Stability of C ∞ Mappings: II. Infinitesimal Stability Implies Stability

Abstract

The purpose of this paper is to prove that a proper, infinitesimally stable mapping of one finite dimensional Co manifold into another is stable. This result is stated as Theorem 1 of ? 3. The method we use to show this result proves somewhat more; two things which it proves are stated as Theorems 2 and 3 in ? 3. The interest of this is the following. Stable mappings play an important role in the theory of singularities of differentiable mappings (cf. [8] and [9]). Therefore it is desirable to have methods for proving that particular mappings are stable. The result of this paper is a step towards such methods, since it is generally easier to prove that a mapping f is infinitesimally stable than it is to prove that f is stable. This paper is organized as follows. In ? 1, we set down our notation and terminology. All of this is quite standard except that manifolds have corners. However it seemed desirable to be explicit, since the precise meaning of the terminology varies from author to author. In ? 2, we introduce the topology W = WO, on the space of Co mappings of one manifold into another. The main results (? 3) are formulated in terms of this topology. In general, composition is not continuous with respect to the topology W. A major part of ? 2 is devoted to proving that various restrictions of the composition mapping are continuous. In ? 3, we state the main results. Note that Theorem 3 trivially implies Theorems 1 and 2. We will refer to the previous paper [5] in this series as I. In ? 4, we apply the results of I. Proposition 1 is essentially a restatement of the division theorem in terms of topologies introduced in ? 2. Proposition 2, on the other hand, introduces something new, in that it enlarges the set of functions that we can divide by. In ? 5, we interpret a result of Seeley in terms of the topologies that we