Timeline for Spin structures and divisibility of cohomology classes
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 26, 2013 at 6:25 | comment | added | Samuel Monnier | Are you asking why the intersection form is even on $H^2(X,\mathbb{Z})$ when $X$ is a spin 4-manifold? This follows from the fact that the degree 2 Wu class, given in terms of the Stiefel-Whitney classes by $w_2 + w_1^2$, vanishes for spin manifolds. | |
| Sep 25, 2013 at 18:58 | history | edited | Ryan Thorngren | CC BY-SA 3.0 |
I've rewritten the question somewhat
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| Sep 25, 2013 at 5:13 | comment | added | Chris Gerig | A post of mine from a while ago is related: mathoverflow.net/questions/100746/… | |
| Sep 25, 2013 at 0:41 | comment | added | Chris Gerig | Well first, CS is not a characteristic class, because $X$ has boundary (the Chern class arises from closed $X$). And I can choose my $X$ to be simply-connected, because the corresponding cobordism group is trivial. | |
| Sep 25, 2013 at 0:21 | comment | added | Ryan Thorngren | I don't think I'm really switching gears. The Chern-Simons invariant is a sort of characteristic class of the $U(1)$ gauge bundle. Perhaps I should have chosen a more simple motivating example. By the way, does this definition of the Chern-Simons invariant only work for simply connected manifolds? | |
| Sep 24, 2013 at 21:40 | answer | added | johndoe | timeline score: 2 | |
| Sep 24, 2013 at 21:28 | comment | added | Chris Gerig | I'm having trouble following this. For your question, what are you referring to by "what the spin structure gives you"? In relation to the motivation, the spin structure gives $w_2(M)=0$ which on a simply-connected 4-manifold is equivalent to having an even intersection form. I don't see why you switch gears to "characteristic classes of bundles other than the tangent bundle". | |
| Sep 24, 2013 at 20:00 | history | asked | Ryan Thorngren | CC BY-SA 3.0 |