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.

How did you define flatness (curvature) in a non-smooth manifold?
– Nikita KalininApr 26 '12 at 16:27

3

@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 ZuddasApr 26 '12 at 16:36