Advanced Studies in Pure Mathematics

Some remarks on profinite completion of spaces

Gereon Quick

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We study profinite completion of spaces in the model category of profinite spaces and construct a rigidification of the completion functors of Artin–Mazur and Sullivan which extends also to non-connected spaces. Another new aspect is an equivariant profinite completion functor and equivariant fibrant replacement functor for a profinite group acting on a space. This is crucial for applications where, for example, Galois groups are involved, or for profinite Teichmüller theory where equivariant completions are applied. Along the way we collect and survey the most important known results of Artin–Mazur, Sullivan and Rector about profinite completion of spaces from a modern point of view. So this article is in part of expository nature.

Article information

Galois–Teichmüller Theory and Arithmetic Geometry, H. Nakamura, F. Pop, L. Schneps and A. Tamagawa, eds. (Tokyo: Mathematical Society of Japan, 2012), 413-448

Received: 24 April 2011
Revised: 5 November 2011
First available in Project Euclid: 24 October 2018

Permanent link to this document euclid.aspm/1540417825

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P60: Localization and completion
Secondary: 14F35: Homotopy theory; fundamental groups [See also 14H30] 55Q90

Profinite homotopy profinite completion equivariant completion


Quick, Gereon. Some remarks on profinite completion of spaces. Galois–Teichmüller Theory and Arithmetic Geometry, 413--448, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06310413.

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