Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 037, 20 pages      arXiv:2411.00587      https://doi.org/10.3842/SIGMA.2025.037

The Stacey-Roberts Lemma for Banach Manifolds

Peter Kristel a and Alexander Schmeding b
a) Cyberagentur, Große Steinstraße 19, 06108 Halle (Saale), Germany
b) NTNU Trondheim, Alfred Getz' vei 1, Trondheim, Norway

Received November 27, 2024, in final form May 07, 2025; Published online May 18, 2025

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.

Key words: manifold of mappings; submersion; connection; Stacey-Roberts lemma; spray; anchored Banach bundle; Banach manifold.

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