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.