An Example of Banach and Hilbert manifold: the universal Teichmüller space

Abstract: For $s >\frac{3}{2}$, the group of Sobolev class s diffeomorphisms of the circle is a smooth manifold modeled on the space of Sobolev class s sections of the tangent bundle of the circle. It is a topological group in the sense that multiplication given by the composition of applications is well-defined and continuous, the inverse is continuous, left translation is continuous and right translation is smooth. These results are consequences of the Sobolev Lemma. For the same reasons, the subgroup of Sobolev class s diffeomorphisms of the circle preserving three points, is, for $s >\frac{3}{2}$ a smooth manifold and a topological group modeled on the space of Sobolev class s vector fields vanishing at these three points. One may ask what happens for the critical value $\frac{3}{2}$ and look for a group with some regularity and a manifold structure such that the tangent space at the identity is isomorphic to the space Sobolev class $\frac{3}{2}$ vector fields vanishing at three given points. The universal Teichm\"uller space verify these conditions.