Classification of Singularities. Part II: The Local Ring of a Singularity

Abstract

The Thom-Boardman theory gives us a way of breaking up a map into simple constituent pieces; however, from the Thom-Boardman data alone we usually cannot reassemble the constituent pieces and see what the map itself looks like. Consider for example the maps

$$f:{R^2} \to R,\quad ({x_1},{x_2}{\text{)}} \mapsto x_1^2 + x_2^2$$

and

$$g:{R^2} \to R,\quad ({x_1},{x_2}{\text{)}} \mapsto x_1^2 - x_2^2$$

. f and g have isolated S₁ singularities at the origin and are regular everywhere else. However, their map germs at the origin are not equivalent, even under homeomorphisms of R² and R, since f has an extremum at 0 and g does not. From the Thom-Boardman data alone there is no way of computing the Hessian of f at 0; and, of course, it is the signature of the Hessian which distinguishes f from g. (See II, Theorem 6.9.)