We find multipullback quantum odd-dimensional spheres equipped with natural U.1/-actions that yield the multipullback quantum complex projective spaces constructed from Toeplitz cubes as noncommutative quotients. We prove that the noncommutative line bundles associated to multipullback quantum odd spheres are pairwise stably non-isomorphic, and that the K-groups of multipullback quantum complex projective spaces and odd spheres coincide with their classical counterparts.We show that theseK-groups remain the same for more general twisted versions of our quantum odd spheres and complex projective spaces.
Hajac, P. M., Nest, R., Pask, D., Sims, A. & Zielinski, B. (2018). The K-theory of twisted multipullback quantum odd spheres and complex projective spaces. Journal of Noncommutative Geometry, 12 (3), 823-863.