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

Abstract

We construct an embedding of the full braid group on $$m+1$$ strands $$B_{m+1}$$ , $$m \ge 1$$ , into the contact mapping class group of the contactization $$Q \times S^1$$ 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 $$Q \times {\mathbb {R}}$$ , reducing the proof to Floer homology. As corollaries we obtain a contribution to the contact isotopy problem for $$Q \times S^1$$ , 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 $$Q \times S^1$$ .