Uniqueness of surface diagrams of smooth 4-manifolds
  In the author's earlier work there appeared a new way to specify any smooth closed 4-manifold by a surface diagram, which consists of an orientable surface decorated with simple closed curves. These curves are cyclically indexed, and each curve has a unique transverse intersection with the next. Each surface diagram comes from a certain type of map from the 4-manifold to the two-sphere. The aim of this paper is to give a uniqueness theorem stating that surface diagrams coming from maps within a fixed homotopy class are unique up to four moves: stabilization, handleslide, multislide, and shift. 
Jonathan D. Williams
math.GT