Dehn twists and free subgroups of symplectic mapping class groups

Given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact sequence to arbitrary powers of a fixed Dehn twist. We also show that the Milnor fibre of any isolated degenerate hypersurface singularity contains such pairs of spheres.Comment: 37 pages, 9 figures; v2: corrected proof of Prop. 4.7, and other minor changes following referee report; v3: minor changes only; accepted, Journal of Topolog

The Zariski-Lefschetz principle for higher homotopy groups of nongeneric pencils

We prove a general Zariski-van Kampen-Lefschetz type theorem for higher homotopy groups of generic and nongeneric pencils on singular open complex spaces.Comment: 17 p.; based on the Newton Institute preprint NI01029, June 200