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VOL. 43 | 2006 Semidifférentiabilité et version lisse de la conjecture de fibration de Whitney
C. Murolo, D. J. A. Trotman

Editor(s) Shyuichi Izumiya, Goo Ishikawa, Hiroo Tokunaga, Ichiro Shimada, Takasi Sano


For controlled stratified maps $f : \mathcal{X} \to \mathcal{X}'$ between two stratified spaces, we define what it means for $f$ to be semi-differentiable, horizontally-$C^1$ and $\mathcal{F}$-semi-differentiable (where $\mathcal{F}$ is a foliation).

When $\mathcal{X}'$ is a smooth manifold, $f$ is always semi-differentiable.

In general, semi-differentiability is equivalent to $f$ being horizontally-$C^1$ with bounded differential.

Horizontally-$C^1$ regularity depends on the existence of $(a)$-regular horizontal stratified foliations of $\mathcal{X}$ and $\mathcal{X}'$, which gives a smooth version of the stratified fibration whose existence was conjectured by Whitney for analytic varieties in 1965, and implies a horizontally-$C^1$ version of Thom's first isotopy theorem.

We obtain finally the corresponding theorems for the finer property of $\mathcal{F}$-semi-differentiability.


Published: 1 January 2006
First available in Project Euclid: 3 January 2019

zbMATH: 1128.58006
MathSciNet: MR2325142

Digital Object Identifier: 10.2969/aspm/04310271

Primary: 58A30, 58A35
Secondary: 57R30, 57R52

Rights: Copyright © 2006 Mathematical Society of Japan


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