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$\omega_2(SO(n))=?$, that is, What is the second stiefel whitney class of SO(n)?

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    What vector bundle did you have in mind?2011-04-21

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$SO(n)$ is a Lie group and so its tangent bundle is trivial. Thus $w_2(TSO(n)) = 0$.

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    @Boh Liu : You should ask this as a separate question, as it has little to do with the original one.2011-04-22