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Given a smooth manifold $M$, we define the differentiable structure on $TM$ in the usual way. I would like to know examples of two smooth manifolds which are non-diffeomorphic, but their tangent bundles are. Which is the smaller smallest dimension in which one can find such examples? What if I ask the same question for $k$ pairwise non-diffeomorphic manifolds? Can we have $k=\infty$? |
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Examples of non-diffeomorphic smooth manifolds with diffeomorphic tangent bundleGiven a smooth manifold $M$, we define the differentiable structure on $TM$ in the usual way. I would like to know examples of two smooth manifolds which are non-diffeomorphic, but their tangent bundles are. Which is the smaller dimension in which one can find such examples? What if I ask the same question for $k$ pairwise non-diffeomorphic manifolds? Can we have $k=\infty$?
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