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 L2 harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted L2 harmonic forms are harmonic forms that are almost in the given weighted L2 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. In analogy with that setting, in the unweighted L2 case, the boundary values of the extended harmonic forms de ne a Lagrangian splitting of the boundary space in the long exact sequence relating upper and lower middle perversity intersection cohomology groups.