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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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