The Stacey-Roberts Lemma for Banach Manifolds
Peter Kristel, Alexander Schmeding
TL;DR
This work extends the Stacey–Roberts lemma to Banach manifolds by proving that for a smooth surjective submersion $p:M o N$ between Banach manifolds with smooth partitions of unity, the pushforward $p_*: C^ ty(X,M) o C^ ty(X,N)$ is a submersion for any $ ext{$ ext{sigma}$-compact}$ source $X$ in the Bastiani calculus. The authors develop a robust geometric framework based on sprays on anchored Banach bundles and Ehresmann connections to construct local additions compatible with submersions, enabling submersion charts for $p_*$ and a complete, gap-free proof. This generalization broadens the applicability of the Stacey–Roberts lemma to Banach-manifold targets, underpinning constructions such as Lie groupoids of mappings and mapping stacks in infinite dimensions, and provides a systematic methodology for transferring submersion properties to spaces of smooth mappings. The result has significant implications for infinite-dimensional differential geometry, geometric hydrodynamics, shape analysis, and higher-categorical differential geometry by enabling robust transfer of submersion properties to mapping spaces.
Abstract
The Stacey-Roberts lemma states that a surjective submersion between finite-dimensional manifolds gives rise to a submersion on infinite-dimensional manifolds of smooth mappings by pushforward. This result is foundational for many constructions in infinite-dimensional differential geometry such as the construction of Lie groupoids of smooth mappings. We generalise the Stacey-Roberts lemma to Banach manifolds which admit smooth partitions of unity.The new approach also remedies an error in the original proof of the result for the purely finite-dimensional setting.