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We show that the Nieh-Yan topological invariant breaks projective symmetryand loses its topological character in presence of non vanishing nonmetricity.The notion of the Nieh-Yan topological invariant is then extended to thegeneric metric-affine case, defining a generalized Nieh-Yan term, which allowsto recover topologicity and projective invariance, independently. As a concreteexample a class of modified theories of gravity is considered and its dynamicalproperties are investigated in a cosmological setting. In particular, bouncingcosmological solutions in Bianchi I models are derived. Finite timesingularities affecting these solutions are analysed, showing that the geodesiccompleteness and the regular behavior of scalar perturbations in thesespace-times are not spoiled.

Big-Bounce in projectively invariant Nieh-Yan models: the Bianchi I case