Smooth geometry of the noncommutative pillow, cones and lens spaces

TL;DR

This paper introduces a new concept of differential smoothness for algebras, demonstrating that various noncommutative spaces, including orbifolds and deformations, are smooth in this sense, and explores their Riemannian properties.

Contribution

It defines differential smoothness combining top forms and Poincaré duality, and applies it to noncommutative geometries like pillow orbifolds and cones, showing their smoothness and Riemannian structures.

Findings

Quantum spheres and tori are differentially smooth.

Noncommutative orbifolds like pillow and lens spaces are also smooth.

Spectral triples satisfy orientability, unlike classical orbifolds.

Abstract

This paper proposes a new notion of smoothness of algebras, termed differential smoothness, that combines the existence of a top form in a differential calculus over an algebra together with a strong version of the Poincar\'e duality realized as an isomorphism between complexes of differential and integral forms. The quantum two- and three-spheres, disc, plane and the noncommutative torus are all smooth in this sense. Noncommutative coordinate algebras of deformations of several examples of classical orbifolds such as the pillow orbifold, singular cones and lens spaces are also differentially smooth. Although surprising this is not fully unexpected as these algebras are known to be homologically smooth. The study of Riemannian aspects of the noncommutative pillow and Moyal deformations of cones leads to spectral triples that satisfy the orientability condition that is known to be broken for classical orbifolds.