Introduction

The idea that stratified vector fields controlled by tubular neighbourhoods of the strata are integrable is due to René Thom [11], more detail having been added by John Mather [3] and Klaus Wirthmüller [2, Chapter II]. The construction of such stratified vector fields by lifting a smooth vector field over a mapping submersive on strata is also treated in the articles cited; our contribution here is to show that the construction can be made to yield continuous controlled vector fields in the case where the stratification is C-regular in the sense of Karim Bekka [1], so in particular (see [1, p.52, Remarques 5]) when it is Whitney regular.

The result in the case of a Whitney regular stratification is announced by Masahiro Shiota in [8], where the result in the case where there are just two strata is proved. However, the extension to the general case, which is the most delicate construction in this article, is dismissed as trivial there. Shiota has, very recently, offered an easy construction in the general case, in his book [9, pp.10–11]; unfortunately, his construction is not sufficient to ensure the required continuity.

Continuous lifts of vector fields had previously been constructed for stratifications satisfying stronger regularity conditions. Jean-Louis Verdier [14] found “rugose” lifts for his notion of W-regular stratification, while Adam Parusinski [6] and Tadeusz Mostowski [4] discussed regularity conditions allowing construction of Lipschitz lifts.