Pontryagin numbers on a fiber bundle over $S^1$ 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.
 A: 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. 
A: The answer is 0. The proof of the first case is as follows. Consider a family of product Riemannian metrics  $ g_{t}=t^{2}ds^{2}+h$, $0<t<1$, on E, where h is  a Riemannian metric on F. The straight forward  calculation shows that the sectional  curvature (or the norm of curvature operator)  of each $g_{t}$ is bounded by  a constant C independent of t, i.e. $|Rm(g_{t})|<C$.  Now the Chern-Weil theory says that for any Pontryagin number P we have 
     $$ P= \int_{E} R_{t} dvol_{g_{t}} ,$$ 
where $R_{t}$ is a polynomial of curvature  operators. When t tends to 0, we have    $$  |\int_{E} R_{t} dvol_{g_{t}} | < C Vol(E, g_{t})  \rightarrow 0.$$
 However the  Pontryagin number P is an integer independent of t, and thus it must be 0. 
