Transversality in families of mappings

Abstract

By a classical result of Thom [20], if N and P are smooth manifolds, and Q is a smooth submanifold of P , the set of maps f : N → P transverse to Q is residual in C∞(N,P ) (if Q is a closed subset of P , the set of transverse maps is also open). However if U is a further smooth manifold, and {fu} a family of smooth mappings, we cannot expect to be able to deform the family to make each fu transverse to Q. This situation was considered by Bruce [2], who showed that, for a residual set of maps F : N × P → U , we have (a) for each u ∈ U , the set of points in N at which fu is not transverse to Q is discrete; and (b) at each such point, the class of the contact of fu(N) with Q has finite codimension. (This is the result he established: the formal statement given was more explicit but weaker.) This result is not entirely satisfactory: we may hope also, for example, that each contact class presented is universally unfolded in the family. We address this problem, and show that this hope is justified under a hypothesis of ‘nice dimensions’. After recalling Mather’s theory of topological stability, we establish a result valid in all dimensions with ‘topologically versal’ unfoldings. Since the proofs of these results involve numerous translations between equivalent versality conditions, we begin with a section developing a notation and approach to facilitate such proofs. We then consider the corresponding question with the submanifold Q replaced by a stratified subset of P . This requires an extensive study of theories of contact equivalence relative to a subset of the target. Good algebraic conditions and criteria are only known under an analyticity hypothesis; for some results this needs to be strengthened to a holonomic condition. With these preliminaries, we obtain a parallel to the theory of the earlier section. However a full account of a topological analogue remains beyond our reach.