Galois action on algebraic matrix groups, Chern classes, and the Euler class (Q794770)

Let \(\Gamma\) be an arbitrary discrete group, G a complex algebraic matrix group defined over some field \(k\subset {\mathbb{C}}\), and \(\rho:\Gamma \to G\) a representation. For \(\sigma \in Gal({\mathbb{C}}/k)\) we denote by \(\rho^{\sigma}\) the Galois conjugate representation, and by \(B\rho\) and \(B\rho^{\sigma}\) the corresponding maps of classifying spaces \(B\Gamma \to BG^{{\hat{\;}}}\) where {\^{\ }} denotes profinite completion. We describe an action \(\psi^{\sigma}\) on \(BG^{{\hat{\;}}}\) and prove: (*) \(B\rho^{\sigma}\) is homotopic to \(\psi^{\sigma}B\rho.\) Combining this result with the effect of \(\psi^{\sigma}\) on the cohomology of \(BG^{{\hat{\;}}}\) (with profinite integer coefficients) yields relations for the profinite characteristic classes associated with \(\rho\). In particular, if \(\rho\) is realizable over a number field \(K\supset k\) one obtains universal bounds for the order of these classes, depending on K and the dimensions only. This concerns, e.g., the Chern classes of representations \(\rho:\Gamma \to GL_ m({\mathbb{C}})\) or the Euler class of orthogonal representations \(\rho:\Gamma \to SO_ m({\mathbb{R}}),\) realizable over K. Some arithmetic properties of these bounds are given. The method for proving the main fact (*) is to replace the profinite completion \(BG^{{\hat{\;}}}\) by the ''etale homotopy completion'' of BG as described by \textit{E. Friedlander} [Ann. Math. Stud. 104 (1982)]; this allows to define and discuss \(\psi^{\sigma}\) in terms of the data of the algebraic group G.