On the contact mapping class group of the contactization of the A_m -Milnor fiber

We construct an embedding of the full braid group on m+ 1 strands B_{m + 1}, m≥ 1 , into the contact mapping class group of the contactization Q× S¹ of the A_m-Milnor fiber Q. The construction uses the embedding of B_{m + 1} into the symplectic mapping class group of Q due to Khovanov and Seidel, and a natural lifting homomorphism. In order to show that the composed homomorphism is still injective, we use a partially linearized variant of the Chekanov–Eliashberg dga for Legendrians which lie above one another in Q× R, reducing the proof to Floer homology. As corollaries we obtain a contribution to the contact isotopy problem for Q× S¹, as well as the fact that in dimension 4, the lifting homomorphism embeds the symplectic mapping class group of Q into the contact mapping class group of Q× S¹.