[sarosh]

A nice summary of the current understanding (from a lay perspective) on manifold "sameness".

From the article: "But Freedman’s [1981] proof left open the “smooth” four-dimensional Poincaré conjecture, which says that any four-dimensional smooth manifold that is homotopy equivalent to the four-dimensional sphere is also diffeomorphic to the four-dimensional sphere. This is an even stronger statement than the one Freedman proved — since a diffeomorphism is a stronger form of equivalence than a homeomorphism — and one that mathematicians today have no idea how to settle.

This leaves them in the strange position of being unable to perform one of the most basic classification tasks of all: recognizing when a smooth four-dimensional manifold is really a sphere."