Broken Lefschetz fibrations and smooth structures on 4-manifolds
  The broken genera are orientation preserving diffeomorphism invariants of closed oriented 4-manifolds, defined via broken Lefschetz fibrations. We study the properties of the broken genera invariants, and calculate them for various 4-manifolds, while showing that the invariants are sensitive to exotic smooth structures. 
R. Inanc Baykur
math.GT