Is there a topological characterisation of what a (closed irreducible) hyperbolic 3-manifold is? I don't know any Riemannian geometry and still want to understand what an exceptional Dehn surgery is. For definition of a hyperbolic knot we can avoid the real understanding of what a hyperbolic manifold is by simply saying that it is not an unknot, not a torus knot and not a satellite. Is there a similar characterisation for hyperbolic 3-manifolds? Cameron Gordon's Park City lecture notes seem to suggest that such a criterion would be that the manifold is simple. But I couldn't find a clear statement of this. If there is such a criterion, a reference for a survey discussing these things would be very desirable(and in general, is there a survey - suitable for early stage PhD students - about the 3-manifold topology after Perelman?). Thanks!
A clear statement is the following:
You may remember that by saying that $M$ is hyperbolic unless there is some clear obstruction, and the obstrucion is $\pi_1$ finite or the existence of some surface with $\chi \geqslant 0$. This statement is pretty general and works for any compact 3-manifold with any kind of (possibly empty) boundary, using the appropriate notions of "hyperbolic" and "essential".
More concretely, an orientable manifold $M$ is hyperbolic unless one of the following obstructions occurs:
This characterization implies the one for knots that you mentioned, in virtue of the following: