The étale homotopy type of varieties over $\textbf {R}$

Abstract:

For a variety X over ${\text {Spec}}({\mathbf {R}})$, the étale homotopy type of X is computed in terms of the action of complex conjugation on the complex points $X({\mathbf {C}})$. This enables one to show that $X({\mathbf {R}}) \ne \emptyset$ is equivalent to various conditions on the étale cohomology of X, and, when X is a smooth, geometrically connected, proper curve over ${\text {Spec}}({\mathbf {R}})$, to compute the étale cohomology. Finally, there is a negative result, showing that étale cohomology cannot be used to compute the topological degree of a mapping germ $f:({{\mathbf {R}}^n},0) \to ({{\mathbf {R}}^n},0)$ .