For an oriented manifold, we have Pontryagin classes. For a manifold with complex structure, we have Chern classes. For a orientable fiber bundle over $S^1$ (ie for orientable mapping tori), do we have the corresponding characteristic classes?
We know that the bordism group for 4-dimensional oriented manifold is $Z$, detected by the first Pontryagin class. What is the bordism group of orientable mapping tori?