Timeline for Classifying abelian (but non-central) group extensions using homotopy theory
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 1, 2023 at 16:22 | comment | added | archipelago | The map $BAut(BA)\rightarrow BAut(A)$ is given by the action on $\pi_1(BA)$. Note that $A$ is abelian, so $\pi_1(BA)$ is functorial in unpointed maps. | |
| Nov 1, 2023 at 16:09 | comment | added | Tim Campion | @archipelago I'm a little confused trying to work out your claim that $B Aut (BA) = (B^2 A)_{h Aut A}$, which suggests there should be a fiber sequence $B^2 A \to B Aut(BA) \to B Aut A$, but what I'm getting is a map in the other direction $B Aut A = B Aut_{E_0}(BA) \to B Aut(BA)$ (given by including the basepoint-preserving maps) with fiber $BA$. What am I missing? | |
| Oct 31, 2023 at 8:26 | comment | added | Maxime Ramzi | @archipelago : this seems like a complete answer :) | |
| Oct 31, 2023 at 3:44 | history | became hot network question | |||
| Oct 30, 2023 at 21:42 | answer | added | Dmitri Pavlov | timeline score: 5 | |
| Oct 30, 2023 at 20:43 | comment | added | archipelago | Now one can compute $BAut(K(A,1))\simeq K(A,2)_{hAut(A)}$ and under this equivalence those fibrations whose monodromy over the basepoint agrees with the given action correspond to those pointed maps $BG\rightarrow K(A,2)_{hAut(A)}$ that induce the given action on fundamental groups. | |
| Oct 30, 2023 at 20:43 | comment | added | archipelago | $H^2(G;A)$ is the group of pointed homotopy classes of maps $BG\rightarrow K(A,2)_{hAut(A)}$ that induce the given action $G\rightarrow Aut(A)$ on fundamental groups (here the h-subscript denotes homotopy orbits). Fibrations over $BG$ with fibre $BA\simeq K(A,1)$ and an identification of the fibre over a fixed basepoint are classified by pointed maps $BG\rightarrow BhAut(K(A,1))$ where $Aut(K(A,1))$ is the topological monoid of self-homotopy-equivalences of $K(A,1)$. | |
| Oct 30, 2023 at 20:04 | comment | added | Qiaochu Yuan | Not a complete description but roughly how it should go: we can talk about principal $G$-bundles not only for $G$ a suitable kind of group but, over a base $X$, $G$ itself can be a suitable kind of group over $X$. When the action of $G$ on $A$ is nontrivial it defines a twisted version of $BA$ over $BG$ and then the claim is that $B \Gamma$ is the space of global sections of a twisted $BA$-bundle, classified by cohomology with local coefficients. | |
| Oct 30, 2023 at 19:51 | comment | added | Jon Pridham | Section 1.1.2 of arxiv.org/abs/1704.03021 is one such description. | |
| Oct 30, 2023 at 19:40 | history | asked | Andy Putman | CC BY-SA 4.0 |