An Index Formula on Manifolds with Fibered Cusp Ends

We consider a compact manifold X whose boundary is a locally trivial fiber bundle, and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups we first express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior of X and a term that comes from the boundary. This leads to an abstract answer to the index problem formulated in [11]. We give a more precise answer for first-order differential operators when the base of the boundary fiber bundle is S. In particular, for Dirac operators associated to a metric of the form g = dx 2 x4 + dθ2 x2 + g F near ∂X = {x = 0} with twisting bundle T we obtain