Extended Hodge Theory for Fibred Cusp Manifolds

For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted $L^2$ harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted $L^2$ harmonic forms are harmonic forms that are almost in the given weighted $L^2$ space for the metric in question, but not quite. This result is akin to the representation of absolute and relative cohomology groups for a manifold with boundary by extended harmonic forms on the associated manifold with cylindrical ends. As in that setting, in the unweighted $L^2$ case, the boundary values of the extended harmonic forms define a Lagrangian splitting of the boundary space in the long exact sequence relating upper and lower middle perversity intersection cohomology groups.