Timeline for Intuitive reason for periods of 2 and 8 in Bott periodicity?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 14 at 14:57 | comment | added | Andrew Lee | @SimonHenry Thanks! To be fair my question is somewhat vague; I was mainly wondering if there was some semi-magical numerical fact that explains the 8 in particular, and your answer helps a lot. | |
| Aug 13 at 1:16 | comment | added | user509184 | @SimonHenry Atiyah-Bott-Shapiro write in "Clifford Modules" (1963) that "one should look for a proof of the periodicity theorem using Clifford algebras. Since this paper was written a proof on these lines has in fact been found by R. Wood. See also the proof given in: J. Milnor, Morse Theory". Not sure if the proof made it into any of Wood's papers, but Milnor's is in section IV.24 of the Morse theory book. | |
| Aug 12 at 18:43 | comment | added | Simon Henry | @QiaochuYuan I might be wrong on this, but I am fairly sure I have seen proof of Bott periodicity relying on Clifford algebra in the context of the K-theory of C* algebra (which immediately imply bott periodicity in topology K-theory, and then the periodicity of these spectrum), but that was a long time ago, so I'd have to do some research to remind myself how this works... | |
| Aug 12 at 18:06 | comment | added | Qiaochu Yuan | @Simon: I was under the impression that this does not actually give a proof of Bott periodicity, and one has to use Bott periodicity itself to relate Clifford algebras to Bott periodicity? | |
| Aug 12 at 14:30 | history | became hot network question | |||
| Aug 12 at 11:29 | answer | added | Carlo Beenakker | timeline score: 9 | |
| Aug 12 at 8:03 | comment | added | Simon Henry | I've always thought about these as coming from the "Bott periodicity" in the classification of real and complex Clifford algebras - which I tend to think as much less surprising as it just corresponds to the fact that $Cl_2(\mathbb{C})$ and $Cl_8(\mathbb{R})$ happen to be isomorphic to matrix algebras (which I'm happy to attribute to chances as there isn't that many option for what these algebras can be) - I'm not sure this answer your question, but maybe you'll want to comment on what sort of thing in that line of thought you would consider an answer to your question. | |
| S Aug 12 at 6:30 | review | First questions | |||
| Aug 12 at 7:04 | |||||
| S Aug 12 at 6:30 | history | asked | Andrew Lee | CC BY-SA 4.0 |