  The operation of crushing a normal surface has proven to be a powerful tool
in computational $3$-manifold topology, with applications both to triangulation
complexity and to algorithms. The main difficulty with crushing is that it can
drastically change the topology of a triangulation, so applications to date
have been limited to relatively simple surfaces: $2$-spheres, discs, annuli,
and closed boundary-parallel surfaces. We give the first detailed analysis of
the topological effects of crushing closed essential surfaces of positive
genus. To showcase the utility of this new analysis, we use it to prove some
results about how triangulation complexity interacts with JSJ decompositions
and satellite knots; although similar applications can also be obtained using
techniques of Matveev, our approach has the advantage that it avoids the
machinery of almost simple spines and handle decompositions.
