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

TL;DR

This paper demonstrates that the Whitney--de Rham complex accurately captures the real homotopy type of semi-analytic sets, linking differential forms, cohomology, and loop space properties.

Contribution

It proves a Poincaré Lemma for the Whitney--de Rham complex on semi-analytic sets and shows this complex determines the real homotopy type when the set is simply connected.

Findings

Cohomology of the Whitney--de Rham complex is isomorphic to the real cohomology of the set.

The isomorphism between complexes preserves the algebraic structure.

Hochschild homology of the complex relates to the cohomology of the free loop space.

Abstract

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 $\mathcal{L} X$.