Timeline for Lower dimensional Pin cobordisms
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Jun 25, 2014 at 21:32 | vote | accept | Zitao Wang | ||
| Jun 25, 2014 at 21:32 | vote | accept | Zitao Wang | ||
| Jun 25, 2014 at 21:32 | |||||
| Jun 25, 2014 at 14:14 | answer | added | user43326 | timeline score: 3 | |
| Jun 25, 2014 at 9:58 | comment | added | user43326 | Yes, they say that $MPin$ splits as a wedge ($\vee $) of spectra that are smash product ($\wedge $) of $\Sigma ^nHZ/2$ etc., with $RP^{\infty}$. So the homotopy groups (or any generalized homology groups) of $MPin$ is a direct sum of those spectra. On the other hand, what is not clear in the paper "Pin cobordism and related topics" is the number of summands $ko<8n>$'s and $ko<8n+2>$'s given $n$. This can be read off from "Structure of Cobordism ring" paper Theorem 2.2, and probably this will explain what happens in dimension 22. But $BO<2>$ seems to be there... | |
| Jun 25, 2014 at 9:00 | comment | added | Zitao Wang | So I think the Kirby and Taylor results are correct. Somehow, for dimensions 1-7, only BO<8n> is contributing to the pin groups. I don't know about higher dimensional cases. It would be interesting to know when and in what ways do the other 2 parts, especially BO<8n+2> contribute. | |
| Jun 25, 2014 at 8:55 | comment | added | Zitao Wang | I think you are right about the BO part. I'm a bit confused. Is the homotopy group of MSpin equal to the direct sum of homotopy groups of the three constituent parts? Theorem 1 of retro.seals.ch/cntmng?pid=comahe-003:1990:65::32 computes the homotopy groups of $\mathbb{RP}^{\infty}$ wedging with $K(\mathbb{Z}_2,n)$, $BO<8n>$, $BO<8n+2>$ respectively. And then follows Cor. 2 which says that the 1st and 2nd line of the 1st table gives you the Pin and Pin+ bordisms up to dim 7. Naively, one expects that the Pin and Pin+ groups should be the sum of tables 1 and 2, instead of table 1 alone. | |
| Jun 25, 2014 at 8:26 | comment | added | user43326 | It looks like by $BO<2>$ they mean $kO<2>$ (1-connective cover of the real $K$-theory {\it spectrum}, which also is denoted bo<2> in the litterature). So $\pi _2(RP^{\infty }\wedge kO<2>)$ is actually $Z/2$. This would mean that there is a contradiction between the paper jstor.org/discover/10.2307/… ( The structure of the Spin cobordism ring ) and the results of Kirby and Taylor. | |
| Jun 25, 2014 at 5:32 | comment | added | user43326 | I see. I guess $\pi _2(RP^{\infty }\wedge kO<2>)$ (or do they really mean $BO<2>$ ?) can be computed easily without Adams SS, so this is probably the first thing to check. | |
| Jun 24, 2014 at 19:59 | history | edited | Zitao Wang | CC BY-SA 3.0 |
typos fixed
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| Jun 24, 2014 at 19:50 | comment | added | Zitao Wang | Well, the Z/2 here is from BO<8n+2>, which contributes $\mathbb{Z}_{2^{4k+1}}$ in dim 8n+2+8k. If you take n=k=0, you get Z/2. | |
| Jun 24, 2014 at 17:49 | comment | added | user43326 | It looks like Anderson, Brown and Peterson say that the contribution of $\pi _*(RP^{\infty }\vee HZ/2)$ is either $Z/2$ or $0$ (and not $Z/2$), this takes care of the problem in dimension 2. | |
| Jun 24, 2014 at 15:54 | comment | added | Zitao Wang | Yes. Kirby and Taylor give them up to dimension 4 math.berkeley.edu/~kirby/papers/… | |
| Jun 24, 2014 at 7:31 | comment | added | user43326 | Could you give a source for "known results"? | |
| Jun 23, 2014 at 8:06 | history | edited | Zitao Wang | CC BY-SA 3.0 |
edited tags; added 56 characters in body
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| Jun 23, 2014 at 8:01 | history | edited | Zitao Wang |
edited tags
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| Jun 23, 2014 at 7:54 | history | asked | Zitao Wang | CC BY-SA 3.0 |