Noncommutative CW-complexes arising from crystallographic groups and their K-theory.

摘要

In this thesis we will construct a new class of examples of the so-called noncommutative CW-complexes (NCCW-complexes). First we show that if G is a finite group acting on a CW-complex X by homeomorphisms that permute the cells of the complex, then the crossed product C( X) ⋊ G is an NCCW-complex. This construction applies to the reduced C*-algebra of a group G = Zn⋊H , where H is a finite group and the induced action of H on Zn&d14; =Tn makes Tn into an H-CW-complex.; The second thing we give is a technique to systematically compute the K-theory of an n-dimensional NCCW-complex when n equals one or two. This is done using the abstract machinery of algebraic topology. In dimension ≥3 a spectral sequence in K-theory is derived, which converges to the K-theory of the NCCW-complex. Finally explicit computations are made with C*-algebras associated to crystallographic groups.