Remarks on cycle classes of sections of the arithmetic fundamental group

Given a smooth and separated $K(\pi,1)$ variety $X$ over a field $k$, we associate a “cycle class” in étale cohomology with compact supports to any continuous section of the natural map from the arithmetic fundamental group of $X$ to the absolute Galois group of $k$. We discuss the algebraicity of this class in the case of curves over $p$-adic fields. Finally, an étale adaptation of Beilinson's geometrization of the pronilpotent completion of the topological fundamental group allows us to lift this cycle class in suitable cohomology groups.