On Singularities of Mappings from the Plane into the Plane

Abstract

The emphasis in this chapter will be on the key paper by H. Whitney published in 1955, which should be regarded as a landmark in the development of the theory of singularities of mappings. In this paper, Whitney not only proved the remarkable theorem which I stated in the last chapter but also did something else which turned out to be very important in later developments in singularity theory. Namely, he found that information about the behavior of differentiable functions is contained in the values of its derivatives, and he was also able to formulate a very useful concept in this regard. Thus to extract information about a map f it makes sense to consider as a separate mathematical object a certain space, which will be called a Jet Space, which possesses as its points the values of the r^th order derivatives of a function, for some r. It was also in this paper that Whitney observed that the non-degeneracy criterion which the Morse function must satisfy is merely the condition that, when the first order partials all vanish, the second order partials will not lie in a certain proper algebraic subset of the jet space, referred to as the “bad” set, and defined by the vanishing of a finite set of polynomials.