  We define generalized currents associated with immersions of abstract
solenoids with a transversal measure. We realize geometrically the full real
homology of a compact manifold with these generalized currents, and more
precisely with immersions of minimal uniquely ergodic solenoids. This makes
precise and geometric De Rham's realization of the real homology by only using
a restricted geometric subclass of currents. These generalized currents do
extend Ruelle-Sullivan and Schwartzman currents. We extend Schwartzman theory
beyond dimension 1 and provide a unified treatment of Ruelle-Sullivan and
Schwartzman theories via Birkhoff's ergodic theorem for the class of immersions
of controlled solenoids. We develop some intersection theory of these new
generalized currents that explains why the realization theorem cannot be
achieved only with Ruelle-Sullivan currents.
