A homotopical revisit to group extensions (v2.0)
\[ \gdef\Bund{\mathbf{B}} \gdef\Aut{\mathrm{Aut}} \gdef\fib{\mathrm{fib}} \gdef\Ani{\mathrm{An}} \gdef\AniB{{\mathrm{An}_\ast}} \gdef\Grp{\mathrm{Grp}} \gdef\Loop{\mathrm{\Omega}} \] I've written a more elaborate Chinese version of this blogpost, hosted on 香蕉空间 Bananaspace. There I've reviewed more of the group theory and higher algebra for a clearer presentation. Feel free to make suggestions and please do point out any other error you notice! This post tries to explain from a homotopist's perspective the following statement: I. Central extensions of groups \(1 \to A \to G \to Q \to 1\) are classified by (homotopy classes of) maps \(\Bund Q \to \Bund ^2 A.\) It's a basic result in group theory/group cohomology that the second cohomology of \(Q\) with coefficients in a trivial module \(A\) classifies such central extensions. More generally, if \(A\) carries some \(Q\)-action, the cohomology with twisted coefficients \(\ma...
