Timeline for Has anyone seen a nice map of multiplicative cohomology theories?
Current License: CC BY-SA 4.0
25 events
| when toggle format | what | by | license | comment | |
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| Aug 25 at 4:40 | comment | added | მამუკა ჯიბლაძე | Unfortunately several links have died, I wonder if it is possible to revive them... | |
| Sep 12, 2018 at 14:50 | history | edited | skd | CC BY-SA 4.0 |
updated broken link
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| Mar 1, 2018 at 23:23 | comment | added | Dylan Wilson | @skd but the fact that you have an E_2-map before taking Thom spectra means you're at least E_2 but you could be more (on accident), and then it takes some argument to show that you're at most E_2... but ok, I'll stop quibbling | |
| Mar 1, 2018 at 22:47 | comment | added | skd | @DylanWilson (i): I think I mean at best, since it's not E_3 (unless my understanding of English is subpar); (ii): it's not relevant, just thought I'd mention that fun fact :P ; (iii): I do mean h(n-1). You're right, of course: it isn't much of a reduction; reducing the reduction is the most interesting part of the whole story. | |
| Mar 1, 2018 at 22:45 | comment | added | Dylan Wilson | also (for the OP), aren't there lots of silly things "above" the sphere? like if $X$ is an $E_{\infty}$-space then $\Sigma^{\infty}_+X \to S^0$ by killing $X$ seems to work. | |
| Mar 1, 2018 at 22:40 | comment | added | Dylan Wilson | @skd penultimate paragraph: (i) maybe you mean "at least" not "at best" E_2, (ii) I don't see how the E_2-structure etc. is relevant?, and (iii) at the end you mean "h(n-1)" not "h(n+1)", and "reduces to" is technically correct but not much of a reduction... :) | |
| Mar 1, 2018 at 14:09 | history | edited | skd | CC BY-SA 3.0 |
added some references
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| Oct 23, 2017 at 8:03 | vote | accept | მამუკა ჯიბლაძე | ||
| Jul 12, 2017 at 20:50 | comment | added | მამუკა ჯიბლაძე | @skd I don't even know what integral K-theory is, I meant just complex K-theory. Its Chern character goes to HQ, and HZ is somewhere nearby, so I just lack vision of how are they situated wrt each other... | |
| Jul 12, 2017 at 20:36 | comment | added | skd | @მამუკაჯიბლაძე I don't know, but that's an interesting question. By "global" K, do you mean integral K-theory? | |
| Jul 12, 2017 at 20:35 | comment | added | skd | @NicholasKuhn Thanks for pointing that out; I should've said that the restriction of the source to $\Sigma^n_+ \mathbf{RP}^\infty$ is nullhomotopic for every $n$. | |
| Jul 12, 2017 at 19:58 | comment | added | Nicholas Kuhn | The comment about the Kahn-Priddy map (aka the transfer associated to the inclusion of the trivial group into the group of order 2) says something that is very very wrong: each of those maps is definitely not nullhomotopic. Also, restricted to the unit, it is multiplication by 2, so it isn't a map of ring spectra, as it isn't unital. | |
| Jul 12, 2017 at 9:13 | comment | added | მამუკა ჯიბლაძე | @skd OK here is the first question that was born in me during reading your exciting answer. There are few "global" ring spectra that I see on your map - S, MSpin, MString, MU, KO, TMF, HQ, and those mysterious X(n). Can you tell how MSpin, MString, etc. and those X(n) relate to each other? Could you also indicate where do things like global K, HZ reside? Are there any others? There must also be a global such guy at each height. | |
| Jul 12, 2017 at 9:03 | comment | added | მამუკა ჯიბლაძე | @RobertBruner Exciting! Would not it be shocking if every ring spectrum has an algebraic version (something lake AEll(S), AMU(S), ..., AS(S)) and they all sit above S repeating the whole pattern! (Sorry for shameless wishful thinking) | |
| Jul 12, 2017 at 5:43 | comment | added | Robert Bruner | There is a very interesting ring above S: the Waldhausen K-theory, A(*) = K(S) augments onto S. | |
| Jul 11, 2017 at 22:45 | comment | added | მამუკა ჯიბლაძე | Great! Is it also a map of ring spectra? | |
| Jul 11, 2017 at 22:36 | comment | added | skd | @მამუკაჯიბლაძე Sorry; I'd understood the question differently. There are nontrivial maps $X\to S$, of course. For instance, there's a map $\Sigma^\infty_+\mathbf{RP}^\infty\to S$, called the Kan-Priddy map. (This is defined via the composition of the maps $\mathbf{RP}^{n-1}_+ \to O(n)$ sending a line to the reflection it defines and $O(n)\to \Omega^n S^n$, which adjuncts to $\Sigma^n_+ \mathbf{RP}^{n-1}\to S^n$. Each of these maps is nullhomotopic --- but when we send $n\to \infty$, we get a nontrivial map $\Sigma^\infty_+ \mathbf{RP}^\infty\to S$.) | |
| Jul 11, 2017 at 22:35 | comment | added | skd | @RobertBruner Thanks! I'll edit that into my notes. | |
| Jul 11, 2017 at 22:00 | comment | added | მამუკა ჯიბლაძე | And yes, most probably after having digested your information I will come up with some further questions, thanks. | |
| Jul 11, 2017 at 21:59 | comment | added | მამუკა ჯიბლაძე | Concerning "above $S$" - $\mathbb Z$ is initial in rings but some other rings still may have homomorphisms to it, right? | |
| Jul 11, 2017 at 21:56 | comment | added | მამუკა ჯიბლაძე | Many thanks for your brilliantly illuminating answer (for me, so it is hardly flattering, as it is not so difficult to illuminate somebody as submerged into darkness as me). And even more thanks for many links to extremely informative sources. I will try to study them. | |
| Jul 11, 2017 at 18:34 | history | edited | skd | CC BY-SA 3.0 |
fixed typos
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| Jul 11, 2017 at 16:27 | comment | added | Robert Bruner | In your notes you say it is unfortunate that a lot of people use ASS to refer to the Adams spectral sequence. Actually, a lot of us scrupulously say Adams ss to avoid this ugly abbreviation. | |
| Jul 11, 2017 at 16:22 | history | edited | skd | CC BY-SA 3.0 |
added references
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| Jul 11, 2017 at 16:13 | history | answered | skd | CC BY-SA 3.0 |