Is there any formula for calculate the $\hat{A}$ genus for the total space of a fiber bundle?
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If the fiber is spin and the structure group of this fiber bundle is a compact, connected Lie group, then the $\hat{A}$ genus of the total space is the product of those of the base space and the fiber. Note that $\hat{A}$ genus is a very special case of the elliptic genera, the abovementioned property is still true you we replace $\hat{A}$ gneus by an arbitrary elliptic genus. 

