The Stacey–Roberts Lemma for Banach Manifolds

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.