It is known that a momentum space Chern class obstructs existence of good tight-binding models in position space. This is a T-duality, and holds in more general geometric contexts, e.g. crystallographic, defective, and non-Euclidean effective interacting topological phases. Namely, good Wannier bases correspond to free modules over pre-C*-algebras of the symmetry group. Thus K-theory and T-duality give precise tools to find topological insulators and understand their boundary gapless modes; explicit new examples will be given. Joint with M. Ludewig.