Relations between Stiefel-Whitney classes on mapping torus

In question Relations between Stiefel-Whitney classes the relations between Stiefel-Whitney classes on manifold are obtained.

My question is that do we have additional relations between Stiefel-Whitney classes on oriented mapping torus? If yes, what are they?

Also do we have relations between Stiefel-Whitney classes and Pontryagin classes (mod 2) on mapping torus?

(Mapping torus is a fiber bundle over $S^1$)

• The tangent bundle is essentially $(n-1)$-dimensional, so $w_n=0$, and $w_*$ restrict to $w_*$ of the fiber (hence, corresponding relations there), with the kernel given by the Wang exact sequence. That's all I can say right out of the box, unless you are interested in something more specific. – Alex Degtyarev May 10 '14 at 6:30