Transversality in families of mappings

If N and P are smooth manifolds, and Q is a smooth submanifold of P, the set of maps of N to P transverse to Q is residual. However given a family F of smooth mappings, we cannot expect to be able to deform the family to make each member of the family transverse to Q. The development of generic transversality conditions runs in parallel with that of stability theory. We develop a convenient notation, and establish equivalence of different transversality conditions. The first main result expresses equivalence between F inducing a versal unfolding of any multi‐germ, multi‐transversality of F to contact orbits relative to projection on the parameter space U, and local stability of the projection to U of the pre‐image of Q. For global stability we need Q to be closed, the deformation F to be proper, and Q or N compact. Corresponding results also hold for topological stability in the Thom–Mather sense, and an appropriate notion of topological versality. To obtain results with Q a stratified subset of P requires an extensive study of theories of contact equivalence relative to a subset of the target. Here we need an analyticity hypothesis on Q; for deeper study of finite determinacy, we need a holonomic condition; even so, the theory is much more complicated. In general, there are properties corresponding to the former ones, but many are weaker. Our second main result, a direct analogue of the first, is only obtained under a hypotheses that F has property (G) at all points and is transverse to Q. In the topological case, while some results can be obtained, the failure of direct analogues to some basic tools is a bar to further progress.