Spun normal surfaces in 3-manifolds, II: The toroidal case

Spun normal surfaces are a useful way of representing proper essential surfaces using ideal triangulations for 3-manifolds with tori boundaries.Here we consider spinning surfaces in the case of a 3-manifold with a nontrivial JSJ decomposition, where each of the JSJ components is hyperbolic.We prove that a proper essential surface can be spun, so long as none of the JSJ components are bundles with fiber a subsurface of and the ideal triangulation satisfies similar properties to a taut structure. There are no embedded normal 2-spheres, projective planes, or Klein bottles. Every embedded normal torus is peripheral.