  We study how the spin structures on finite-volume hyperbolic n-manifolds
restrict to cusps. When a cusp cross-section is a (n-1)-torus, there are
essentially two possible behaviours: the spin structure is either bounding or
Lie. We show that in every dimension n there are examples where at least one
cusp is Lie, and in every dimension n <= 8 there are examples where all the
cusps are bounding.
  By work of C. Bar, this implies that the spectrum of the Dirac operator is R
in the first case, and discrete in the second. We therefore deduce that there
are cusped hyperbolic manifolds whose spectrum of the Dirac operator is R in
all dimensions, and whose spectrum is discrete in all dimensions n <= 8.
