The nuclear dimension of C^⁎-algebras associated to topological flows and orientable line foliations

We show that for any locally compact Hausdorff space Y with finite covering dimension and for any continuous flow R↷Y, the resulting crossed product C^⁎-algebra C₀(Y)⋊R has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. As an application, we obtain bounds for the nuclear dimension of C^⁎-algebras associated to one-dimensional orientable foliations. This result is analogous to the one we obtained earlier for non-free actions of Z. Some novel techniques in our proof include the use of a conditional expectation constructed from the inclusion of a clopen subgroupoid, as well as the introduction of what we call fiberwise groupoid coverings that help us build a link between foliation C^⁎-algebras and crossed products.