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The tangent bundle of $S^2$ is stably trivial, so it's already zero in K-theory. To answer this, first let's actually just figure out what all the real vector bundles over $S^2$ are. Bundles over $S^2$ are automatically orientable, and oriented bundles are in bijection with $[S^2,BSO(n)] = \pi_1(SO(n))$, which is $\Bbb Z$ for $n=2$ and $\Bbb Z/2$ for ...


A space is called ultraparacompact if every open cover can be refined to a cover by disjoint open sets. Note that clearly any fiber bundle on an ultraparacompact space is trivial (just apply the definition to an open cover on which it is trivial). So it suffices to show that $\mathbb{Q}P^n$ is ultraparacompact (with either the standard topology or the ...

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