Diffeology and Arithmetic of Irrational Tori
Abstract
The irrational torus, , originally introduced as a geometric model for quasicrystals, is a foundational object in the theory of diffeology. This paper, after recalling its main algebraic properties, provides a comprehensive analysis of a new geometric invariant for this singular space: the group of flows, . This invariant, which is trivial for all manifolds, arises as the core of the obstruction to the de Rham theorem in the diffeological setting. We provide a complete computation and geometric interpretation of this group, proving the isomorphism .
Keywords
Cite
@article{arxiv.2508.07460,
title = {Diffeology and Arithmetic of Irrational Tori},
author = {Patrick Iglesias-Zemmour},
journal= {arXiv preprint arXiv:2508.07460},
year = {2026}
}
Comments
25 pages, 2 figures. This paper introduces a new geometric invariant, the group of flows, which is specific to diffeology as it is identically trivial for all manifolds. We provide the complete theory and a detailed application to the irrational torus, revealing a deep connection to arithmetic (fixed two misprints an improved a piece of proof)