Poincare for $n=2$ is contained in the classification theorem for surfaces (and that phrase should get you started, if you want to search for a proof), which says that every compact surface is homeomorphic to a sphere with some number of handles or cross-caps attached.
I once heard an expert "explain" the difficulty of the $n=3$ case to a general audience by saying something like this: when $n\le2$, there isn't enough room for anything to go wrong, while for $n\ge4$, there's enough room to fix anything that goes wrong; for $n=3$, there's enough room for something to go wrong, and (this was 15 years ago) it's not clear whether there's enough room to fix things when they go wrong.