Geometry of compact lifting spaces
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Geometry of compact lifting spaces Gregory R. Conner1 · Wolfgang Herfort2
· Petar Paveši´c3
Received: 16 October 2019 / Accepted: 14 August 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We study a natural generalization of inverse systems of finite regular covering spaces. A limit of such a system is a fibration whose fibres are profinite topological groups. However, as shown in Conner et al. (Topol Appl 239:234–243, 2018), there are many fibrations whose fibres are profinite groups, which are far from being inverse limits of coverings. We characterize profinite fibrations among a large class of fibrations and relate the profinite topology on the fundamental group of the base with the action of the fundamental group on the fibre, and develop a version of the Borel construction for fibrations whose fibres are profinite groups. Keywords Covering projection · Lifting projection · Fundamental group · Inverse system · Deck transformations · Profinite group · Group completion Mathematics Subject Classification Primary 55R05; Secondary 57M10 · 54D05
Communicated by Andreas Cap. G. R. Conner is supported by Simons Foundation collaboration Grant 246221 and P. Paveši´c was partially supported by the Slovenian Research Agency program P1-0292 and Grants N1-0083, N1-0064.
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Wolfgang Herfort [email protected] Gregory R. Conner [email protected] Petar Paveši´c [email protected]
1
Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
2
Institute for Analysis and Scientific Computation Technische Universität Wien, Wiedner Hauptstraße 8-10/101, Vienna, Austria
3
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, Ljubljana, Slovenia
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G. Conner et al.
1 Introduction A foundational result states that every Hausdorff, compact and totally disconnected group is profinite, i.e., an inverse limit of finite groups. We show that under suitable assumptions a Hurewicz fibration with profinite groups as fibres is an inverse limit of regular finite coverings, which may be viewed as a fibrewise version of the abovementioned theorem. The crucial role in the proof is played by the action of the fundamental group of the base on the fibres. An action of a topological group on a fibration can paint an intimate picture of the group, even more so than the analogous covering space action of a discrete group (consider the action of a p-adic group on a solenoid). Our main result will make this relation clearly visible. The notion of finite covering spaces is a basic tool in a wide array of mathematical topics including group theory, algebraic geometry, combinatorics, manifold theory and analysis. A natural generalization of covering spaces are Hurewicz fibrations with unique path lifting property but they also include quite pathological examples (cf. [3]), and does not reveal in general a natural canonical group action like in covering spaces. Alternatively, one could consider inverse limits of finite covering spaces. These are very amen
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