Abstract: The contact mapping class group of a contact manifold V is the set of contact isotopy classes of its contactomorphisms. When V is the 2n-dimensional (n at least 2) Am Milnor fiber times the circle, with a natural contact structure, we show that the full braid group Bm+1 on m+1strands embeds into the contact mapping class group of V. We deduce that when n=2, the subgroup Pm+1 of pure braids is mapped to the part of the contact mapping class group consisting of smoothly trivial classes. This solves the contact isotopy problem for V. The construction is based on a natural lifting homomorphism from the symplectic mapping class group of the Milnor fiber to the contact mapping class group of V,
and on a remarkable embedding of the braid group into the former due to Khovanov and Seidel. To prove that the composed homomorphism remains injective, we use a variant of the Chekanov-Eliashberg dga for Legendrian links in V. This is joint work with Sergei Lanzat.