Timeline for Atiyah's paper on complex structures on $S^6$
Current License: CC BY-SA 3.0
19 events
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| Aug 12, 2019 at 10:19 | history | edited | YCor |
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| Nov 3, 2016 at 22:38 | comment | added | David Roberts♦ | Regarding the map you wish to understand, I think that the KSp should be seen as coming from one specific KR. Btw web.archive.org/web/20230113145507/https://www.math.umd.edu/… should be rather useful, but I haven't been able to figure out the dimension correctly from there. Rosenberg uses a different grading to Atiyah. | |
| Nov 3, 2016 at 21:55 | comment | added | Willie Wong | For your final question, the section titled "Ambient Space Construction" on this blog entry of mine is related. Basically it is a way to fix one conformal structure on $\mathbb{S}^6$. | |
| Nov 3, 2016 at 19:59 | history | edited | David C | CC BY-SA 3.0 |
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| Nov 1, 2016 at 21:28 | comment | added | Mingcong Zeng | @RyanBudney I suspect that the sentence "integers goes to 0" means integrable ACS goes to 0 in KR, while non-integrable ones goes to 1. It seems that he was using this map to separate integrable and non-integrable ACSs, it cannot be done if it is trivial. I feel that this might be the most crucial step, but do not have a clear idea what's going on... | |
| Nov 1, 2016 at 8:19 | comment | added | Tyrone | Karoubi briefly treats KR-theory in "K-Theory: An Introduction" (page 177). The book is a good reference for K-Theory in general. | |
| Nov 1, 2016 at 2:56 | comment | added | Igor Belegradek | I think basics of $KR$ theory is easy to learn from [1]. What is puzzling is the claim on p.5 of arxiv.org/abs/1610.09366 that "Because of the Atiyah-Singer theory, the linear algebra acquires a topological meaning, and that is embodied in $KR$ theory." | |
| Nov 1, 2016 at 1:42 | comment | added | David Roberts♦ | And the involution on the base space would be trivial. | |
| Nov 1, 2016 at 1:41 | comment | added | mme | @DenisNardin You go from $E$ to $E \oplus \overline{E}$; your involution is the obvious one that swaps factors. | |
| Nov 1, 2016 at 0:20 | comment | added | Denis Nardin | @RyanBudney Can you explain a little bit more how to treat a complex bundle as a real bundle? You need to throw in a C_2-action and I really don't see how to do that (this is a bit embarassing because I should know KR very well). | |
| Oct 31, 2016 at 21:22 | comment | added | Ryan Budney | I think it's just a straight-forward computation, isn't it? I've only scanned the paper and this is my first time thinking about $KR$, but I suspect what's going on is that the generator for complex k-theory is the bundle with Euler class $1$. In $KR$ theory this is twice the generator, because the generator is one where the fixed-point set of the involution is a Moebius bundle. I haven't thought about this in detail but that's what I suspect is going on. | |
| Oct 31, 2016 at 21:08 | comment | added | David C | Yes of course, but do you know why it is trivial? | |
| Oct 31, 2016 at 21:07 | comment | added | Ryan Budney | The integers he's talking about is 6-dimensional complex k-theory of $S^6$, i.e. this group is infinite cyclic. | |
| Oct 31, 2016 at 21:02 | comment | added | David C | This sentence is on p4 of Atiyah's preprint. | |
| Oct 31, 2016 at 20:45 | comment | added | Ryan Budney | The forgetful map appears to be that you can treat a complex bundle as a real bundle, that gives your map from complex k-theory to this "KR" variant. Where is the sentence you are quoting? | |
| Oct 31, 2016 at 20:43 | history | edited | YCor |
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| Oct 31, 2016 at 20:22 | comment | added | David C | I am reading [1], it is beautifully written. However as I am a complete amateur, I would like to understand the relationships between topological complex K-theory and KR-theory. In particular this sentence: "There are natural forgetful maps from complex K-theory to KR-theory and in dimension 6 the integers go to 0". | |
| Oct 31, 2016 at 20:05 | comment | added | Ryan Budney | It looks like you implicitly do not accept Atiyah's reference [1] in his paper. What isn't "nice" about that? | |
| Oct 31, 2016 at 20:01 | history | asked | David C | CC BY-SA 3.0 |