Most mammalian species present topological maps of preference for the ori- entation of edges in their early visual cortex. However, despite the evidence that evolution has favoured topological map organisation in most species, a clear explanation for the functional role of such maps is lacking. Traditional approaches based on wiring minimisation are incomplete, showing how topology can reduce metabolic costs, but only after assuming a functional role of short- ranged like-tuned connectivity. In this work, we claim that topological maps are for reducing the complexity of the propagated information in a robust and wiring-efficient way through dimensionality reduction. In particular, we use com- putational approaches based on activity dependant synaptic plasticity to show that topological maps reflect the wiring minimisation of a component of lateral connectivity, short-range excitation, that reduces the effective dimensionality and increases the noise-robustness of neural codes, in line with expectations from effi- cient coding and the free energy principle. We also show how this idea can explain phenomena observed in the visual cortex of animal species