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Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds. (English) Zbl 1143.22016
The authors study Lie group structures on groups of the form $$C^{\infty}(M,K)$$, where $$M$$ is a non-compact smooth manifold and $$K$$ is a possibly infinite-dimensional Lie group. The main results of the paper are the following: Theorem A. Let $$K$$ be a connected regular real Lie group and $$M$$ a real finite-dimensional connected manifold. Then the group $$C^{\infty}(M,K)$$ carries a Lie group structure compatible with evaluations in the following cases: (1) The universal covering of $$K$$ is diffeomorphic to a locally convex space. If, in addition, $$\pi_1(M)$$ is finitely generated, then the Lie group structure is compatible with the smooth compact open topology. (2) $$M$$ has dimension one. (3) $$M\equiv \mathbb{R}^k\times C$$, where $$C$$ is compact. Theorem B. Let $$K$$ be a regular complex Lie group and $$M$$ a finite-dimensional connected complex manifold without boundary. Then the group of all holomorphic maps on $$M$$ with values in $$K$$ carries a Lie group structure with Lie algebra compatible with evaluations in the following cases: (1) The universal covering of $$K$$ is diffeomorphic to a locally convex space. If, in addition, $$\pi_1(M)$$ is finitely generated, then the Lie group structure is compatible with the compact open topology. (2) $$M$$ has complex dimension one, $$\pi_1(M)$$ is finitely generated, and $$K$$ is a Banach-Lie group.

MSC:
 2.2e+66 Infinite-dimensional Lie groups and their Lie algebras: general properties 2.2e+68 Loop groups and related constructions, group-theoretic treatment 2.2e+16 General properties and structure of real Lie groups 2.2e+31 Analysis on real and complex Lie groups
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