\(K\)-theory of noncommutative Bieberbach manifolds (Q891369)

Bieberbach manifold is the quotient space \(\mathcal{B}:=\mathbb T^3/G\) of the three-dimensional torus by a finite group freely acting on it by isometries and the fundamental group of \(\mathcal{B}\) is a crystallographic group, i.e., a particular discrete subgroup of isometries group from the Euclidean space onto itself. First of all, the authors recall that there are ten kind of explicit (non)orientables \(\mathcal{B}\) (Tables 1 and 2). Then, the authors state eight explicit \(K\)-theory groups of some appropriate noncommutative Bieberbach manifolds (Propositions 3.9,3.12,3.15, and 3.18) and their proofs are essentially based on using Pimsner-Voiculescu exact sequences allowing to compute \(K\)-groups.