Timeline for What is an example of a compact smooth manifold whose K-theory and Cech cohomology are not isomorphic?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Jul 4, 2010 at 19:28 | vote | accept | Efton Park | ||
| Jul 3, 2010 at 4:39 | comment | added | Dan Ramras | One way to think about this is that the Chern character isomorphism implies that the Atiyah-Hirzebruch spectral sequence collapses rationally. So you're just looking for an example in which something interesting happens to the torsion in this spectral sequence. Tom's answer shows that there can be non-trivial extensions. I suppose it's then natural to ask for an example with non-zero differentials. | |
| Jul 3, 2010 at 3:38 | comment | added | Tom Goodwillie | Yes. Perhaps it's the more natural recipient of the Chern character, but for manifolds and many other spaces it's equivalent to singular cohomology. | |
| Jul 3, 2010 at 3:35 | comment | added | Kevin H. Lin | Is it ok to replace "Cech cohomology" with just "cohomology" here? | |
| Jul 3, 2010 at 2:26 | answer | added | Tom Goodwillie | timeline score: 23 | |
| Jul 2, 2010 at 21:07 | comment | added | Chris Schommer-Pries | $E \wedge HQ$ always splits (up to equivalence) as a product of Eilenberg-Maclane spectra. This is true for any spectrum E. | |
| Jul 2, 2010 at 20:38 | comment | added | Greg Friedman | I don't know the answer to the question, but I wonder if the rational case is due to a spectrum level effect. Does anyone know if $K\wedge Q$ splits as a product of Eilenberg-MacLane spectra? | |
| Jul 2, 2010 at 20:22 | history | edited | Qiaochu Yuan |
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| Jul 2, 2010 at 19:55 | history | asked | Efton Park | CC BY-SA 2.5 |