Abstract: In 2018, Watanabe disproved the 4-dimensional Smale conjecture by showing that the diffeomorphism group of a 4-dimensional disk relative to its boundary is noncontractible. A central tool in Watanabe's argument is a version of Kontsevich's characteristic classes for smooth families of disk bundles. One may wonder what's the role played by the smooth structure. In a recent project, we show that Kontsevich's characteristic class can be redefined just using a formal smooth structure (i.e., a lift of the tangent microbundle to a vector bundle). As an application, we show that for arbitrary compact 4-manifold (with or without boundary), the space of smooth structure is noncontractible. And we show that the homeomorphism group of the 4-dimensional Euclidean space has infinitely many nontrivial rational homotopy groups. (This is a joint work with Yi Xie.)
