Now that the Poincare conjecture appears to be solidly proven, I'd like to see the known corollaries. Over the years a number of theorems have proved "Either there's a counterexample to the Poincare conjecture, or else something very strange happens in 4 or 5 dimensions" (with the something very strange being specified). Very weird stuff is already known to happen in 4 dimensions. E.g., it's the unique dimension whose Euclidean space R^4 carries more than one differentiable structure, up to equivalence (diffeomorphism). There is a continuum of them! Also, given any finitely-presented group G, there is a 4-dimensional manifold fundamental group M whose fundamental group pi_1(M) is isomorphic to G (using topological operations that mimic the presentation using glueing of 1-handles & 2-handles, then capping it off with a conractible 4-manifold with the same boundary -- I think). Since a theorem of Markov says there is no algorithm which, given two arbitrary finite group presentations, can determine if they define isomorphic groups, it follows that there is no algorithm to classify all (even compact) 4-manifolds up to homeomorphism. (Okay, okay, this is also true in dimensions > 4, but 4 is the lowest case where it happens.) --Dan