Timeline for Do elements of every order occur in homotopy groups of spheres?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Aug 2, 2022 at 8:59 | vote | accept | Arshak Aivazian | ||
| Jul 31, 2022 at 20:16 | answer | added | user164898 | timeline score: 18 | |
| Jul 26, 2022 at 19:01 | comment | added | Arshak Aivazian | @A.S. Thank you! If you post this comment as an answer, I'll accept it. | |
| Jul 26, 2022 at 7:57 | comment | added | Mark Grant | @A.S. This seems more like an answer than a comment. | |
| Jul 25, 2022 at 15:38 | comment | added | user164898 | Sure. Let n be a positive integer. Let N be the product of 2n and Euler phi(2n). Then for each prime p that divides n, the number p-1 divides 2N. So for each prime divisor p of n, the denominator of the Bernoulli number B_N is div'l by p. So none of the prime factors of n cancel with factors in the numerator of B_N/N. So denom(B_N/N) is divisible by n. Now denom(B_N/N) or 2*denom(B_N/N) is the order of a cyclic summand in the image of the J-homomorphism in the (2N-1)st stable homotopy group of S^0. So by Freudenthal, there is an unstable homotopy group of a sphere with an element of order n. | |
| Jul 25, 2022 at 14:49 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
added 4 characters in body; edited title
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| Jul 25, 2022 at 14:49 | comment | added | Arshak Aivazian | I just don't know English well yet, I'm correcting, thanks. | |
| Jul 25, 2022 at 14:48 | comment | added | LSpice | I always have a little bit of difficulty parsing "any" as a quantifier. I think "every" might be clearer here. (Unless that's not what's meant!) | |
| Jul 25, 2022 at 14:19 | comment | added | Arshak Aivazian | Theorem 5.30, page 44 pi.math.cornell.edu/~hatcher/AT/ATch5.pdf | |
| Jul 25, 2022 at 14:18 | review | Close votes | |||
| Jul 25, 2022 at 15:03 | |||||
| Jul 25, 2022 at 14:13 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
edited body
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| Jul 25, 2022 at 14:12 | comment | added | Noam D. Elkies | What's "Serra's [Serre's?] classical result"? | |
| Jul 25, 2022 at 14:12 | comment | added | Arshak Aivazian | I'm aware of this question, thanks, but I'm asking a much weaker question. There may not be a group isomorphic to $Z_5$, but it is easy to prove that every sphere has homotopy groups containing elements of order $5$. | |
| Jul 25, 2022 at 14:10 | comment | added | Noam D. Elkies | Duplicates part of mathoverflow.net/questions/377545/… namely the special case of a cyclic group. That question has a lot of discussion but no answers. The discussion suggests there might never be an element of order 5. | |
| Jul 25, 2022 at 13:45 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |