Math. Proc. Camb. Phil. Soc. (2000), 128, 233 Printed in the United Kingdom c  2000 Cambridge Philosophical Society 233 Compact coGalois groups By EDGAR E. ENOCHS† Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, U.S.A. e-mail: enochs@ms.uky.edu J. R. GARC ´ IA ROZAS,†‡ and LUIS OYONARTE †‡ Departamento de ´ Algebra y An´ alisis Matem´ atico, Universidad de Almer´ ıa, 04120 Almer´ ıa, Spain e-mail: jrgrozas@ualm.es loyonart@ualm.es and OVERTOUN M. G. JENDA Department of Discrete and Statistical Sciences, 120 Mathematical Annex, Auburn University, Auburn, AL 36849-5307, U.S.A. e-mail: jendaov@mail.auburn.edu (Received 23 December 1998; revised 11 February 1999) Abstract In this paper we extend the concept of the group of covering automorphisms associated to a universal covering space φ: U → X (where X is a connected topolog- ical manifold), to the case of left (or right) minimal approximations. In the case of torsion-free coverings of abelian groups we exhibit a topology on these groups which makes them into topological groups and we give necessary and sufficient conditions for these groups to be compact. Finally we prove that when these groups are com- pact they are pronilpotent (Theorem 5·3). We also characterize when these groups are torsion-free (Proposition 5·4). 1. Introduction If k ⊂ Ω is an algebraic closure of the field k, then the group of automorphisms of Ω over k is a compact topological group and is the Galois group of the extension k ′ ⊂ Ω where k ′ is the fixed field of this Galois group. Given a connected topological manifold X (i.e. X is locally Euclidean) and a uni- versal covering space φ: U → X we have the group of covering automorphisms, i.e. the morphisms f : U → U such that φ ◦ f = φ. The (dual) analogy between these groups is well known. Recently many more examples of such groups have appeared. † Partially supported by grant no. CRG 971543 from NATO. ‡ Supported by grant no. PB91-1068 from DGES. www.DownloadPaper.ir www.DownloadPaper.ir