Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 31 (2021), No. 3, 871--884Copyright Heldermann Verlag 2021 On Epimorphisms in some Categories of Infinite-Dimensional Lie Groups Vladimir G. Pestov Dep. de Matemática, Universidade Federal de Santa Catarina, Trindade, Florianópolis, Brazil and: Dept of Mathematics and Statistics, University of Ottawa, Ontario, Canada vladimir.pestov@uottawa.ca Vladimir V. Uspenskij Dept. of Mathematics, Ohio University, Athens, Ohio 45701, U.S.A. uspenski@ohio.edu [Abstract-pdf] Let $X$ be a smooth compact connected manifold. Let $G=\text{Diff}\,X$ be the group of diffeomorphisms of $X$, equipped with the $C^\infty$-topology, and let $H$ be the stabilizer of some point in $X$. Then the inclusion $H\to G$, which is a morphism of two regular Fr\'echet-Lie groups, is an epimorphism in the category of smooth Lie groups modelled on complete locally convex spaces. At the same time, in the latter category, epimorphisms between finite dimensional Lie groups have dense range. We also prove that if $G$ is a Banach-Lie group and $H$ is a proper closed subgroup, the inclusion $H\to G$ is not an epimorphism in the category of Hausdorff topological groups. Keywords: Epimorphism, locally convex Lie group, Frechet-Lie group, Banach-Lie group, Hausdorff topological group. MSC: 18A20, 22E65, 58D05. [ Fulltext-pdf  (143  KB)] for subscribers only.