I had not realized the 4-dimensional case was so hard that the much more famous 3-dimensional case was "another" contender for the hardest. I had thought it was sort of like the 4 color map theorem on stereoids, where first it was proved in 5+ dimensions (Smale, 1961), then 4 dimensions (Freedman, 1982), and finally 3 (Perelman, 2006), getting harder and harder at each step.
And the proofs of this one fact are all completely different depending on the dimension. I wonder if there are any prospects of a unified proof for all dimensions.
Smale's proof for d>=5 was unified. The d=4 and d=3 cases were special and those are special dimensions for a lot of reasons. "Low-dimensional topology", i.e. d=3 or d=4, is a big subject in its own right, because of those issues. At least for now, the prospect of unifying them with higher dimensions is somewhat remote. Basically in higher dimensions, there is more space to maneuver and things don't get in each other's way as much, so many proofs become easier. As well, many things are true in high dimensions and false in low dimensions, in which case unification is obviously impossible.. See also "curse of dimensionality" about why things are different for high dimensions.
