Almost toric manifolds form a class of singular Lagrangian fibered
symplectic manifolds that include both toric manifolds and the K3
surface. We classify closed almost toric four-manifolds up to diffeomorphism
and indicate precisely the structure of all almost toric fibrations
of closed symplectic four-manifolds. A key step in the proof is a geometric
classification of the singular integral affine structures that can
occur on the base of an almost toric fibration of a closed four-manifold.
As a byproduct we provide a geometric explanation for why a generic
Lagrangian fibration over the two-sphere must have 24 singular fibers.
We introduce filtered cohomologies of differential forms on symplectic
manifolds. They generalize and include the cohomologies
discussed in Papers I and II as a subset. The filtered cohomologies
are finite-dimensional and can be associated with differential
elliptic complexes. Algebraically, we show that the filtered
cohomologies give a two-sided resolution of Lefschetz maps, and
thereby, they are directly related to the kernels and cokernels of
the Lefschetz maps. We also introduce a novel, non-associative
product operation on differential forms for symplectic manifolds.
This product generates an A-infinity algebra structure on forms that
underlies the filtered cohomologies and gives them a ring structure.
As an application, we demonstrate how the ring structure of
the filtered cohomologies can distinguish different symplectic four-manifolds
in the context of a circle times a fibered three-manifold.