Date：September 25th, 16:00-18:00

Location：东7一层报告厅Lecture hall

Abstract: 

Watanabe disproved the 4-dimensional Smale conjecture by establishing many disk bundles which are topologically trivial but not smoothly so. Amazingly, Watanabe used Kontsevich's characteristic classes, which are very different from previous invariants that can detect exoticness in dimension 4 (e.g. the Seiberg-Witten invariants and the Donaldson invariants). So one may wonder what's the role played by the smooth structure in this story. In this talk, I will sketch our proof that Kontsevich's characteristic classes only depend on a formal smooth structure (i.e. a vector bundle structure on the topological tangent bundle). This makes the invariant more flexible and allows several new applications. For example, we show that the homeomorphism group of the 4-dimensional sphere or Euclidian space has nontrivial rational homotopy/homology group in infinitely many dimensions. And we show that for any compact orientable 4-manifold, the natural inclusion from the diffeomorphism group to the homeomorphism group is not a homotopy equivalence. Furthermore, we discovered a new MMM (Miller-Morita-Mumford) class, which can obstruct smoothings of 4-dimensional topological bundles. The talk is based on a joint work with Yi Xie.

Welcome!