Second cohomology classes of the group of \(C^1\)-flat diffeomorphisms (Q442120)

Let \(\text{Diff}^{\infty}_1({\mathbb R})\) denote the group of all orientation-preserving \(C^{\infty}\)-diffeomorphisms of the real line \({\mathbb R}\) which fix the origin and are \(C^1\)-flat to the identity at the origin. In the 2-dimensional cohomology of \(\text{Diff}^{\infty}_1({\mathbb R})\), the author constructs by clever use of Massey products two non-trivial real cohomology classes and uncountably many integral cohomology classes.