Stability of transversality to a stratification implies Whitney (a)-regularity

Abstract

In this note we show that Whitney (a)-regularity is precisely the condition to impose on a stratification in order that the maps transverse to the strata form an open set, i.e. that transversality be stable, as well as being generic (the transverse maps always form a dense set). (a)-regularity was introduced by Whitney in [ 15] as a sufficient condition for this to be true; at the time it was thought that (t)regularity (defined in [10]) was the condition required, and that (a) was only useful in that it implied (t) (see the introduction to [15]). This is true in the analytic case, since then (t) and (a) are equivalent (proved in [10]), however we have given examples (see [10, 11] and [12]) showing that (t) is in general weaker than (a). In fact (a) is both sufficient and necessary for openness: sufficiency was proved in detail by E.A. Feldman in [3] and we prove necessity here. The only difficulty in the proof is to find a transverse map with a given 1-jet at a given point: for this we choose a nonempty weakly closed subspace of the space of maps containing the given jet at the given point, then quote the result of [6] and [5] that any weakly closed subspace of the space of smooth maps is a Baire space in the strong topology, and finally use the standard arguments to prove that, in this chosen Baire subspace, transverse maps are dense. This trick should have applications elsewhere.