Deformable registration, or nonlinear alignment of images, is a fundamental preprocessing tool in medical imaging. State-of-the-art algorithms restrict to diffeomorphisms to regularize an otherwise ill-posed problem. In particular, such models assume that a one-to-one matching exists between any pair of images. In a range of real-life-applications, however, one image may contain objects that another does not. In such cases, the one-to-one assumption is routinely accepted as unavoidable, leading to inaccurate preprocessing and, thus, inaccuracies in the subsequent analysis. We present a novel, piecewise-diffeomorphic deformation framework which models topological changes as explicitly encoded discontinuities in the deformation fields. We thus preserve the regularization properties of diffeomorphic models while locally avoiding their erroneous one-to-one assumption. The entire model is GPU-implemented, and validated on intersubject 3D registration of T1-weighted brain MRI. Qualitative and quantitative results show our ability to improve performance in pathological cases containing topological inconsistencies.