  We propose a topological approach suitable to establish a connection between
thermodynamics and topology in the microcanonical ensemble. Indeed, we report
on results that point to the possibility of describing {\it interacting
classical spin systems} in the thermodynamic limit, including the occurrence of
a phase transition, using topology arguments only. Our approach relies on Morse
theory, through the determination of the critical points of the potential
energy, which is the proper Morse function. Our main finding is to show that,
in the context of the studied classical models, the Euler characteristic
$\chi(E)$ embeds the necessary features for a correct description of several
magnetic thermodynamic quantities of the systems, such as the magnetization,
correlation function, susceptibility, and critical temperature. Despite the
classical nature of the studied models, such quantities are those that do not
violate the laws of thermodynamics [with the proviso that Van der Waals loop
states are mean field (MF) artifacts]. We also discuss the subtle connection
between our approach using the Euler entropy, defined by the logarithm of the
modulus of $\chi(E)$ per site, and that using the {\it Boltzmann}
microcanonical entropy. Moreover, the results suggest that the loss of
regularity in the Morse function is associated with the occurrence of unstable
and metastable thermodynamic solutions in the MF case. The reliability of our
approach is tested in two exactly soluble systems: the infinite-range and the
short-range $XY$ models in the presence of a magnetic field. In particular, we
confirm that the topological hypothesis holds for both the infinite-range ($T_c
\neq 0$) and the short-range ($T_c = 0$) $XY$ models. Further studies are very
desirable in order to clarify the extension of the validity of our proposal.
