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I am looking for interesting applications of the 1/4-pinched sphere theorem. The theorem says: A compact, simply connected riemannian manifold whose sectional curvature K satisfies $1/4 < K \leq$ 1 (possibly after multiplying the metric by a constant) is homeomorphic (note: not diffeomorphic) recently extended to "diffeomorphic") to the sphere. I just wanted to know: is it just a beautiful theorem or can you use it in concrete situations to derive some conclusions difficult to see otherwise? I am interested in this just because I am curious, I do not have any specific purpose in mind.

Post Reopened by Mark Sapir, Emerton, Anton Petrunin, Anton Geraschenko♦♦
Post Closed as "not a real question" by Andres Caicedo, Anton Geraschenko♦♦, Ryan Budney, Zev Chonoles, Simon Thomas
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