If a quantum space X can be obtained by “gluing” two subspaces Y and Z, there is a Mayer-Vietoris sequence relating the K-theory of X to that of Y and Z. This is particularly useful for quantum spaces with a CW complex structure, such as the projective spaces coming from Vaksman-Soibelman quantum spheres, since it allows to get generators of K-theory iteratively from lower dimension. In this talk I will explain how to adapt this idea to multipullback quantum projective spaces, that have some kind of CW complex structure but only “up to homotopy”. Finally, I will show how, using an equivariant weak-isomorphism from Vaksman-Soibelman to multipullback quantum spheres, one can construct generators of K_0 of the latter represented by equivariant vector bundles (projective modules). Based on a joint work with P. Hajac, T. Maszczyk, A. Sheu, B. Zielinski.

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