohomology group of this space is important because elements in this group one-to-one correspond to characteristic classes for smooth bundles with fiber X. A celebrated result of Harer states that homology groups of the moduli spaces of Riemann surfaces stabilize if one fixes a degree and increases the genus. Galatius and Randal-Williams established analogous homological stability for moduli spaces of manifolds of even dimension at least 6. In this talk, we will show that homological stability fails for the moduli space of any simply-connected closed smooth 4-manifold in any degree of homology. The central tool is a characteristic class constructed using Seiberg-Witten equations, which detects the subtle difference between the topological category and the smooth category of 4-manifolds. This is a joint work with Hokuto Konno.