The index of Dirac operators on manifolds with fibered boundaries

Abstract

Let $X$ be a compact manifold with boundary $\partial X$, and suppose that $\partial X$ is the total space of a fibration \[ Z\rightarrow \partial X \rightarrow Y\, . \] Let $D_\Phi$ be a generalized Dirac operator associated to a $\Phi$-metric $g_\Phi$ on $X$. Under the assumption that $D_\Phi$ is fully elliptic we prove an index formula for $D_\Phi$. 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 $g_b (\epsilon)$. Next we write the $b-$ (i.e. the APS) index formula for $g_b(\ep)$; the $\Phi$-index formula follows by analyzing the limiting behaviour as $\epsilon\searrow 0$ 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.