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


Author: David A. Cox
Journal: Proc. Amer. Math. Soc. 76 (1979), 17-22
MSC: Primary 14F20
DOI: https://doi.org/10.1090/S0002-9939-1979-0534381-4
MathSciNet review: 534381
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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)$ .


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DOI: https://doi.org/10.1090/S0002-9939-1979-0534381-4
Article copyright: © Copyright 1979 American Mathematical Society

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