Mapping Groups Associated with Real-Valued Function Spaces and Direct Limits of Sob olev-Lie Groups

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>2m (M, G) be the group of all mappings 7: M-+ G which are Hs for some s > m2 . We show that H>m2 (M, G) can be made a regular Lie group in Milnor's sense, modelled on the Silva spaceH> m2 (M, g) := lim Hs(M, g), -? s> m such that 2 H> m2 (M, G) = lim Hs(M, G) -? s>m 2 as a Lie group (where Hs(M, G) is the Hilbert-Lie group of all G-valued Hs-mappings on M). We also explain how the (known) Lie group structure on Hs(M, G) can be obtained as a special case of a general construction of Lie groups F(M, G) whenever function spaces F(U, 1[8) on open subsets U C 1[8m are given, subject to simple axioms. Mathematics Subject Classification: Primary 22E65; Secondary 22E67, 46A13, 46E35, 46M40. Key Words: Sobolev space, Banach space-valued section functor, mapping group, direct limit, pushforward, superposition operator, Nemytskij operator.