A relative Poincar\'e Theorem
Jean-Paul Brasselet (Institut de Math\'ematiques de Luminy, France)

This is a join work with Bernard Teissier.
The origin of the work is a question by Fran\c cois Treves :
Consider an analytic map $g:{\bf S }^n \to {\bf R}$ where ${\bf S}^n$
is the $n$-dimensional sphere, and a $r$-differential form $\omega$ of 
class ${\cal C}^{\infty}$ on ${\bf S}^n$ whose restriction to each 
non-singular fibre of $g$ is exact. Does there exist a H\"older
$(r-1)$-differential form $H$ on ${\bf S}^n$ such that
$dg \wedge (\omega - dH)=0$ ? Here $dH$ is considered as a distribution.
We show the result in the more general situation of proper sub-analytic maps
between non-singular spaces and for sub-analytic differential forms.
The proof uses entertaining results in combinatorics and concerning 
Whitney forms. The lecture will be explicit and full of pictures.