Curves are algebraic K ( π , 1 ) $K(\pi,1)$ : Theoretical and practical aspects

We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(\pi ,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible étale sheaf of $Z/nZ$-modules, with $n$ invertible in $k$, is canonically isomorphic to the cohomology of its corresponding ${\pi }_1\left(X\right)$-module. To this end, we explicitly construct some Galois coverings of $X$ corresponding to Galois coverings of the normalisation of its irreducible components. When $k$ is finite or algebraically closed, we precisely describe finite quotients of ${\pi }_1\left(X\right)$ that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products $H^1\times H^1\to H^2$ and $H^1\times H^2\to H^3$ in étale cohomology in terms of finite group cohomology.