  It is well known that the Fock quantization of field theories in general
spacetimes suffers from an infinite ambiguity, owing to the inequivalent
possibilities in the selection of a representation of the canonical commutation
or anticommutation relations, but also owing to the freedom in the choice of
variables to describe the field among all those related by linear
time-dependent transformations, including the dependence through functions of
the background. In this work we remove this ambiguity (up to unitary
equivalence) in the case of a massive Dirac free field propagating in a
spacetime with homogeneous and isotropic spatial sections of spherical
topology. Two physically reasonable conditions are imposed in order to arrive
to this result: (a) The invariance of the vacuum under the spatial isometries
of the background, and (b) the unitary implementability of the dynamical
evolution that dictates the Dirac equation. We characterize the Fock
quantizations with a non trivial fermion dynamics that satisfy these two
conditions. Then, we provide a complete proof of the unitary equivalence of the
representations in this class under very mild requirements on the time
variation of the background, once a criterion to discern between particles and
antiparticles has been set.
