Timeline for Coboundary matrix of bar resolution for group cohomology: do the elementary divisors always divide $|G|$?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 2 at 0:16 | comment | added | Joshua Grochow | I knew about that application of Hopf, but hadn't thought about the (|G|-1)^2 as the number of relators in the multiplication table. That's a nice way to think about it. | |
| Oct 1 at 23:08 | comment | added | Benjamin Steinberg | Notice by Hopf's formula the second cohomology should be generated by at most r generators where e is the minimal number of relators in a presentation. But $(|G|-1)^2$ is the number of relators in the multiplication table presentation (with 1 removed). So you expect many 1s | |
| Oct 1 at 14:26 | comment | added | Benjamin Steinberg | Notice the number of 1s really depends on the resolution while the other elementary divisors depend only on the cohomology. | |
| Oct 1 at 14:09 | comment | added | Benjamin Steinberg | I don't know much about the distribution. Presumably many are 1 because the bar resolution is too big and often you can use a smaller space/resolution to compute the cohomology. | |
| Oct 1 at 14:06 | comment | added | Joshua Grochow | Ah nice, thanks! The part I was missing was that the image was free so the sequence splits. And then that means my question about the distribution of elementary divisors becomes equivalent to asking about the distribution of orders in the torsion part of group cohomology, which I guess is a well-known question. | |
| Oct 1 at 13:24 | history | answered | Benjamin Steinberg | CC BY-SA 4.0 |