We study simple wrinkledfibrations, a variation of the simplified purely wrinkled fibrations of Williams
(Geom. Topol.14:2 (2010), 1015–1061), and their combinatorial description in terms
of surface diagrams. We show that simple wrinkled fibrations induce handle
decompositions of their total spaces which are very similar to those obtained from
Lefschetz fibrations. The handle decompositions turn out to be closely related to
surface diagrams and we use this relationship to interpret some well known
operations on 4-manifolds in terms of surface diagrams. This, in turn, allows us
classify all closed 4-manifolds which admit simple wrinkled fibrations of genus one,
the lowest possible fiber genus.
