  In this second paper, we prove a necessity Theorem about the topological
origin of phase transitions. We consider physical systems described by smooth
microscopic interaction potentials V_N(q), among N degrees of freedom, and the
associated family of configuration space submanifolds {M_v}_{v \in R}, with
M_v={q \in R^N | V_N(q) \leq v}. On the basis of an analytic relationship
between a suitably weighed sum of the Morse indexes of the manifolds {M_ v}_{v
\in R} and thermodynamic entropy, the Theorem states that any possible unbound
growth with N of one of the following derivatives of the configurational
entropy S^{(-)}(v)=(1/N) \log \int_{M_v} d^Nq, that is of |\partial^k
S^{(-)}(v)/\partial v^k|, for k=3,4, can be entailed only by the weighed sum of
Morse indexes.
  Since the unbound growth with N of one of these derivatives corresponds to
the occurrence of a first or of a second order phase transition, and since the
variation of the Morse indexes of a manifold is in one-to-one correspondence
with a change of its topology, the Main Theorem of the present paper states
that a phase transition necessarily stems from a topological transition in
configuration space. The proof of the Theorem given in the present paper cannot
be done without Main Theorem of paper I.
