
Journal of Lie Theory 31 (2021), No. 3, 871884 Copyright Heldermann Verlag 2021 On Epimorphisms in some Categories of InfiniteDimensional 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 [Abstractpdf] 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\'echetLie 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 BanachLie 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, FrechetLie group, BanachLie group, Hausdorff topological group. MSC: 18A20, 22E65, 58D05. [ Fulltextpdf (143 KB)] for subscribers only. 