2008/01/12 by Helge Glockner, Glockner, Helge
Mathematics · #22E65 (Primary) #58B10 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Group Theory (math.GR) #math.DG #math.GR #msc:22E65 #msc:58B10
paper · pdf · doi:10.48550/arxiv.0801.1919
23 pages
arxiv created 2008/01/12 · arxiv updated 2009/12/01
We solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb: (1) There exists a subgroup of some infinite-dimensional Lie group G which does not admit an initial Lie subgroup structure; (2) The pathology cannot occur if G is a direct limit of an ascending sequence of finite-dimensional Lie groups; (3) Every such direct limit group is a ``topological group with Lie algebra'' in the sense of Hofmann and Morris. Moreover, we prove a version of Borel's Theorem announced in the survey, ensuring the existence of compactly supported smooth diffeomorphisms with given Taylor series around a fixed point p (provided the tangent map at p has positive determinant).