The Stacey-Roberts Lemma for Banach Manifolds

Abstract

The Stacey-Roberts lemma states that a surjective submersion between finite-dimensional manifolds induces a submersion of smooth mappings on infinite-dimensional manifolds 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 that 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.