Diffeology and Arithmetics of Irrational Tori
April 1, 2025
 
We present the classifying group Fl(X,R) for(R,+)princi-pal bundles over a diffeological space X. As it is trivial for manifolds, this invariant is specific to diffeology. It reveals intricate structure within thediffeology of X. As an example, we compute Fl(Tα,R)for the irrational torus. We show how it distinguishes whether α is Diophantine or not, demonstrating a direct link between the arithmetic properties of α and the geometry of Tα.
Submitted for publication.http://math.huji.ac.il/~piz/documents/DAAOIT.pdfshapeimage_3_link_0