  Exotic heat equations that allow to prove the Poincar\'e conjecture and its
generalizations to any dimension are considered. The methodology used is the
PDE's algebraic topology, introduced by A. Pr\'astaro in the geometry of PDE's,
in order to characterize global solutions. In particular it is shown that this
theory allows us to identify $n$-dimensional {\em exotic spheres}, i.e.,
homotopy spheres that are homeomorphic, but not diffeomorphic to the standard
$S^n$.
