  We generalize Abel's classical theorem on linear equivalence of divisors on a
Riemann surface. For every closed submanifold $M^d \subset X^n$ in a compact
oriented Riemannian $n$--manifold, or more generally for any $d$--cycle $Z$
relative to a triangulation of $X$, we define a (simplicial) $(n-d-1)$--gerbe
$\Lambda_{Z}$, the Abel gerbe determined by $Z$, whose vanishing as a Deligne
cohomology class generalizes the notion of `linear equivalence to zero'. In
this setting, Abel's theorem remains valid. Moreover we generalize the
classical Inversion Theorem for the Abel--Jacobi map, thereby proving that the
moduli space of Abel gerbes is isomorphic to the harmonic Deligne cohomology;
that is, gerbes with harmonic curvature.
