On stratifying 3-jets of functions

Abstract

The crucial step in the proof of the Topological Stability Theorem is the construction of a certain contact-invariant Whitney stratification of the sufficient jets in the jet space Jk(n, p). There is therefore a certain intrinsic interest in the construction of natural contact-invariant Whitney stratifications of the jet space. It has long been known that if there are only finitely many contact orbits then these provide a Whitney stratification, but in general there is the double problem of finding a candidate for a Whitney stratification, and then establishing that it has the desired properties. Virtually nothing is known about this problem, so it seems sensible to start by attempting to understand the first non-trivial examples which arise. Since functions are rather better understood than mappings it seems natural to try the case p = 1 first. The complexity of the problem now increases with k. When k = 1, 2 there are only finitely many contact orbits, so we start with k = 3. Having fixed p = 1, k = 3 we turn our attention to n. When n = 1, 2 we still obtain only finitely many contact orbits, as easy computations verify, so we start with n = 3. Thus our first example is J3(3, 1). It is a relatively easy matter to list the contact orbits in this case, and in so doing one is presented with an obvious candidate for a Whitney stratification. Our objective is to show that this candidate is indeed a Whitney stratification.