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I'm hoping someone can help me out with finding an example of the following:

a nontrivial fiber bundle $Y \hookrightarrow Z \rightarrow X$ where $X,Y,$ and $Z$ are all compact even dimensional spin manifolds with first Pontryagin classes satisfying $p_1(Z)=0$ and $p_1(X)\neq 0$. I'd also like dim $Y\geq8$.

Thanks in advance.

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$X=\mathbb CP^3$, $Z=S^7\times S^1$ mapping to $X$ by product projection on $S^7$ followed by the usual circle-bundle $S^7\to\mathbb CP^3$. So $Y=S^1\times S^1$. Oh, you wanted $dim(Y)$ to be at least $8$, so cross it with six more circles.

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  • $\begingroup$ Thanks! I figured this should be easy for people who know what they're doing. Is it possible to cook up an example where $p_n(Y)\neq 0$ for some $n\geq 2$? $\endgroup$
    – charris
    Jul 20, 2011 at 23:01
  • $\begingroup$ Sure. Consider $\mathbb CP^m$ for other values of $m$. $\endgroup$
    – Tom Goodwillie
    Jul 21, 2011 at 5:11

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