I will speak about joint work in progress with Ian Putnam and Karen Strung. The goal of the project is to study the existence of minimal dynamical systems and more generally minimal equivalence relations. In particular, I will discuss the following question: given a compact Hausdorff space does there exist a minimal homeomorphism on it? Although the answer is no, a similar question has a positive answer for any finite CW-complex and this leads to a more systematic approach to the construction of minimal homeomorphisms with prescribed K-theoretic properties.