zbMATH — the first resource for mathematics

Semidifferentiability and smooth version of the fibration conjecture of Whitney. (SemidiffĂ©rentiabilitĂ© et version lisse de la conjecture de fibration de Whitney.) (French. English summary) Zbl 1128.58006
Izumiya, Shyuichi (ed.) et al., Singularity theory and its applications. Papers from the 12th MSJ International Research Institute of the Mathematical Society of Japan, Sapporo, Japan, September 16–25, 2003. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-32-7/hbk). Advanced Studies in Pure Mathematics 43, 271-309 (2006).
A stratification of a manifold \(M\) is an expression of \(M\) as the disjoint union of a locally finite set of connected analytic manifolds, called strata, such that the frontier of each stratum is the union of a set of lower-dimensional strata. Let \(\chi\) and \(\widetilde\chi\) be two stratified spaces and let \({\mathcal F}\) be a foliation. For controlled stratified maps \(f:\chi\to\widetilde\chi\) the authors define semi-differentiable, horizontally-\(C^ 1\), and \({\mathcal F}\)-semi-differentiable maps. If \(\widetilde\chi\) is a smooth manifold, then \(f\) is always semi-differentiable. It is shown that semi-differentiability of \(f\) is equivalent to \(f\) being horizontally-\(C^ 1\) with bounded differential. H. Whitney [Differ. and Combinat. Topology, Sympos. Marston Morse, Princeton, 205–244 (1965; Zbl 0129.39402)] conjectured that semi-analytic fibrations do exist for suitable stratifications. In this paper, the authors show that horizontally-\(C^ 1\) regularity depends on the existence of \((\alpha)\)-regular horizontal stratified foliations of \(\chi\) and \(\widetilde\chi\), which gives a smooth version of the stratified fibration proposed by Whitney for analytic manifolds. It also implies a horizontally-\(C^ 1\) version of Thorn’s first isotopy theorem.
For the entire collection see [Zbl 1110.32001].

58A35 Stratified sets
58A30 Vector distributions (subbundles of the tangent bundles)
57R30 Foliations in differential topology; geometric theory
57R52 Isotopy in differential topology