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Sep 29 at 14:00 comment added HJRW I totally agree with this answer. Classification theorems are, in many ways, the fundamental task of pure mathematics. (Cf. the classification of finite simple groups.) The Poincaré Conjecture is the key ingredient needed to classify 3-manifolds. (Indeed, a complete classification is now known, as a result of Perelman's work.) And smooth Poincaré in dimension 4 is the most basic thing one might need to know in order to dream of classifying smooth (simply connected, say) 4-manifolds.
Sep 27 at 16:44 history edited Student CC BY-SA 4.0
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Sep 25, 2010 at 20:45 vote accept Topologieee
Aug 30, 2010 at 11:49 history answered rpotrie CC BY-SA 2.5