Timeline for Analogue of Bockstein for crossed module extensions and higher Steenrod square
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 24, 2022 at 8:22 | comment | added | Bertram Arnold | The action of $A$ on $D$ defines a principal $D$-bundle $EA\times_A D\to BA$, and since degree $1$ cohomology classes in $H^1(X;A)$ are homotopy classes from $X$ to $BA$, you can pull it back to get a principal $D$-bundle over $X$. Sections of this bundle are then a sheaf which is locally isomorphic to locally constant functions (aka a local system) to $D$, and you can define its sheaf cohomology as usual. A cocycle in $H^k(BA;D)$ then defines a degree $k$ cohomology classpulled back from the universal example $X = BA$, whose sheaf cohomology gives group cohomology. | |
| Jan 23, 2022 at 13:53 | vote | accept | Andi Bauer | ||
| Jan 23, 2022 at 13:53 | comment | added | Andi Bauer | Are you saying that one can construct a cohomology operation $H^1(-, A)\rightarrow H^k(-, D)$ also for group cocycles in $H^k(A, D)$ with a non-trivial action of $A$ on $D$? I thought $H^k(K(A, 1), D)=H^k(BA, D)$, and the latter are only the group cocycles with trivial action? Or are you saying in order to get an element of $H^k(-, D)$, we need some extra data which is related to what you call "local system"? | |
| Jan 23, 2022 at 13:53 | comment | added | Andi Bauer | Thanks a lot for your detailed answers! It'll take a while till I'm able to understand everything you say in detail, but I already got some idea. One last thing I wanted to clarify concerning your answer to 1): | |
| Jan 21, 2022 at 8:29 | comment | added | Bertram Arnold | I've tried to answer your questions in the comments. Let me just also mention that for general (unstable!) cohomology operations, one can investigate its "delooping" (operations one degree higher which give the operation after conjugation with the suspension isomorphism). It's easy to see that the operation mist be additive for a delooping to exist. For ordinary cohomology, it turns out that this is sufficient, and that there is a unique delooping in this case. | |
| Jan 20, 2022 at 17:55 | history | edited | Bertram Arnold | CC BY-SA 4.0 |
added 2795 characters in body
|
| Jan 20, 2022 at 10:31 | comment | added | Andi Bauer | If we apply the non-trivial group 3-cocycle in $H^3(B\mathbb{Z}_2, \mathbb{Z}_2)$ to a 1-cocycle $C$, we get $C\cup C\cup C$ which is a non-trivial cohomology operation. However, $Sq^2$ is trivial on 1-cocycles (as also remarked in a comment to the question). Is that another way to see that the crossed module of $Sq^1$ has to be trivial? (I was hoping that maybe $Sq^2$ might correspond to a crossed module only when applied to certain degrees, but from what you're writing it doesn't sound like this would make sense.) | |
| Jan 20, 2022 at 10:21 | comment | added | Andi Bauer | 1) Given a 2-group/crossed module, is there a way to construct a cohomology operation from that? 2) You say $Sq^2$ gives rise to a crossed module, but do I understand it correctly that later you say that this crossed module is trivial (i.e. equivalent to the trivial crossed module), since it has to "deloop once"? 3) Are you saying that even though the 2-group is trivial for $Sq^2$, there is another algebraic object (the "braided Picard groupoid" you mention) which describes $Sq^2$? Is there a way to explicitly construct a cohomology operation from such a Picard groupoid? | |
| Jan 20, 2022 at 10:21 | comment | added | Andi Bauer | Thanks a lot for your great answer! As a physicist who doesn't know much about spectra etc. I have some very basic questions: | |
| Jan 19, 2022 at 16:02 | history | answered | Bertram Arnold | CC BY-SA 4.0 |