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Regarding the parallelizability of the Milnor's seven dimensional exotic spheres: Parallelizability of the Milnor's exotic spheres in dimension 7 The following question naturally arises: Suppose that a manifold has some exotic structures,and is parallelizable with respect to one of them ,is that manifold also parallelizable with respect to other smooth structures?

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You are asking good, interesting questions. However, I would advise you to slow down. Give each question a few days or a week to collect interesting answers. This will give you time to reflect, and to perhaps look at some of the literature to answer some of these questions yourself. At the very least, a delay between questions will ensure that you give the MO community adequate time to consider your questions. Too many, too fast, will simply overwhelm some of us. –  Spiro Karigiannis Mar 11 '11 at 13:42

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