  We prove a finiteness result for the $\partial$-patterned guts decomposition
of all 3-manifolds obtained by splitting a given orientable, irreducible and
$\partial$-irreducible 3-manifold along a closed incompressible surface. Then
using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds
dominated by a given compact 3-manifold belong, up to homeomorphism, to a
finite collection of compact 3-manifolds. We show also that any closed
orientable 3-manifold dominates only finitely many integral homology spheres
and any compact 3-manifolds orientable 3-manifold dominates only finitely many
exterior of knots in $S^3$.
