FAMILIES OF DIRAC OPERATORS, BOUNDARIES AND THE B-CALCULUS

A version of the Atiyah-Patodi-Singer index theorem is proved for gene ral families of Dirac operators on compact manifolds with boundary. Th e vanishing of the analytic index of the boundary family, in K-1 of th e base, allows us to define, through an explicit trivialization, a smo oth family of boundary conditions of generalized Atiyah-Patodi-Singer type. The calculus of b-pseudodifferential operators is then employed to establish the family index formula. A relative index formula, descr ibing the effect of changing the choice of the trivialization, is also given. In case the boundary family is invertible the form of the inde x theorem obtained by Bismut and Cheeger is recovered.