The index of Dirac operators on manifolds with fibered boundaries

Let X be a compact manifold with boundary ∂X, and suppose that ∂X is the total space of a fibration Z → ∂X → Y. Let D Φ be a generalized Dirac operator associated to a Φ-metric gΦ on X. Under the assumption that DΦ is fully elliptic we prove an index formula for DΦ. The proof is in two steps: first, using results of Melrose and Rochon, we show that the index is unchanged if we pass to a certain b-metric gb(ε). Next we write the b- (i.e. the APS) index formula for gb(ε); the Φ-index formula follows by analyzing the limiting behaviour as ε ↘ 0 of the two terms in the formula. The interior term is studied directly whereas the adiabatic limit formula for the eta invariant follows from work of Bismut and Cheeger.