The Virtual Fibring theorem provides the topological classification of closed 3-manifolds.

Very roughly, after passing to finite covers, we can build 3-manifolds by gluing together simpler pieces that we understand, in close analogy wiht the topological classification of surfaces. So the main extra subtlety as we pass to the 3-dimensional case is having to pass to finite covers.

Likewise, of course, the elevator pitch for the geometrisation theorem is that it's the 3-dimensional analogue of the uniformisation theorem for surfaces.

There are a few caveats:

• The statement is subtle, because not all 3-manifolds virtually fibre and so it's crucial that we also have an exact gluing description of the ones that don't.

• Arguably the Virtual Haken theorem provides another version of the same thing: after passing to finite covers, every irreducible closed 3-manifold has a Haken hierarchy. I think the point is that, from the "long range" point of view needed for an elevator pitch, the Virtual Haken and Virtually Fibred theorems are difficult to distinguish.

• A sceptic might argue that the mere existene of a Heegaard decomposition is a sort of topological classification. To this I'd respond that gluing along inessential things gives a much less useful classification. Heegaard decompositions are analgous to the statement that every surface is triangulable, which is a bit weaker than the topological classification of surfaces.

The Virtual Fibring theorem provides the topological classification of closed 3-manifolds.

Very roughly, after passing to finite covers, we can build 3-manifolds by gluing together simpler pieces that we understand, in close analogy with the topological classification of surfaces. So the main extra subtlety as we pass to the 3-dimensional case is having to pass to finite covers.

Likewise, of course, the elevator pitch for the geometrisation theorem is that it's the 3-dimensional analogue of the uniformisation theorem for surfaces.

There are a few caveats:

• The statement is subtle, because not all 3-manifolds virtually fibre and so it's crucial that we also have an exact gluing description of the ones that don't.

• Arguably the Virtual Haken theorem provides another version of the same thing: after passing to finite covers, every irreducible closed 3-manifold has a Haken hierarchy. I think the point is that, from the "long range" point of view needed for an elevator pitch, the Virtual Haken and Virtually Fibred theorems are difficult to distinguish.

• A sceptic might argue that the mere existence of a Heegaard decomposition is a sort of topological classification. To this I'd respond that gluing along inessential things gives a much less useful classification. Heegaard decompositions are analogous to the statement that every surface is triangulable, which is a bit weaker than the topological classification of surfaces.