Noncommutative bundles over the multi-pullback quantum complex projective plane

We equip the multi-pullback $C^*$-algebra $C(S^5_H)$ 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 non-triviality 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 K\"unneth formula, we also compute the $K$-groups of $C(S^5_H)$.