Hodge theory on Cheeger spaces

Abstract

Abstract We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary conditions and the notion of mezzoperversity, which intermediates between the standard lower and upper middle perversities in intersection theory, as interpreted in this de Rham setting, and show that the de Rham operator with these boundary conditions is Fredholm and has compact resolvent. We also prove an isomorphism between the resulting Hodge and L 2 L^{2} de Rham cohomology groups, and that these are independent of the choice of iterated edge metric. On spaces which admit ideal boundary conditions of this type which are also self-dual, which we call ‘Cheeger spaces’, we show that these Hodge/de Rham cohomology groups satisfy Poincaré duality.