Topology and phase transitions: Paradigmatic evidence

We report upon the numerical computation of the Euler characteristic chi (a topologic invariant) of the equipotential hypersurfaces Sigma(v), of the c onfiguration space of the two-dimensional lattice phi(4) model. The pattern chi(Sigma(v)) versus v (potential energy) reveals that a major topology ch ange in the family {Sigma(v)}(v is an element of R) is at the origin of the phase transition in the model considered. The direct evidence given here-o f the relevance of topology for phase transitions-is obtained through a gen eral method that can be applied to any other model.