STRATIFICATIONS AND FINITE DETERMINACY

Abstract

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 $(t^k )$-regular at $y_o$ in $Y$ if for every $C^k$-submanifold $S$ transverse to $Y$ at $y_o \in Y \cap S$, there is a neighbourhood of $y_o$ in which $S$ is transverse to $X$.

The very general microlocal conditions of this paper include $(t^k)$ for transversals of arbitrary h\‘olderian continuity and differentiability classes. Special cases of our conditions are equivalent to Verdier's $(w)$ condition, Whitney's $(a)$ condition, and their relative versions $(w_f)$ and $(a_f)$, 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 \S 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.