We prove a finiteness result for the -patterned guts decomposition of all -manifolds obtained by splitting a given orientable, irreducible and -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 orientable 3-manifold dominates only finitely many exteriors of knots in .
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Michel Boileau, J. Hyam Rubinstein, Shicheng Wang, Finiteness of 3-manifolds associated with non-zero degree mappings. Comment. Math. Helv. 89 (2014), no. 1, pp. 33–68DOI 10.4171/CMH/312