Nontrivial Deformation of a Trivial Bundle

The ${\rm SU}(2)$-prolongation of the Hopf fibration $S^3\to S^2$ is a trivializable principal ${\rm SU}(2)$-bundle. We present a noncommutative deformation of this bundle to a quantum principal ${\rm SU}_q(2)$-bundle that is not trivializable. On the other hand, we show that the ${\rm SU}_q(2)$-bundle is piecewise trivializable with respect to the closed covering of $S^2$ by two hemispheres intersecting at the equator.