Georgia Institute of Technology College of Sciences

I'll report on an ongoing project, partly joint work with J. Hillman, aimed at finding criteria for the existence of sections on a given Lefschetz fibration over a surface. We will start by presenting a nice algebraic criterion for the existence of sections in a surface bundle and explain what goes wrong if we try to apply it to the more general Lefschetz fibration case. The question of when a nullhomotopic loop in the boundary of a Lefschetz fibration over the disk can be extended to a section over the whole disk is one such subtle issue. Our computations suggest that working with continuous or smooth sections leads to different answers.