Uniqueness of surface diagrams of smooth 4-manifolds

Abstract

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.

Informal material added for readability in numerous places and many details and intermediate steps added to the later arguments in response to referee comments; references updated. 62 pages, many figures