Classification and equivalences of noncommutative tori and quantum lens spaces

Abstract

In noncommutative geometry, one studies abstract spaces through their, possibly noncommutative, algebras of continuous functions. Through these function algebras, and certain operators interacting with them, one can derive much geometrical information of the underlying space, even though this space does not exist as a classical topological space if the algebra is noncommutative. In this thesis the classification of certain geometrical structures, spin structures, is extended to several kinds of noncommutative spaces: noncommutative tori and quantum lens spaces. It is found that this classification yields very similar results to the classical geometrical classification, even though the methods are completely different. Also, equivalences between these spaces and spinstructures are considered. It is proven that a certain proposed equivalence, Morita equivalence of spectral triples, is really an equivalence in the case of noncommutative tori. In general, it is not known when a Morita equivalence of spectral triples is really an equivalence relation.