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

TL;DR

This paper develops and compares strong Whitney-type topologies for spaces of smooth maps into possibly infinite-dimensional manifolds, proving their properties and applications such as the topological group structure of bisection groups.

Contribution

It introduces refined Whitney and $ ext{FD}$-topologies for smooth map spaces and demonstrates their continuity properties and applications to Lie groupoids.

Findings

Established continuity of composition maps in these topologies.

Proved the bisection group of a Lie groupoid is a topological group.

Confirmed Whitney topologies via jet bundles coincide with those via local charts.

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 $\mathcal{FD}$-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.