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I was wondering whether there might be anything similar to Thurston's Geometrization Conjecture, except for four manifolds instead of three manifolds. (Essentially, this conjecture (iiuc) states that it is always possible to take an arbitrary 3-manifold and break it up into smaller "pieces" of which there are eight types, each of which is pretty well understood -- see the explanation in mathworld [1].) I think that in two dimensions it is called the "uniformization theorem" [2], and ref [2] states that "It would be very important if similar results could be proved in higher dimensions." So I was wondering whether anyone has even *stated* the idea in dim=4?

David

[1] http://mathworld.wolfram.com/ThurstonsGeometrizationConjecture.html

[2] http://pdg.cecm.sfu.ca/~warp/java/uniform/node2.html [Broken]

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# Extension of Thurston's Geometrization Conjecture to four dimensions?

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