L¹ cohomology of bounded subanalytic manifolds

2010/11/09 by Guillaume Valette, Valette, Guillaume
Mathematics · #14F40 #32B20 #55N33 #57P10 #58A10 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14F40 #msc:32B20 #msc:55N33 #msc:57P10 #msc:58A10

paper · pdf · doi:10.48550/arxiv.1011.2023

36 pages

arxiv created 2010/11/09 · arxiv updated 2010/11/10

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 Poincaré dual to L^∞ cohomology (in dimension j <m-1). In general, Poincaré 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^∞ forms. This is the counterpart of a theorem proved by Cheeger on L² forms.

Related

• Poincaré duality for L^p cohomology on subanalytic singular spaces
• L^p Hodge theory for bounded subanalytic manifolds
• L^∞ cohomology is intersection cohomology
• Isomorphisms in l¹-homology
• On the de Rham theorem in the globally subanalytic setting