I will discuss a non-trivial twisting in the global patching of local supersymmetric solutions in the context of M-theory and Heterotic String Theory. In particular, I will apply the previous twisting to M-theory compactifications to three-dimensional Minkowski space-time preserving N=2 supersymmetries, showing that the standard supersymmetry analysis allows for topologies different than the Calabi-Yau one.