Timeline for Third page differential in the Lyndon–Hochschild–Serre Spectral Sequence
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 23 at 3:15 | history | edited | Sutirtha Datta | CC BY-SA 4.0 |
deleted 133 characters in body
|
| Oct 22 at 17:44 | comment | added | LSpice | Whenever possible, it is better to use (searchable) text than (un-searchable) images. I have transcribed your images as best as I could. Once you have checked that the transcription is correct, and/or fixed any errors, you can delete the images. | |
| Oct 22 at 17:40 | history | edited | LSpice | CC BY-SA 4.0 |
Link to book; transcribed images
|
| Oct 22 at 15:55 | answer | added | user509184 | timeline score: 4 | |
| Oct 22 at 8:09 | comment | added | Sutirtha Datta | Thanks a lot, this helps. However could you elaborate how $E_2 \cong \mathbb{F_2}[x,y,e]$ implies $E_3 \cong \mathbb{F_2}[x,y]/ (d_2(e)) \otimes \mathbb{F_2}[e^2]$ | |
| Oct 22 at 6:27 | comment | added | user509184 | Yes, try the version of the Kudo transgression theorem stated in appendix 1, Theorem A1.5.7, of Ravenel's book "Complex cobordism and stable homotopy groups of spheres." It is for the spectral sequence of an extension of Hopf algebroids, rather than the Serre SS of a fiber sequence, but the Lyndon-Hochschild-Serre SS you are running is a special case of both. | |
| Oct 22 at 6:10 | comment | added | Sutirtha Datta | Does “Steenrod operations commute with Transgressions” imply $d_3( Sq^1 (e))= Sq^1 ( d_2(e))$? I am not sure how to apply Kudo’s Theorem since its statement starts with a transgressive element from $E_2^{0,2k}$ (for example, the statement in McCleary), but here $e$ is apparently in $E_2^{0,1}$. Could you please explain this? I am new to all these. | |
| Oct 22 at 5:28 | comment | added | user509184 | For getting the $E_3$-page: you know the $d_2$-differential on generators for $E_2$ as a ring, so you can use the Leibniz rule to propagate the differentials and calculate $d_2$ on everything. Then take cohomology of that cochain complex, and that's $E_3$. Drawing the SS chart in a range of degrees might help to get a feel for what's going on. For the $d_3$ differential on $e^2$, i.e., on $Sq^1(e)$: this is the Kudo transgression theorem, i.e., under appropriate hypotheses, the transgression in the Serre SS commutes with Steenrod operations. | |
| Oct 22 at 4:18 | history | edited | Sutirtha Datta |
edited tags
|
|
| S Oct 22 at 3:54 | review | First questions | |||
| Oct 22 at 6:00 | |||||
| S Oct 22 at 3:54 | history | asked | Sutirtha Datta | CC BY-SA 4.0 |