  In the present work we provide a cosmological description of the Palatini
action with the addition of three well-known topological invariants, each of
which is attached to a scalar field. As a consequence, torsion plays a central
role, yielding an effective action with three dynamical scalar fields coupled
to gravity subjected to the influence of a potential. A special heed will be
addressed to inflation scenarios and quintom theory, although other noteworthy
cases will be analyzed. Two out of three scalars can be thought as the
Barbero-Immirzi scalar field coupled to the Holst and Nieh-Yan terms with a
potential whereby they shape the evolution of the universe. The other field
turns out to be a phantom that we couple to the inflaton that at late times
holds the stage inducing a fast expansion
