On the contact mapping class group of the contactization of the $A_m$- 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 $Q\times S^1$.

15 pages, the results were generalized from the case of 4-dimensional A_m-Milnor fiber to the general case of 2n-dimensional A_m-Milnor fiber, the title was changed accordingly, new references were added

References in corpus (1)

• On the contact mapping class group of Legendrian circle bundles