Thom–Boardman singularities

Summary

The Thom–Boardman singularities sort the singular points of a smooth map by a sequence of numbers $(i_1, i_2, \ldots)$. The first number is the dimension of the kernel of the differential. The second is the same dimension for the restriction of the map to the set of points with that first number, and so on. Thom introduced the first steps, and Boardman made the construction work in general by defining the corresponding subsets of jet space. For a generic map the sets $\Sigma^{i_1, i_2, \ldots}(f)$ are submanifolds of known codimension, and they give a first coarse classification of singularities.