Hodge and signature theorems for a family of manifolds with fibration boundary

The Hodge theorem and Hirzebruch signature theorem form an important bridge between geometric and topological properties of compact smooth manifolds. There has been a great deal of work over the past thirty years aimed at understanding how to generalize these theorems to L2 results in the noncompact and singular settings. Early and important work was done by Atiyah, Patodi and Singer [2]. Their work concerned both manifolds with the simplest sort of singularities, namely boundaries, and noncompact manifolds with cylindrical ends, that is, manifolds which off a compact set are isometric to (0,∞)×N for some compact manifold N . They proved both a Hodge result and a signature result. Their Hodge result says that the space of L2 harmonic forms on a manifold, M̂ , with cylindrical end is canonically isomorphic to the image of relative cohomology of M̂ in its absolute cohomology, i.e.