# Characteristic class for a fiber bundle over $S^1$

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?

• I notice that you use "Chern class" and "Pontryagin class" in the singular... I just want to point out that there are many Chern classes, and many Pontryagin classes. For example, the first Chern class $c_1$ is an element in the second cohomology group $H^2(M,\mathbb Z)$, the second Chern class $c_2$ is an element in $H^4(M,\mathbb Z)$, etc. – André Henriques Apr 28 '14 at 4:07
• You asked about cobordism in a separate question, and it was answered: mathoverflow.net/questions/164509/… – Danny Ruberman Apr 28 '14 at 12:34
• In the other article, the question "What are the cobordism groups of orientable mapping tori?" is answered. Here I try to ask "What are the differential forms (or characteristic classes) that can detect the cobordism group of orientable mapping tori?" Just like what the Pontryagin classes do for the cobordism group of orientable manifolds. – Xiao-Gang Wen Apr 28 '14 at 21:28

A fiber bundle over $S^1$ is a mapping torus. Characteristic classes of mapping tori are studied here: http://arxiv.org/abs/math/0611612 (Ebert 2006) and here: http://www.math.ku.dk/~xvd217/szymik.families.pdf (Szymik)