Is it possible to find a complex vector bundle on $S^2$ with nonzero first Chern class, which is trivial as a real vector bundle?
Is Hatcher's proof of thom isomorphism theorem flawed?: I don't believe that $H^n(E,E_0)\cong H^n(R^n,R^n-0)$
Is there a description of the Grassmannian as a homogeneous space where the principal bundle is one associated to the universal vector bundle?
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