Timeline for Geometric theory for cohomology groups $H^p(M;\mathbb{Z})$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 11, 2020 at 11:43 | comment | added | Praphulla Koushik | @SebastianGoette I mean $H^2$ is a first step..the authors hoping for quantization of $H^p(M,\mathbb{Z})$... I hope I conveyed it correctly... | |
| Apr 11, 2020 at 10:21 | comment | added | Sebastian Goette | I see. I thought you wanted to quantise $H^p(M;\mathbb Z)$. | |
| Apr 10, 2020 at 13:25 | comment | added | Praphulla Koushik | @SebastianGoette That is my usage. The book does not use that phrase at that point.. suppose I have a differential $2$-form $K$ on a manifold $M$, then, I can ask if this $2$-form is curvature of some connection on some line bundle over $M$.. This they called "quantization condition" in the book.. Is it not suitable word here? | |
| Apr 10, 2020 at 13:19 | comment | added | Sebastian Goette | And I wondered how geometric quantisation enters the picture. | |
| Apr 10, 2020 at 12:58 | comment | added | Praphulla Koushik | @SebastianGoette Ok. I will read and try to realte to the above mentioned question :) Thank you :) | |
| Apr 10, 2020 at 11:03 | comment | added | Sebastian Goette | Yes, that's the paper. "Differential characters" are a simultaneous refinement of (say) integral cohomology and de Rham forms (cf. Deligne cohomology in arithmetic geometry). Characteristic classes of vector bundles with connections can be refined to differential characters. Easiest example: the first Chern class of a complex line bundle with connection on $S^1$ as a differential character is equivalent to $\frac\log{2\pi i}\in\mathbb{C/Z}$ of the holonomy of the connection. Note that both the integral and the de Rham version vanish for degree reasons, so the refinement is nontrivial. | |
| Apr 8, 2020 at 16:23 | comment | added | Praphulla Koushik | @SebastianGoette Hi.. Just to confirm, are you referring to numr.wdfiles.com/local--files/differential-cohomology/…? I have not read that, I want to read that... Can you please say in 1/2 lines in your view what refinement they are mentioning... | |
| Apr 8, 2020 at 15:18 | comment | added | Sebastian Goette | Did you try to read about Cheeger-Simons characters? In their original paper, Cheeger and Simons provide a refinement of both descriptions of Chern classes at once. | |
| Apr 7, 2020 at 11:36 | history | asked | Praphulla Koushik | CC BY-SA 4.0 |