De Rham isomorphism for $L^p$-cohomology and intersection homology

Andrzej Weber

24 pages

The purpose of this paper is to present a connection between the intersection homology theory and $L^p$-cohomology. We investigate the simplest case: riemannian pseudomanifolds. These are piecewise linear singular spaces with a riemannian metric on the nonsingular part which is concordant with a triangulation. We show that there exists a perversity (the maximal smaller then $i\over p$) such that intersection homology with respect to this perversity and $L^p$-cohomology coincide for $p \ge 2$. We prove this isomorphism using shadow forms. For $p < 2$ we need to assume that the codimension of the singular set is at least $q$, where $1/p+1/q=1$. It's a version of the de Rham isomorphism. The condition of $L^p$-growth in the neighborhood of singularities for forms corresponds to restrictions for intersections of chains with singular strata. Next we consider an important abstract functional condition of a negligible boundary. For example it allows us to define a pairing between $L^p$- and $L^q$-cohomology. It turns out that in our case it is a purely topological condition concerning the structure of singularities. The negligible boundary condition is equivalent to vanishing of intersection homology of links in certain dimensions (around $s/p$, where $s$ is the dimension of the link). The spaces with this property for $p=2$ were discovered by Cheeger and independently by Goresky and MacPherson and called Witt spaces. In the end we show that the negligible boundary condition is equivalent to the duality in the derived category between the sheaves of $L^p$- and $L^q$- cohomology.