General perversities and [formula omitted] de Rham and Hodge theorems for stratified pseudomanifolds

Abstract

Given a compact stratified pseudomanifold X with a Thom–Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the L2 de Rham and Hodge cohomology of the regular part of X and the intersection cohomology of X associated to some perversities. More precisely, to a kind of metric which we call quasi edge with weights, we associate two general perversities in the sense of G. Friedman, pg and its dual qg. We then show that:1.The absolute L2 Hodge cohomology is isomorphic to the maximal L2 de Rham cohomology and this is in turn isomorphic to the intersection cohomology associated to the perversity qg.2.The relative L2 Hodge cohomology is isomorphic to the minimal L2 de Rham cohomology and this is in turn isomorphic to the intersection cohomology associated to the perversity pg. Moreover we give a partial answer to the inverse question: given p, a general perversity in the sense of Friedman on X, is there a riemannian metric g on reg(X) such that an L2 de Rham and Hodge theorem holds for g and p? We then show that the answer is positive in the following two cases: if p is greater or equal to the upper middle perversity or if it is smaller or equal to the lower middle one. Finally we conclude giving several corollaries about the properties of these L2 Hodge and de Rham cohomology groups.