Pullbacks, C(X)-algebras, and their Cuntz semigroup

In this paper we analyse the structure of the Cuntz semigroup of certain C(X)-algebras, for compact spaces of low dimension, that have no K1-obstruction in their fibres in a strong sense. The techniques developed yield computations of the Cuntz semigroup of some surjective pullbacks of C⁎-algebras. As a consequence, this allows us to give a complete description, in terms of semigroup valued lower semicontinuous functions, of the Cuntz semigroup of C(X,A), where A is a not necessarily simple C⁎-algebra of stable rank one and vanishing K1 for each closed, two-sided ideal. We apply our results to study a variety of examples.