Abstract
We recall the fundamental groupoid of a connected, quasi-compact scheme X as in Grothendieck (Documents Mathématiques, vol. 3, 2003) Exposé V, with special attention towards the effect of a k-structure in case of a variety X ∕ k. Galois invariant base points are discussed and related to the profinite Kummer map. In Sect. 2.6, we address the reformulation of the section conjecture in terms of higher étale homotopy theory.
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References
Bousfield, A.K., Kan, D.M.: Homotopy limits, completions and localizations. Lecture Notes in Mathematics, vol. 304, vi + 348 pp. Springer, Berlin (1972)
Carlsson, G.: Equivariant stable homotopy and Sullivan’s conjecture. Invent. Math. 103, 497–525 (1991)
Harpaz, Y., Schlank, T.M.: Homotopy obstructions to rational points, preprint. http://arxiv.org/abs/1110.0164 arXiv: 1110.0164v1 [math.AG] (October 2011)
Pál, A.: Homotopy sections and rational points on algebraic varieties. http://arxiv.org/abs/1002.1731v2 arXiv: 1002. 1731v2 [math.NT] (March 2010)
Quick, G.: Profinite homotopy theory. Doc. Math. 13, 585–612 (2008)
Quick, G.: Continuous group actions on profinite spaces. J. Pure Appl. Algebra 215, 1024–1039 (2011)
Schmidt, A.: Motivic aspects of anabelian geometry, in: Galois-Teichmüller Theory and Arithmetic Geometry, Proceedings for a conference in Kyoto (October 2010), H. Nakamura, F. Pop, L. Schneps, A. Tamagawa eds., Advanced Studies in Pure Mathematics 63, 503–517 (2012)
Grothendieck, A.: Séminaire de Géométrie Algébrique du Bois Marie (SGA 1) 1960–1961: Revêtements étales et groupe fondamental. Documents Mathématiques vol. 3, xviii + 327 pp. Société Mathématique de France (2003)
Sullivan, D.: Geometric topology, Part I: Localization, periodicity, and Galois symmetry. Massachusetts Institute of Technology, 432 pp., revised and annotated version, xiii + 284 pp. Cambridge. http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf www.maths.ed.ac.uk/ ∼ aar/surgery/gtop.pdf (1971)
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Stix, J. (2013). The Fundamental Groupoid. In: Rational Points and Arithmetic of Fundamental Groups. Lecture Notes in Mathematics, vol 2054. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30674-7_2
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DOI: https://doi.org/10.1007/978-3-642-30674-7_2
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