K-theory of noncommutative Bieberbach manifolds

Abstract

We compute K-theory of noncommutative Bieberbach manifolds, which quotients of a three-dimensional noncommutative torus by a free action of a cyclic group Z N , N = 2, 3, 4, 6.

Real spectral triples over noncommutative Bieberbach manifolds

“…The classical limit of presented noncommutative geometries might have a place in the studies of models of flat compact space geometries in the context of cosmology [8]. The construction of different inequivalent spectral triples might suggest that there might be some differences between particle models built using them, in particular, twisting the obtained geometries with the projective modules, which represent nontrivial torsion in the K-theory of these objects [9].…”

“…Proof. In [9] we showed that the crossed product algebra A(T 3 θ ) ⋊ Z N is isomorphic to the matrix algebra of the noncommutative Bieberbach manifold algebra:…”

“…We have shown in [9] that the actions of the cyclic groups Z N , N = 2, 3, 4, 6 on the noncommutative three-torus, as given in the table 1 is free. The aim of this paper is to study and classify flat (i.e.…”

“…In this section we shall briefly recall the description of three-dimensional noncommutative Bieberbach manifolds as quotients of the three-dimensional noncommutative tori by the action of a finite discrete group. For details we refer to [9], here we present the notation and the results. Out of 10 different Bieberbach three-dimensional manifolds, (six orientable, including the three-torus itself and four nonorientable ones) only six have noncommutative counterparts.…”

“…Definition 1.1 (see [9]). Let A(T 3 θ ) be an algebra of smooth elements on a three-dimensional noncommutative torus, which contains the polynomial algebra generated by three unitaries U, V, W satisfying relations,…”