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!



Topological characterisation for a (closed irreducible) hyperbolic 3-manifold
A clear statement is the following:



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> A compact 3-manifold $M$ is hyperbolic if and only if it has infinite fundamental group and does not contain any essential surface with $\chi \geqslant 0$.
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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".


* *Hyperbolic*  means that, after removing tori and Klein bottles from $\partial M$, the manifold admits a complete finite-volume metric with constant curvature $-1$ and with geodesic boundary.
* *Essential*  means that the surface is immersed in $M$ with a map that is $\pi\_1$-injective and not homotopic to some map whose image is in some component of $\partial M$. In addition, a 2-sphere that does not bound a 3-disc is essential.


More concretely, an orientable manifold $M$ is hyperbolic unless one of the following obstructions occurs:


* $\pi\_1(M)$ is finite,
* $M$ contains a 2-sphere which does not bound a 3-disc (hence $M$ is *reducible*)
* $M$ contains an essential 2-disc (hence $M$ is *$\partial$-reducible*)
* $M$ contains an essential 2-torus whose $\pi\_1$ injects but which is not $\partial$-parallel. Note that the 2-torus may be immersed and not embedded: this occurs precisely in the *small Seifert spaces*
* $M$ contains an essential annulus (for instance in $\Sigma \times [0,1]$ for any surface $\Sigma$)


This characterization implies the one for knots that you mentioned, in virtue of the following:


* every knot complement has infinite $\pi\_1(M)$ and is irreducible,
* a knot complement contains an essential 2-disc if and only if it is trivial,
* a knot complement contains an essential annulus or torus if and only if it is not prime, a torus knot, or a satellite.


