On the cohomology of almost-complex and symplectic manifolds and proper surjective maps

Let (X,J) be an almost-complex manifold. In [Comparing tamed and compatible symplectic cones and cohomological properties of almost-complex manifolds, Comm. Anal. Geom. 17 (2009) 651-683], Li and Zhang introduce HJ(p,q),(q,p)(X) as the cohomology subgroups of the (p + q)th de Rham cohomology group formed by classes represented by real pure-Type forms. Given a proper, surjective, pseudo-holomorphic map between two almost-complex manifolds, we study the relationship among such cohomology groups. Similar results are proven in the symplectic setting for the cohomology groups introduced in [Cohomology and Hodge Theory on Symplectic manifolds: I, J. Differ. Geom. 91(3) (2012) 383-416] by Tseng and Yau and a new characterization of the hard Lefschetz condition in dimension 4 is provided.