  The topological hypothesis claims that phase transitions in a classical
statistical mechanical system are related to changes in the topology of the
level sets of the Hamiltonian. So far, the study of this hypothesis has been
restricted to continuous systems. The purpose of this article is to explore
discrete models from this point of view. More precisely, we show that some form
of the topological hypothesis holds for a wide class of discrete models, and
that its strongest version is valid for the Ising model on $\mathbb{Z}^d$ with
the possible exception of dimensions $d=3,4$.
