$L^1$ cohomology of bounded subanalytic manifolds

Abstract

We prove some de Rham theorems on bounded subanalytic submanifolds of ${\mathbb{R}}^n$ (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 Poincaré dual to $L^{\infty }$ cohomology (in dimension $j<m-1$). In general, Poincaré 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.

36 pages

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