$L^1$ cohomology of bounded subanalytic manifolds

TL;DR

This paper establishes de Rham theorems linking $L^1$ cohomology and singular homology for bounded subanalytic manifolds, extending duality results and analyzing the $L^1$ Stokes' property in relation to Poincaré duality.

Contribution

It proves isomorphisms between $L^1$ cohomology and singular homology for bounded subanalytic manifolds, and characterizes the $L^1$ Stokes' property in terms of duality.

Findings

$L^1$ cohomology is isomorphic to singular homology.

Poincaré duality holds under certain singularity conditions.

The $L^1$ Stokes' property characterizes duality between $L^1$ and $L^ Infty$ forms.

Abstract

We prove some de Rham theorems on bounded subanalytic submanifolds of $\R^n$ (not necessarily compact). We show that the $L^1$ cohomology of such a submanifold is isomorphic to its singular homology. In the case where the closure of the underlying manifold has only isolated singularities this implies that the $L^1$ cohomology is Poincar\'e dual to $L^\infty$ cohomology (in dimension $j <m-1$). In general, Poincar\'e duality is related to the so-called $L^1$ Stokes' Property. For oriented manifolds, we show that the $L^1$ Stokes' property holds if and only if integration realizes a nondegenerate pairing between $L^1$ and $L^\infty$ forms. This is the counterpart of a theorem proved by Cheeger on $L^2$ forms.