Timeline for Kervaire invariant: Why dimension 126 especially difficult?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 4, 2015 at 0:42 | vote | accept | Joseph O'Rourke | ||
| Aug 4, 2013 at 10:55 | comment | added | Oliver Nash | I am over a year late but it seems worth mentioning that you are of course correct about the construction of the 30-dimensional manifold. According to these notes of Jones: math.rochester.edu/people/faculty/doug/otherpapers/jones3.pdf, the 30-dimensional case can be obtained as $M = Y\times S^7\times S^7\times S^7\times S^7/D_8$ where $Y$ is a genus-5 surface, $D_8$ is the dihedral group and the octonionic framing of $S^7$ determines the one used for M. | |
| Jul 2, 2012 at 12:07 | comment | added | André Henriques | I realize that this is not a complete argument. Atiyah's hope was that there should exist a geometric construction that starts with the quateroctonionic projective plane and constructs a 62-manifold of Kervaire invariant one, and that the same construction, applied to the octooctonionic projective plane would produce a 126-manifold of Kervaire invariant one. | |
| Jul 1, 2012 at 21:30 | comment | added | Peter May | I should probably add that I hope you are right, since I have a student who hopes to prove that you are (homotopically of course). | |
| Jul 1, 2012 at 20:41 | comment | added | Peter May | Interesting. But I think to give it substance you would have to display a manifold of dimension 62 and Kervaire invariant one. | |
| Jul 1, 2012 at 19:26 | comment | added | André Henriques | I disagree with the statement that there is no reason for guessing which way the answer will go. There is a good reason, which I learned from Atiyah, to guess that the answer in dimension 126 is the same as in the lower dimensions 62, 30, 14,... Namely, there exist remarkable manifolds in dimensions 128, 64, 32, and 16. These are the octooctonionic projective plane (see math.ucr.edu/home/baez/octonions/node19.html), the quateroctonionic projective plane (see math.ucr.edu/home/baez/octonions/node18.html), the bioctonionic projective plane, and finally $OP^2$ itself. | |
| Jul 1, 2012 at 13:58 | history | edited | Peter May | CC BY-SA 3.0 |
deleted 3 characters in body
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| Jul 1, 2012 at 1:26 | history | answered | Peter May | CC BY-SA 3.0 |