A geometrisation of N-manifolds

This paper proposes a geometrisation of N-manifolds of degree n as n-fold vector bundles equipped with a (signed) Sn-symmetry. More precisely, it proves an equivalence between the categories of [n]-manifolds and the category of (signed) symmetric n-fold vector bundles, by finding that symmetric n-fold vector bundle cocycles and [n]-manifold cocycles are identical. This extends the already known equivalences of [1]-manifolds with vector bundles, and of [2]-manifolds with involutive double vector bundles, where the involution is understood as an S2-action. (c) 2024 The Authors. Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).