In this paper, the authors study two ”strong” topologies for spaces of smooth functions from a finite dimensional manifold \(M\) to a (possibly infinite-dimensional) manifold \(N\) modelled on a locally convex space.
We recall some of the topologies on spaces of smooth functions between finite-dimensional manifolds. For \(0 \leq r \leq \infty\), let \(C^r(M, N)\) denote the set of \(r\)-times continuously differentiable functions between manifolds \(M\) and \(N\). In the case where \(r\) is finite, the standard choice for a topology on \(C^r(M,N)\) is the well known Whitney \(C^r\)-topology. For \(r = \infty\) and \(M\) non-compact there are several choices for a suitable topology. One can for example choose the topology generated by the union of all Whitney \(C^r\)-topologies. We call this topology the strong \(C^{\infty}\)-topology on \(C^{\infty}(M,N)\). Note that each basic neighborhood of the strong \(C^{\infty}\)-topology allows one to control derivatives of functions only up to a fixed upper bound. However, in applications one wants to control the derivatives of up to arbitrary high order. To achieve this one has to refine the strong topology, obtaining the very strong \(C^{\infty}\)-topology in the process. The space of smooth functions with the very strong \(C^{\infty}\)-topology is fine enough for many questions arising from differential topology.
Unfortunately, as P. W. Michor has argued in his book [Manifolds of differentiable mappings. Shiva Mathematics Series, 3. Orpington, Kent (U.K.): Shiva Publishing Limited. (1980; Zbl 0433.58001)], this topology is still not fine enough, if one wants to obtain manifold structures on \(C^{\infty}(M,N)\) (and subsequently on the group of diffeomorphisms \(\text{Diff}(M\)). Hence Michor constructed a further refinement of the very strong \(C^{\infty}\)-topology, which he termed the \(\mathcal{FD}\)-topology, and which in the present paper is called the fine very strong \(C^{\infty}\)-topology on \(C^{\infty}(M,N)\).
If the source manifold \(M\) is compact all of the above topologies coincide with the compact-open \(C^{\infty}\)-topology. The compact-open \(C^{\infty}\)-topology for infinite-dimensional target manifolds is already well understood and has been used for example in infinite-dimensional Lie theory. Hence the investigations in this paper only give new results for non-compact source manifolds \(M\) and infinite-dimensional target manifolds \(N\).
The paper presents a careful and systematic treatment of the topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modeled on a locally convex space. In particular, the authors establish the continuity of certain mappings between spaces of smooth mappings, e.g. the continuity of the joint composition map. As an application they prove that the bisection group of an arbitrary Lie groupoid (with finite-dimensional base) is a topological group. The paper also includes a proof of the folklore fact that the Whitney topologies defined via jet bundles coincide with the ones defined via local charts.