Noncommutative flat manifolds in 2 and 3 dimensions

Speaker: 
Anndrzej Sitarz, Jagiellonain University, Krakov

Abstract: 
Flat compact manifolds are usually defined as quotient manifolds of tori by an action of a finite group. Clasically, in 2 dimensions, there are only 2 such objects (torus and Klein bottle) and in 3 dimensions there are 10 (6 of them orientable). 
Passing to noncommutative geometry with the noncommutative torus as the basic flat compact manifold one can study 4 nontrivial orientable noncommutative flat manifolds in 3 dimensions. However, it appears that there are (possibly) more 2-dimensional flat manifolds. This is illustrated by an example of the "noncommutative pillow with 4 courners" (quotient of the noncommutative torus by an action of Z/2Z group), which is more regular (not an orbifold) in the noncommutative case.