Diffeology and Arithmetic of Irrational Tori

TL;DR

This paper studies the irrational torus within diffeology, introducing a new geometric invariant called the group of flows, and provides a complete computation and interpretation of this invariant, revealing its role as an obstruction to the de Rham theorem.

Contribution

It introduces and computes the group of flows as a new geometric invariant for irrational tori, linking it to the cokernel of a specific operator and clarifying its geometric significance.

Findings

The group of flows for irrational tori is isomorphic to R times the cokernel of Δ_α.

This invariant measures the obstruction to the de Rham theorem in the diffeological setting.

Complete algebraic and geometric characterization of the group of flows for irrational tori.

Abstract

The irrational torus, $\mathrm{T}_\alpha$, 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, $\mathbf{Fl}(\mathrm{T}_\alpha, \mathbf{R})$. 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 $\mathbf{Fl}(\mathrm{T}_\alpha, \mathbf{R}) \simeq \mathbf{R} \times \mathrm{coker}(\Delta_\alpha)$.