Reconstructing 4-manifolds from Morse 2-functions
  Given a Morse 2-function $f: X^4 \to S^2$, we give minimal conditions on the fold curves and fibers so that $X^4$ and $f$ can be reconstructed from a certain combinatorial diagram attached to $S^2$. Additional remarks are made in other dimensions. 
David T. Gay, Robion Kirby
math.GT