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Galois actions on homotopy groups of algebraic varieties. (English) Zbl 1217.14017
Homotopy groups for an algebraic variety $$X$$ over a number field $$K$$ may be defined in a number of different ways: as the classical homotopy groups of $$X_{\mathbb C}$$ for an embedding $$X \to \mathbb C$$; as the étale or $$\ell$$-adic homotopy groups of $$X_{\overline{K}}$$ for the algebraic closure $$\overline{K}$$ of $$K$$; and as the crystalline homotopy groups of $$X_{\hat{K}}$$ for the various local completions $$\hat{K}$$ of $$K$$.
In this paper, the author obtains, under appropriate hypotheses on $$X$$, comparison theorems that relate all these homotopy theories. He further shows that the Galois actions on $$\ell$$-adic groups are mixed representations of a determined type. Combined with his comparison theorems, this shows that a similar statement applies to the Galois action on the étale homotopy groups.
The comparison theorems are obtained by constructing the various homotopy types of which the various homotopy groups of $$X$$ are the corresponding groups, and then exhibiting comparisons among these types of which his comparison theorems are instances. This requires a careful examination of the types from first principles, and it is to this examination that the bulk of this substantial paper is devoted.

##### MSC:
 14F35 Homotopy theory and fundamental groups in algebraic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies 14F30 $$p$$-adic cohomology, crystalline cohomology
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