  Exotic dark spinor fields are introduced and investigated in the context of
inequivalent spin structures on arbitrary curved spacetimes, which induces an
additional term on the associated Dirac operator, related to a Cech cohomology
class. For the most kinds of spinor fields, any exotic term in the Dirac
operator can be absorbed and encoded as a shift of the electromagnetic vector
potential representing an element of the cohomology group H^1(M, Z_2). The
possibility of concealing such an exotic term does not exist in case of dark
(ELKO) spinor fields, as they cannot carry electromagnetic charge, so that the
full topological analysis must be evaluated. Since exotic dark spinor fields
also satisfy Klein-Gordon propagators, the dynamical constraints related to the
exotic term in the Dirac equation can be explicitly calculated. It forthwith
implies that the non-trivial topology associated to the spacetime can
drastically engender --- from the dynamics of dark spinor fields ---
constraints in the spacetime metric structure. Meanwhile, such constraints may
be alleviated, at the cost of constraining the exotic spacetime topology.
Besides being prime candidates to the dark matter problem, dark spinor fields
are shown to be potential candidates to probe non-trivial topologies in
spacetime, as well as probe the spacetime metric structure.
