We equip the multi-pullback C*-algebra C(S⁵_H) of a noncommutative deformation of the 5-sphere with a free U(1)-action, and show that its fixed-point subalgebra is isomorphic with the C*-algebra of the multi-pullback quantum complex projective plane. Our main result is the stable nontriviality of the dual tautological line bundle associated to the action. We prove it by combining Chern-Galois theory with the Milnor connecting homomorphism in K-theory. Using the Mayer-Vietoris six-term exact sequences and the functoriality of the Künneth formula, we also compute the K-groups of C(S⁵_H).

This work was partially supported by NCN grant 2012/06/M/ST1/00169.