Timeline for Cokernel of the stable J-homomorphism at odd primes
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 7, 2014 at 23:06 | answer | added | Igor Belegradek | timeline score: 2 | |
| Mar 6, 2014 at 22:21 | comment | added | Drew Heard | I can also recommend Mark Behrens' talk recently at MSRI, which can be found msri.org/workshops/685. He explicitly shows the periodic families on Hatcher's charts | |
| Mar 6, 2014 at 22:19 | comment | added | Drew Heard | Perhaps it is worth saying that you can cross check with Ravenel's tables at math.rochester.edu/people/faculty/doug/mybooks/ravenelA3.pdf. For example $\sigma^2$ is actually in the 14-stem. Likewise the other dot in the 2nd band is (I think) the element $x_{28}$ is the 28-stem. Hatcher's "essentially" refers to the fact that there are some elements in coker J in the bottom band, for example $\eta^2 \in \pi_2$ that Craig mentions below | |
| Mar 6, 2014 at 14:17 | comment | added | Igor Belegradek | I did read Hatcher's explanations of the table but it does not answer all the questions. Here are some for the top table. What does "essentially the same as the image of $J$" mean? Does "essentially" refer to some abbreviation reflecting the size of $2$-components? If a graph in the 2nd band sits over several stems, is its group present in all of them. Does the isolated $\sigma^2 dot on the left of the 3rd band live both in 15th and 11th (it is kind of between them but closer to 15). Same for the isolated dot in the 2nd band. | |
| Mar 6, 2014 at 6:02 | comment | added | Drew Heard | Each dot represents a copy of $Z/p$, whilst connected vertical dots are meant to represent a non-trivial extension (e.g. at $p=2,\eta^3$ corresponds to a $\mathbb{Z}/8$) | |
| Mar 6, 2014 at 4:45 | comment | added | Igor Belegradek | @DrewHeard: I still need to learn how to read the diagrams, how does one see the order of $\mathrm{coker} J$? | |
| Mar 6, 2014 at 4:13 | answer | added | Craig Westerland | timeline score: 4 | |
| Mar 6, 2014 at 2:49 | comment | added | Drew Heard | There are some nice pictures on Alan Hatcher's webpage: math.cornell.edu/~hatcher/stemfigs/stems.html. In particular the bottom forms the image of the $J$ homomorphism, and you can see how large the coker J is! | |
| Mar 6, 2014 at 2:01 | history | asked | Igor Belegradek | CC BY-SA 3.0 |