##
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.