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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.

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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.

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  • $\begingroup$ Do you mean the Pontryagin numbers of some closed orientable mapping tori are not zero? $\endgroup$
    – Xiao-Gang Wen
    Commented Apr 28, 2014 at 2:55
  • $\begingroup$ Yes, that's apparently what Kreck's theorem implies; it seems a bit counterintuitive to me. Have you looked in the referenced papers? I don't know a specific example, but perhaps they have one. Kreck also wrote a book (Lecture Notes in Mathematics, 1069) on the subject. $\endgroup$
    – Danny Ruberman
    Commented Apr 28, 2014 at 11:44
  • $\begingroup$ Indeed, Kreck's theorem is very helpful. Thanks for refs. I asked the question to make sure that I understand Kreck's theorem (I am a physicist). Kreck has obtained the cobordism group of mapping tori for dimension greater then 4. The cobordism group of closed 4-dim mapping tori is calculated in another paper by Melvin, and is found to be 0. I wonder do you know what is the cobordism group of closed 3-dim mapping tori (or any refs)? Thanks! $\endgroup$
    – Xiao-Gang Wen
    Commented Apr 28, 2014 at 15:49
  • $\begingroup$ Let's be careful about the terminology. What you call the cobordism group of n+1 dimensional mapping tori is usually called the cobordism group of diffeomorphisms of n-manifolds. (This is clearer, since in your terminology you might be concerned about the cobordism class of the manifold arising as a mapping torus.) The cobordism group of diffeomorphisms of surfaces was computed by F. Bonahon: Cobordism of automorphisms of surfaces, Ann. Scient. Ec. Norm. Sup. 16 (1983), 237-270. numdam.org/item?id=ASENS_1983_4_16_2_237_0. $\endgroup$
    – Danny Ruberman
    Commented Apr 28, 2014 at 16:04
  • $\begingroup$ Thank you very much for the ref! Do you happen to know the generators of the 3-dim cobordism group? What characteristic classes can detect the cobordism group (in 3-dim and also in higher dimensions)? $\endgroup$
    – Xiao-Gang Wen
    Commented Apr 29, 2014 at 12:29
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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.

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