Lefschetz fibrations over the disc
  We provide a complete set of moves relating any two Lefschetz fibrations over the disk having as their total space the same 4-dimensional 2-handlebody up to 2-equivalence. As a consequence, we also obtain moves relating diffeomorphic 3-dimensional open books, providing a different approach to an analogous previous result by Harer. 
Nikos Apostolakis, Riccardo Piergallini and Daniele Zuddas
math.GT