The Stacey-Roberts Lemma for Banach Manifolds

TL;DR

This paper extends the Stacey-Roberts lemma from finite-dimensional manifolds to Banach manifolds with partitions of unity, correcting a previous proof error and enabling advanced constructions in infinite-dimensional differential geometry.

Contribution

It generalizes the Stacey-Roberts lemma to Banach manifolds with partitions of unity and corrects an earlier proof error in the finite-dimensional case.

Findings

The lemma is valid for Banach manifolds with partitions of unity.

The approach corrects a known proof error in finite-dimensional cases.

Facilitates the construction of Lie groupoids of smooth mappings.

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.