Timeline for Third differential in Atiyah Hirzebruch spectral sequence
Current License: CC BY-SA 3.0
10 events
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| Feb 4, 2014 at 18:22 | history | edited | Lennart Meier | CC BY-SA 3.0 |
Changed k to 2k for RP^k.
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| Apr 27, 2011 at 15:07 | comment | added | Dan Ramras | Tyler: Right; I figured that once I re-read your comment and saw "Chern classes." | |
| Apr 27, 2011 at 6:38 | comment | added | Tyler Lawson |
@Dan: Whoops. Meant $c_2$ and hence $w_4$. My bad.
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| Apr 24, 2011 at 17:24 | comment | added | Dan Ramras | Tyler, I get something a little different. To be specific, if we complexify the tautological bundle $L$ on $RP^n$ and then forget the complex structure, we get the Whitney sum $L\oplus L$ (Milnor-Stasheff 15.7). Now, $w_1(L)$ is the generator of $H^1 (RP^n)$ and $w_2(L) = 0$, so $w_1(L\oplus L)$ is trivial and $w_2 (L\oplus L)$ is the generator of $H^2 (RP^n)$. Now, let's consider the Whitney sum $2(L\otimes C)$. As a real vector bundle, this is $4L = 2L\oplus 2L$, so $w_4 (2(L\otimes C))$ generates $H^4 (RP^n)$ (but $w_2 = 0$). | |
| Apr 24, 2011 at 8:39 | comment | added | Tyler Lawson |
In this case, you can solve the extension problem using Chern classes; namely, you show twice a line bundle is not stably trivial by showing that it has nontrivial $w_2$.
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| Apr 24, 2011 at 6:11 | comment | added | Dan Ramras | For what I wrote above, one really does need to know that the torsion group is $Z/4$ and not $Z/2\oplus Z/2$. I was actually specifically interested in figuring out how to solve this extension problem the day this question appeared; I'll look into it further. | |
| Apr 24, 2011 at 6:09 | comment | added | Dan Ramras | According to this question math.stackexchange.com/questions/26438/… math.stackexchange, there's a computation of $K^*(RP^n)$ on pp. 100-110 of Atiyah's K-theory book. I'm not sure what's involved; I don't think I've ever gone through the computation. | |
| Apr 24, 2011 at 6:07 | comment | added | Dan Ramras | Sam, are you saying that my argument might be circular, because one would need to understand the $d_3$ differential already in order to compute the K-theory of $RP^2$ and $RP^4$? In both cases, there's no room for a non-zero differential (on any page). One does have to solve the extension problem in order to see that $K^0 (RP^4)$ is $Z\oplus Z/4$ and not $Z\oplus Z/2 \oplus Z/2$. It seems unlikely to me that this could require knowing something about $d_3$, since it's zero in this case. | |
| Apr 24, 2011 at 5:06 | comment | added | Sam Nariman | But computing the K group of $\mathbb{R}P^k$ needs to apply AHSS. Actually at the appendix of the paper by Atiyah, titled Analytic cycle on complex manifold, He computed $d_{2p-1}$ in terms of Steenrod p-power by attaching a cell to some $\Sigma^k \matbb{C}P^n$ and using some relation of differentials and chern character which I don't quite understand. But I expect for $d_3$ there should be some straight forward example. | |
| Apr 22, 2011 at 21:39 | history | answered | Dan Ramras | CC BY-SA 3.0 |