The recent development of quantum simulators using optical lattices and ultracold atoms have made it possible to model higher-dimensional quantum systems in the lab, by exploring so-called synthetic dimensions. This is of particular relevance to the investigation of new exotic phases of topological matter: for example, a four-dimensional (4D) analogue of the quantum Hall effect (QHE) has been effectively probed. The topology of 4D quantum systems features an integer, called the second Chern number, which is responsible for this effect in a similar way to how the "usual" first Chern number controls the two-dimensional QHE. In this talk, I will show how the second Chern number arises as an obstruction to the existence of orthonormal bases of localized Wannier functions in 4D crystals, and how relaxing the orthonormality constraint requiring only a Parseval frame removes this topological obstruction. The talk is based on ongoing joint work with Thaddeus Roussigné (ENS Paris Saclay).