  We discuss the behaviour of the Dirac fermions in a general spherically
symmetric black hole background with a non-trivial topology of the event
horizon. Both massive and massless cases are taken into account. The analytical
studies of intermediate and late-time behaviour of massive Dirac hair in the
background of a black hole with a global monopole and dilaton black hole
pierced by a cosmic string will be conducted. It was revealed that in the case
of a global monopole swallowed by a static black hole the intermediate
late-time behaviour depends on the mass of the Dirac field, the multiple number
of the wave mode and the global monopole parameter. The late-time behaviour is
quite independent of these factors and has the decay rate proportional to
$t^{-5/6}$. As far as the black hole pierced by a cosmic string is concerned
the intermediate late-time behaviour depends only on the hair mass and the
multipole number of the wave mode while the late-time behaviour dependence is
the same as in the previous case. The main modification stems from the topology
of the $S^2$ sphere pierced by a cosmic string. This factor modifies the
eigenvalues of the Dirac operator acting on the transverse manifold.
