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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.

flag	asked Jul 20 2011 at 19:29 charris 521●3●8

Tom Goodwillie · Accepted Answer · 2011-07-20 20:15:36Z UTC

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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.

answered Jul 20 2011 at 20:15

Tom Goodwillie
26.8k●2●60●110

	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 $n\geq 2$? – charris Jul 20 2011 at 23:01
	Sure. Consider $\mathbb CP^m$ for other values of $m$. – Tom Goodwillie Jul 21 2011 at 5:11

Example of a nontrivial fiber bundle with total space compact, spin, and $p_1=0$

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