The resolution of the geometrization conjecture is an exciting time for 3-manifold topologists. We are now justified in using geometric intuition to attack the mysteries of the topology of 3-manifolds. Most modern constructions of 3-manifolds use a surface homeomorphism in some way. In all cases we find a natural measure of the complexity of this map, usually defined via the curve complex of the surface. Our goal is to understand basic topological invariants when a 3-manifold is constructed by a "sufficiently complicated" surface map. This gives rise to an understanding of the topology of "generic" 3-manifolds.