Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories

Abstract

Let G be a Lie group which is the union of an ascending sequence G 1 ⊆ G 2 ⊆ ⋯ of Lie groups (all of which may be infinite-dimensional). We study the question when G = → lim G n in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group Diff c ( M ) of compactly supported C ∞ -diffeomorphisms of a σ-compact smooth manifold M; and for test function groups C c ∞ ( M , H ) of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.