THE MULTIPLICATIVE K-THEORY TYPE OF QUANTUM CW-COMPLEXES

We enhance current noncommutative methods in topology to distinguish some very different classical homotopy types (e.g., a disconnected and a connected one) of finite CW-complexes, which cannot be distinguished using the Kasparov KK-theory. We achieve this by constructing a noncommutative counterpart of the cup product in K-theory, which is equivalent to its standard version in the classical case. Noting that all vector bundles are associated with compact principal bundles and their tensor product leads to the cup product in K-theory, we  introduce the notion of k-topology, a noncommutative version of Grothendieck topology with covering families given by compact principal bundles and bases related by continuous maps. By the universal property of the pullback of compact principal bundles, this does not change the topology of compact Hausdorff spaces. As for noncommutative topology, it goes beyond the plain C*-algebraic setting. Combining this with cofibration-weakening-Waldhausen structure on the category of compact quantum spaces, we introduce the concept of multiplicative K-theory type. We show that non-isomorphic quantizations of the standard CW-complex structure of a complex projective space can have 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. (Based on joint work with F. D'Andrea, P. M. Hajac, A. Sheu, and B. Zieliński.)