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I would really appreciate if anyone could provide me with an example of a locally trivial, but globally nontrivial, fiber bundle $Y\hookrightarrow Z \rightarrow X$, where $X$, $Y$, and $Z$ are all compact spin manifolds with even dimension (preferably divisible by $4$) satisfying $p_1(Z)=0$, but $p_1(X)\neq 0$.

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    This question has been answered on mathoverflow http://mathoverflow.net/questions/70840/example-of-a-nontrivial-fiber-bundle-with-total-space-compact-spin-and-p-102011-07-29

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