2008/12/26 by Helge Glöckner, Helge Glockner, Glockner, Helge
Mathematics · #22E65 #55P10 #55P42 #55Q05 (Secondary) #57N20 (Primary) 55N65 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.AT #math.GR #msc:22E65 #msc:55N65 #msc:55P10 #msc:55P42 #msc:55Q05 #msc:57N20
paper · pdf · doi:10.48550/arxiv.0812.4713
44 pages, LaTeX; v2: update of references
openalex publication_date 2008/12/26 · arxiv created 2010/07/02 · arxiv updated 2010/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a topological manifold modelled on topological vector spaces, which is the union of an ascending sequence of such manifolds Mn. We formulate a mild condition ensuring that the k-th homotopy group of M is the direct limit of the k-th homotopy groups of the steps Mn, for each non-negative integer k. This result is useful for Lie theory, because many important examples of infinite-dimensional Lie groups G can be expressed as ascending unions of finite- or infinite-dimensional Lie groups (whose homotopy groups may be easier to access). Information on the k-th homotopy groups of G, for k=0, k=1 and k=2, is needed to understand the Lie group extensions of G with abelian kernels. The above conclusion remains valid if the union of the steps Mn is merely dense in M (under suitable hypotheses). Also, ascending unions can be replaced by (possibly uncountable) directed unions.