  The relation between thermodynamic phase transitions in classical systems and
topological changes in their configuration space is discussed for two physical
models and contains the first exact analytic computation of a topologic
invariant (the Euler characteristic) of certain submanifolds in the
configuration space of two physical models. The models are the mean-field XY
model and the one-dimensional XY model with nearest-neighbor interactions. The
former model undergoes a second-order phase transition at a finite critical
temperature while the latter has no phase transitions. The computation of this
topologic invariant is performed within the framework of Morse theory. In both
models topology changes in configuration space are present as the potential
energy is varied; however, in the mean-field model there is a particularly
"strong" topology change, corresponding to a big jump in the Euler
characteristic, connected with the phase transition, which is absent in the
one-dimensional model with no phase transition. The comparison between the two
models has two major consequences: i) it lends new and strong support to a
recently proposed topological approach to the study of phase transitions; ii)
it allows us to conjecture which particular topology changes could entail a
phase transition in general. We also discuss a simplified illustrative model of
the topology changes connected to phase transitions using of two-dimensional
surfaces, and a possible direct connection between topological invariants and
thermodynamic quantities.
