The K-theory type of quantum CW-complexes

The multipullback quantization of complex projective spaces lacks the naive 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. Better still, we construct a noncommutative counterpart of the cup product in K-theory, which is equivalent to its standard version in the classical case. To this end, we define k-topology, a noncommutative version of Grothendieck topology with covering families given by compact principal bundles and bases related by continuous maps, which leads to the much desired idea of multiplicative K-theory for noncommutative C*-algebras. Combining this with cw-Waldhausen structure on the category of compact quantum spaces, we arrive at the multiplicative K-theory type of finite weak quantum CW-complexes. We show that non-isomorphic quantizations of the standard CW-complex structure of a complex projective space enjoy the same multiplicative K-theory type admitting a noncommutative generalization of the Atiyah--Todd calculation of the K-theory ring in terms of truncated polynomials.