Exact result on topology and phase transitions at any finite N.

We study analytically the topology of a family of submanifolds of the configuration space of the mean-field XY model, computing also a topological invariant (the Euler characteristic). We prove that a particular topological change of these submanifolds is connected to the phase transition of this system, and exists also at finite N. The present result is the first analytic proof that a phase transition has a topological origin and provides a key to a possible better understanding of the origin of phase transitions at their deepest level, as well as to a possible definition of phase transitions at finite N.