Hebrew University Topology and Geometry Seminar



March 9, 2016
1100-1235
Ross building, Seminar Room 70A



Frol Zapolsky

University of Haifa

On the contact mapping class group of the prequantization space over the $A_m$ Milnor fiber




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 $2$) $A_m$ Milnor fiber times the circle, with a natural contact structure, we show that the full braid group $B_{m+1}$ on $m+1$ strands embeds into the contact mapping class group of $V$. We deduce that when $n=2$, the subgroup $P_{m+1}$ 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.