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The elsewhere surmised topological origin of phase transitions is given herenew important evidence through the analytic study of an exactly solvable modelfor which both topology and thermodynamics are worked out. The model is amean-field one with a k-body interaction. It undergoes a second order phasetransition for k=2 and a first order one for k>2. This opens a completely newperspective for the understanding of the deep origin of first and second orderphase transitions, respectively. In particular, a remarkable theoretical resultconsists of a new mathematical characterization of first order transitions.Moreover, we show that a "reduced" configuration space can be defined in termsof collective variables, such that the correspondence between phase transitionsand topology changes becomes one-to-one, for this model. Finally, an unusualrelationship is worked out between the microscopic description of a classicalN-body system and its macroscopic thermodynamic behaviour. This consists of afunctional dependence of thermodynamic entropy upon the Morse indexes of thecritical points (saddles) of the constant energy hypersurfaces of themicroscopic 2N-dimensional phase space. Thus phase space (and configurationspace) topology is directly related to thermodynamics

Topology and phase transitions: From an exactly solvable model to a relation between topology and thermodynamics