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The Poincare conjecture states that a simply-connected, closed 3-manifold is homeomorphic to the 3-sphere. Now that the conjecture has been settled, could someone tell me what this homeomorphism is (if it can be written down explicitly)?

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I don't understand the question. Given a manifold $M$ which is known to be homeomorphic to the $3$-sphere, you don't get a distinguished such homeomorphism; any such homeomorphism can be twisted by an automorphism of $S^3$. Are you actually asking whether / how the proof of the Poincare conjecture exhibits such a homeomorphism? –  Qiaochu Yuan Jan 11 '13 at 3:41
    
Though I have limited knowledge of the subject, it's my impression that the proof involves something of an algorithm. One 'smooths' the manifold to bring it towards constant curvature; it's possible that this leads to singularities, which Perelman dealt with by temporarily chopping off these pieces and continuing the process. Look up 'Ricci flow' to get started. –  Paul VanKoughnett Jan 11 '13 at 4:48
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