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We use Seidel representation for symplectic orbifolds to compute the quantum cohomology ring of a compact symplectic toric orbifold (X , ω).

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.

Let$C$and$C$′ be two smooth self-transverse immersions of$S$^{1}into ℝ^{2}. Both$C$and$C$′ subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes$C$to$C$′ induces a one-to-one correspondence between the disks of$C$and$C$′. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking$C$to$C$′ is that there exists an isotopy for which the area of every disk of$C$equals that of the corresponding disk of$C$′. In this paper we show that this is also a sufficient condition.

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