Relative de Rham Theory on Nash Manifolds

TL;DR

This paper develops a relative de Rham theory for Nash manifolds, establishing homotopy equivalences and properties of Schwartz sections that depend only on the map's homology type, with implications for the topology of these spaces.

Contribution

It introduces Schwartz sections of constructible sheaves on Nash manifolds and proves their homotopy invariance and Hausdorff properties, advancing the understanding of relative de Rham complexes.

Findings

Homotopy equivalence of Schwartz sections to proper push-forward sheaves

Dependence of Schwartz sections on the homology type of the map

Hausdorff property of the homology spaces

Abstract

For a Nash submersion $\phi\colon X\to Y$, we study the complex $\mathcal{SDR}(\phi)$ of Schwartz sections of the relative de Rham complex of $\phi$. We define the notion of Schwartz sections of constructible sheaves on Nash manifolds and prove that $\mathcal{SDR}(\phi)$ is homotopy equivalent to the Schwartz sections of the proper push-forward $\phi_!\mathbb{R}_X$ of the constant sheaf $\mathbb{R}_X$. Using this equivalence, we show that $\mathcal{SDR}(\phi)$ depends (up to homotopy equivalence) only on the homology type of the map $\phi$. We also deduce that $\mathcal{SDR}(\phi)$ has Hausdorff homology spaces.