Not a topologist, but my understanding is that high dimensions can be handled by one set of methods because there's so much freedom, while low dimensions can be handled by another set of methods because they're so constrained. And so there ends up being a nasty point in the middle that isn't constrained enough to be handled by low-dimensional techniques or free enough to be handled by high-dimensional techniques. Not being much of a topologist, though, I couldn't possibly say why that middle point occurs where it does.
Also not a topologist, but thought it was interesting after watching the video from the current top comment, https://youtu.be/mceaM2_zQd8 , where if you look at the size of the "contained sphere", 4D is the only place where the contained sphere is exactly tangent to the containing box. Lower than that, and it's easy to visualize how the contained sphere is smaller, higher than that, the contained sphere is always bigger. So seems like it might be a natural consequence that techniques that work at higher or lower dimensions don't work for the case in 4D where the sphere is exactly tangent.
No, that's almost certainly not related. The relative volumes of the cube and the sphere is a geometric matter, not a topological one. Remember that topology uses much looser equivalences that do not respect such information.
The topologist R. H. Bing described it using more colorful language.
> Dimension 4 is the most difficult dimension. It is too old to spank, the way we might deal with the little dimensions 1, 2, and 3; but it is also too young to reason with, the way we deal with the grown-up dimensions 5 and higher.
Only case in which codimension 2 is dimension 2. Codim 2 means “complement generates topology”. Dim2 is “can avoid points using an arc”. I guess this means a lot.
