In the article about Perelman's putative proof of Bill Thurston's geometrization conjecture that Thane mentioned was in The Princetonian, the writer got things almost exactly backwards when he wrote: << In fact, there are infinitely many distinct two-dimensional surfaces. The ball is different from the donut is different from a two-holed donut is different from a three-holed donut, ad infinitum. But this is not the case in higher dimensions. As was first suggested by former University mathematics professor William Thurston, when dealing with three-dimensional surfaces, there are only eight shapes that are topologically distinct. This statement, known as the Geometrization Conjecture in three dimensions, is currently unproven.
The latter paragraph is exactly wrong. There are assuredly infinitely many topologically distinct 3-manifolds; in some sense each dimension is more complicated than lower dimensions, but 3 dimensions may be in another sense the most complicated of all. What Bill proved for a large class of (compact) 3-dimensional manifolds and conjectured for all of them is that each such manifold can be decomposed into pieces each of which can be given one of only 8 different types of geometries. For surfaces, the analogue is that each compact surface can be given a metric of constant curvature, either = 1, or = 0, or = -1. Furthermore, surfaces don't need to e decomposed into pieces first. Three of the eight geometries in the conjecture are the 3D analogues of the 3 constant curvature metrics in 2D; the other 5 might be called hybrids. --Dan