Timeline for What is the "real" meaning of the $\hat A$ class (or the Todd class)?
Current License: CC BY-SA 3.0
10 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Feb 9, 2016 at 9:59 | comment | added | Paul Siegel | Also, I thought quite a bit about the relation between 1 and 3 in graduate school, but all I was left with was a question: can one prove Bott periodicity using heat kernels? I think this would have to be the first step, though I'm not sure. | |
| Feb 9, 2016 at 9:54 | comment | added | Paul Siegel | Another topological description: the $\hat{A}$ class is the genus associated to the $E_\infty$ ring homomorphism $M Spin \to KO$. This might somehow mediate between your $1$ and $2$. | |
| Feb 9, 2016 at 9:30 | comment | added | AlexE | @SebastianGoette Yes, the Dirac operator of a spin manifold is a KO-fundamental class (and for a spin^c manifold the corresponding Dirac type operator is a K-fundamental class). That it can not be compatible in the naiv way with the homological fundamental class of the induced orientation of the manifold can be seen by noting the following: different spin structures (and there may be many) give rise to different Poincare duality maps, but the homological Poincare duality only depends on the induced orientation (of which there are only two). | |
| S Feb 9, 2016 at 7:47 | history | suggested | Ali Taghavi |
I add a tag
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| Feb 9, 2016 at 7:37 | review | Suggested edits | |||
| S Feb 9, 2016 at 7:47 | |||||
| Feb 8, 2016 at 22:32 | comment | added | Sebastian Goette | @AlexE Isn't the Dirac operator actually a fundamental class for $K$-homology? So the Poincaré dual of $\hat A$ indicates that Poincaré duality in $K$-theory is not compatible with Poincaré duality in cohomology. | |
| Feb 8, 2016 at 20:50 | comment | added | Steve Huntsman | FWIW, in the context of Atiyah-Singer, I liked to think of the Chern character as encoding analytic properties, and the Todd class as encoding geometrical properties. | |
| Feb 8, 2016 at 20:29 | comment | added | AlexE | Similar to 1: The Dirac operator defines a class in K-homology. Apply now the Chern-Connes character to go to de Rham homology. What you will end up with is the Poincare dual of the \hat{A}-class. | |
| Feb 8, 2016 at 20:19 | history | asked | Sebastian Goette | CC BY-SA 3.0 |