TITLE: What makes four dimensions special?
QUESTION [137 upvotes]: Do you know properties which distinguish four-dimensional spaces among the others?

What makes four-dimensional topological manifolds special?
What makes four-dimensional differentiable manifolds special?
What makes four-dimensional Lorentzian manifolds special?
What makes four-dimensional Riemannian manifolds special?
other contexts in which four dimensions or $3+1$ dimensions play a distinguishing role.

If you feel there are many particularities, please list the most interesting from your personal viewpoint. They may be concerned with why spacetime has four dimensions, but they should not be limited to this.

REPLY [2 votes]: 3 dimensions of space are special, because this is the lowest number of dimensions, where a random walk doesn't return to its origin with certainty (probability = 1), see
http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
Similarly I think one time dimension is special, because less than one would mean no evolution at all, and more than one would lead to instabilities of all kinds.