For any finite-dimensional representation $V$ of a compact quantum group acting
freely on a unital $C^*$-algebra $A$, we can form an associated finitely-generated
projective module $A_V$ over the fixed-point subalgebra $B$ for this action. The
module $A_V$ is the section module of the associated noncommutative vector
bundle. Given an equivariant $C^*$-homomorphism $f$ from $A$ to $A'$, we get the
induced K-theory map $f*$ from $K_0(B)$ to $K_0(B')$, where $B'$ is the fixed-point
subalgebra of $A'$. Using Chern-Galois theory, we show that $f*([A_V])=[A'_V]$.
As
an application, we combine this formula with higher-rank graph $C^*$-algebra
technology and index pairing computations to prove that the noncommutative
line bundles associated via the diagonal U(1)-action on the multi-pullback
quantum odd-dimensional spheres are pairwise stably non-isomorphic. In
particular, we conclude that the tautological line bundles over the
multi-pullback quantum complex projective spaces are stably non-trivial. The
same reasoning and conclusions hold for noncommutative line bundles
associated to Vaksman-Soibelman quantum spheres. (Partly based on joint work
with David Pask, Aidan Sims and Bartosz Zieliński.)