© 1999 by London Mathematical Society
Stratifications and Finite Determinacy
UMR 6632 du CNRS, Université de Provence 39 rue Joliot-Curie, Marseille 13453, France. E-mail: trotman{at}gyptis.univ-mrs.fr
Department of Mathematics, University of Hawaii 2565 The Mall, Honolulu, Hawaii 96822, U.S.A. E-mail: les{at}math.hawaii.edu
Received 10 March 1997. Revision received 9 December 1997.
In this paper we define a large class of regularity conditions on a stratified set, which generalise the (t) condition introduced by Thom in 1964. According to the first author's 1977 generalisation, a pair of adjacent strata (X, Y) is said to be (tk)-regular at yo in Y if for every Ck-submanifold S transverse to Y at y0 
Y 
S, there is a neighbourhood of yo in which S is transverse to X.
The very general microlocal conditions of this paper include (tk) for transversals of arbitrary hölderian continuity and differentiability classes. Special cases of our conditions are equivalent to Verdier's (w) condition, Whitney's (a) condition, and their relative versions (wf) and (af), respectively.
We analyse how the new conditions transform under pushforward and pullback by maps defining parametrised families of transversals to a stratum, and we deduce many consequences. In particular, whenever the pullback verifies (w), the Verdier isotopy theorem (of which we prove a strengthened version in 
4) implies stratified topological triviality of the family of transversals. This enables us to prove a conjecture made by the first author in 1981 about bounding the number of topological types of germs of transverse intersections with a subanalytic set, and also to give new and unified proofs, with stronger statements, of most of the known results in topological sufficiency theory for V-equivalence and right-equivalence. 1991 Mathematics Subject Classification: 32B20, 32S15, 58A35, 58C27.