Abstract:
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.
Key words and phrases:étale fundamental group, cycle class map, pronilpotent completion.