Mapping groups associated with real-valued function spaces and direct limits of Sobolev-Lie groups

Abstract

Let M be a compact smooth manifold of dimension m (without boundary) and G be a finite-dimensional Lie group, with Lie algebra g. Let H > m 2 (M, G) be the group of all mappings γ : M → G which are H s for some s > m 2 . We show that H > m 2 (M, G) can be made a regular Lie group in Milnor's sense, modelled on the Silva spaceas a Lie group (where H s (M, G) is the Hilbert-Lie group of all Gvalued H s -mappings on M ). We also explain how the (known) Lie group structure on H s (M, G) can be obtained as a special case of a general construction of Lie groups F(M, G), whenever function spaces F(U, R) on open subsets U ⊆ R m are given, subject to simple axioms.