  We introduce certain homology and cohomology subgroups for any almost complex
structure and study their pureness, fullness and duality properties. Motivated
by a question of Donaldson, we use these groups to relate J-tamed symplectic
cones and J-compatible symplectic cones over a large class of almost complex
manifolds, including all Kahler manifolds, almost Kahler 4-manifolds and
complex surfaces.
