Stable map

Summary

A smooth map between smooth manifolds is stable if every map close enough to it is equivalent to it. Here equivalent means equal after a change of coordinates in the source and in the target. Small perturbations do not change its singularities in kind, only move them a little. Stable maps are the maps whose singularities survive in nature and in experiments, which is why singularity theory aims to classify them. Mather proved that stability is equivalent to a linear condition, infinitesimal stability, for proper maps.