Human brain dynamics are shaped by rare long-range connections over and above cortical geometry.

A fundamental topological principle is that the container always shapes the content. In neuroscience, this translates into how the brain anatomy shapes brain dynamics. From neuroanatomy, the topology of the mammalian brain can be approximated by local connectivity, accurately described by an exponential distance rule (EDR). The compact, folded geometry of the cortex is shaped by this local connectivity, and the geometric harmonic modes can reconstruct much of the functional dynamics. However, this ignores the fundamental role of the rare long-range (LR) cortical connections, crucial for improving information processing in the mammalian brain, but not captured by local cortical folding and geometry. Here, we show the superiority of harmonic modes combining rare LR connectivity with EDR (EDR+LR) in capturing functional dynamics (specifically LR functional connectivity and task-evoked brain activity) compared to geometry and EDR representations. Importantly, the orchestration of dynamics is carried out by a more efficient manifold made up of a low number of fundamental EDR+LR modes. Our results show the importance of rare LR connectivity for capturing the complexity of functional brain activity through a low-dimensional manifold shaped by fundamental EDR+LR modes.