Abstract
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.
Information
Digital Object Identifier: 10.2969/aspm/04310271
