THE K-THEORY TYPE OF QUANTUM CW-COMPLEXES

The CW-complex structure of topological spaces not only reveals how they are built, but also is a natural tool to compute and unravel their K-theory. Therefore, it is desirable to define a noncommutative version of the CW-complex that would play a similar role in noncommutative topology. From some quantizations of CW-complexes, we can extract a straightforward concept of a finite quantum CW-complex. However, there are also important examples of quantizations of CW-complexes that lack the naive (strict) quantum CW-complex structure because the quantization of an embedding of the n-skeleton into the (n+1)-skeleton does not exist. To overcome this difficulty, we introduce the framework of cw-Waldhausen categories, which includes the concept of weak equivalences leading to the notion of a finite weak quantum CW-complex in the realm of unital C*-algebras. Here weak equivalences are unital *-homomorphisms that induce an isomorphism on K-theory. By design, weak quantum CW-complex structures exist in important examples lacking a strict quantum CW-complex structure, and are equally good to compute K-theory as their strict counterparts. In this talk, I will mention many natural examples instantiating our new framework. In particular, I will discuss the Vaksman-Soibelman and multipullback quantizations of complex projective spaces as strict and weak, respectively, quantum CW-complexes of the same K-theory type, and show that our framework leads to extending the Atiyah-Todd description of the K-theory of complex projective spaces to the noncommutative setting. (Based on joint work with F. D'Andrea, T. Maszczyk, A. Sheu, and B. Zieliński.)