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.



Example of a nontrivial fiber bundle with total space compact, spin, and $p_1=0$
$X=\mathbb CP^3$, $Z=S^7\times S^1$ 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.


