Whitney functions determine the real homotopy type of a semi-analytic set

In this paper, we investigate the Whitney--de Rham complex $\Omega^\bullet_\text{W} (X)$ associated to a semi-analytic subset $X$ of an analytic manifold $M$. This complex is a commutative differential graded algebra, that is defined to be the quotient of the de Rham complex of smooth differential forms on $M$ by the differential graded ideal generated by all smooth functions which are flat on $X$. We use Hironaka's desingularization theorem to prove a Poincar\'e Lemma for $\Omega^\bullet_\text{W} (X)$ holds true, which entails that its cohomology is isomorphic to the real cohomology of $X$. Furthermore, we show that this isomorphism is induced by a quasi-isomorphism of differential graded algebras. Thus it preserves the product structure, and is therefore an isomorphism of commutative differential graded algebras. As a consequence we show, when $X$ is simply connected, that the Whitney--de Rham complex determines the real homotopy type of $X$. This allows one further to conclude that the Hochschild homology of the differential graded algebra $\Omega^\bullet_\text{W} (X)$ is isomorphic to the cohomology of the free loop space $\mathcal{L} X$.