The index of Dirac operators on manifolds with fibered boundaries

Abstract

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 be a generalized Dirac operator associated to a -metric g on X. Under the assumption that D is fully elliptic we prove an index formula for D . 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( ). Next we write the b (i.e. the APS) index formula for gb( ); the -index formula follows by analyzing the limiting behaviour as & 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.