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Is there a locally flat torus in some not smoothable topological 4-manifold such that surgering on it produces a smoothable 4-manifold? Surgering means removing a tubular neighborhood and reattaching it in a different way.

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How did you define flatness (curvature) in a non-smooth manifold? –  Nikita Kalinin Apr 26 '12 at 16:27
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@Nikita: locally flat is a purely topological notion, not related with curvature. Roughly speaking, a submanifold is locally flat if admits a tubular neighborhood in the ambient manifold. –  Daniele Zuddas Apr 26 '12 at 16:36
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