Strong topologies for spaces of smooth maps with infinite-dimensional target

Abstract

In this article we study two "strong" topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modeled on a locally convex space. Namely, we construct Whitney type topologies for these spaces and a certain refinement corresponding to Michor's $\mathcal{F}\mathcal{D}$-topology. Then we establish the continuity of certain mappings between spaces of smooth mappings, e.g.\ the continuity of the joint composition map. As a first application we prove that the bisection group of an arbitrary Lie groupoid (with finite-dimensional base) is a topological group (with respect to these topologies). For the reader's convenience the article includes also a proof of the folklore fact that the Whitney topologies defined via jet bundles coincide with the ones defined via local charts.

45 pages, v3: corrected several typos and added some details, results remain unchanged

References in corpus (1)

• (Re)constructing Lie groupoids from their bisections and applications to prequantisation

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