Timeline for Pontryagin numbers on a fiber bundle over $S^1$
Current License: CC BY-SA 3.0
7 events
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| Apr 29, 2014 at 12:29 | comment | added | Xiao-Gang Wen | 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)? | |
| Apr 28, 2014 at 18:02 | vote | accept | Xiao-Gang Wen | ||
| Apr 28, 2014 at 16:04 | comment | added | Danny Ruberman | 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. | |
| Apr 28, 2014 at 15:49 | comment | added | Xiao-Gang Wen | 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! | |
| Apr 28, 2014 at 11:44 | comment | added | Danny Ruberman | 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. | |
| Apr 28, 2014 at 2:55 | comment | added | Xiao-Gang Wen | Do you mean the Pontryagin numbers of some closed orientable mapping tori are not zero? | |
| Apr 27, 2014 at 22:44 | history | answered | Danny Ruberman | CC BY-SA 3.0 |