  We evaluate the parity-odd part of the effective action due to massive Dirac
fermions on a $S^2 \times S^1$ manifold, minimally coupled to an external
Abelian gauge field. We do that for a special class of gauge-field
configurations, which is specially suitable to the study of the behaviour of
the fermionic determinant under large gauge field configurations, which are
allowed by the space-time geometry.
