I know classification of 2 manifolds and geometrization for 3 manifolds.
Why for dimension great or equal to 4, this task become impossible?


edit: Or should I ask "why geometrization won't be possible for 4 or higher dimension?"



Why "Classification" of 4 manifolds is NOT possible?
I'm guessing that you heard this from someone whose reasoning goes "Every finite presentation of a group can be made to give the $\pi\_1$ of a smooth 4-manifold. If we could put any 4-manifold into the Magic List of All, then we could recognize presentations of the trivial group. But no algorithm can do that."


Often people worry about classifications of simply connected manifolds, and don't have to deal with this. (Of course in three dimensions this becomes Perelman's theorem.)


