"it is an open question if the known compact manifolds in 3-D are complete."
This is a quote from Eric Weisstein's
CRC Concise Encyclopedia of Mathematics, Second Edition. 2010, p.480.
(Google Books link)
Is this still the case, post-Perelman? What are the known compact manifolds in $\mathbb{R}^3$?
I ask this as someone (obviously!) naive in these areas.
Thanks for educating me!
Best Answer
Actually, it's very easy to write down a complete list of orientable 3-manifolds (and thus all, since every nonorienatble 3-manifold has its orientation cover), using a tool called Heegaard splitting. This list has the following structure: break it up into $\mathbb{N}$ many sublists corresponding to different genera of 2-dimensional manifolds. Having fixed $g$, look at all mapping classes from a surface with that genus to itself (these form a finitely generated group), and then consider the 3-manifold obtained by gluing two handlebodies (filled genus $g$ surfaces) using that map to identify the boundaries.
The proof that this is a complete list of 3-manifolds is quite easy; it can be explained in a few minutes to a smart undergrad. The hard part isn't listing all 3-manifolds; it's that the list I gave above (and various other lists one can produce with similar techniques, such as surgery) is massively redundant. We even know the rules (analogues of Reidemeister moves) that tell you all the redundancies, but just as Reidemeister moves don't help that much with classifying knots, this doesn't mean you can take two Heegaard splittings and efficiently test if they give you the same 3-manifold.