Let $F$ be a closed manifold. What are the Pontryagin numbers on $E=F\times S^1$? More generaly, let $E$ be a closed manifold which is a fiber bundle over $S^1$ (with fiber $F$). $E$ is also called mapping torus. What are the Pontryagin numbers on $E$?
When $E$ is 4-dimensional, the signature of such a fiber bundle over $S^1$ is zero, which implies that the corresponding Pontryagin number is zero. I wonder if all the Pontryagin numbers of closed orientable mapping tori in any dimensions are always zero.