Évènement

Topology in dimension four exhibits a great deal of poorly understood exotic behavior, where « exotic » means behavior that seems trivial from a continuous (topological) point of view but highly nontrivial from a differentiable (smooth) point of view. The miracle is not so much that such behavior exists but that one can actually prove that it exists; this typically relies on Freedman’s work in the topological category, which led to the proof of the 4-dimensional topological Poincaré conjecture, and gauge theory in the smooth category, beginning with Donaldson’s work, obstructing smooth triviality by counting solutions to PDE’s. Most of the focus in this work has been on the objects, namely the 4-manifolds themselves, but as we all know well, morphisms are just as important as objects! This talk is a survey of some history and some recent work related to understanding the difference between smooth and topological for automorphisms of smooth 4-manifolds.