Noncommutative Geometry and Topology in Prague

On differentiable Cuntz–Pimsner constructions for Hermitian line bimodules with connection

The Cuntz–Pimsner construction is ubiquitous and indispensable in current work on C^\ast-algebras and their noncommutative algebraic topology. Surprisingly (or not), its well-known application to topological quantum circle bundles admits the following formal distillate: the groupoid of all homomorphisms from \mathbb{Z} to a coherent 2-group G canonically deformation retracts onto G itself. Now, apply this proposition to the coherent 2-group of Hermitian line modules with bimodule connection on a unital pre-C^\ast-algebra with \ast-differential calculus. I hope to show that this one application permits the wholesale adaptation of the Cuntz–Pimsner construction with its structural implications to the noncommutative differential and Riemannian geometry of differentiable quantum circle bundles. If time permits, I’ll also sketch how these considerations imply the impossibility of coherently representing the q-monopole of Brzeziński–Majid in terms of spectral triples.