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.



Pontryagin numbers on a fiber bundle over $S^1$
You are asking about vanishing of the Pontryagin numbers, and hence the (rational) cobordism class, of the mapping torus of a diffeomorphism. According to M. Kreck, (Bordism of diffeomorphisms Bull. Amer. Math. Soc. Volume 82, Number 5 (1976), 641-789, with details in Cobordism of odd-dimensional diffeomorphisms. Topology 15 (1976). 353-361) any cobordism class in the kernel of the signature homomorphism contains a mapping torus. So the signature in dimension 4 is the only obstruction, but that's not true in higher dimensions.


The special case of $S^1 \times M$ is the boundary of $D^2 \times M$, so of course its Pontryagin numbers vanish. 


