Formes de Whitney et primitives relatives de formes diff\'erentielles sous-analytiques

Let $X$ be a real-analytic manifold and $g\text{\hspace{0.17em}⁣}:X\to {\mathbf{R}}^n$ a proper triangulable subanalytic map. Given a subanalytic $r$-form $\omega$ on $X$ whose pull-back to every non singular fiber of $g$ is exact, we show tha $\omega$ has a relative primitive: there is a subanalytic $(r-1)$-form $\mathrm{\Omega }$ such that $dg\mathrm{\Lambda }(\omega -d\mathrm{\Omega })=0$. The proof uses a subanalytic triangulation to translate the problem in terms of “relative Whitney forms” associated to prisms. Using the combinatorics of Whitney forms, we show that the result ultimately follows from the subanaliticity of solutions of a special linear partial differential equation. The work was inspired by a question of Fran\c{c}ois Treves.