  The investigation of the Hamiltonian dynamical counterpart of phase
transitions, combined with the Riemannian geometrization of Hamiltonian
dynamics, has led to a preliminary formulation of a differential-topological
theory of phase transitions. In fact, in correspondence of a phase transition
there are peculiar geometrical changes of the mechanical manifolds that are
found to stem from changes of their topology. These findings, together with two
theorems, have suggested that a topological theory of phase transitions can be
formulated to go beyond the limits of the existing theories. Among other
advantages, the new theory applies to phase transitions in small $N$ systems
(that is, at nanoscopic and mesoscopic scales), and in the absence of
symmetry-breaking. However, the preliminary version of the theory was
incomplete and still falsifiable by counterexamples. The present work provides
a relevant leap forward leading to an accomplished development of the
topological theory of phase transitions paving the way to further developments
and applications of the theory that can be no longer hampered.
