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$L^1$ cohomology of bounded subanalytic manifolds (1011.2023v1)
Published 9 Nov 2010 in math.AG
Abstract: We prove some de Rham theorems on bounded subanalytic submanifolds of $\Rn$ (not necessarily compact). We show that the $L1$ 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 $L1$ 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 $L1$ Stokes' Property. For oriented manifolds, we show that the $L1$ Stokes' property holds if and only if integration realizes a nondegenerate pairing between $L1$ and $L\infty$ forms. This is the counterpart of a theorem proved by Cheeger on $L2$ forms.