Families Index for Pseudodifferential Operators on Manifolds with Boundary

An analytic families index is defined for (cusp) pseudodifferential operators on a fibration with fibres which are compact manifolds with boundaries. This provides an extension to the boundary case of the setting of the (pseudodifferential) Atiyah-Singer theorem and to the pseudodifferential case of the families Atiyah-Patodi-Singer index theorem for Dirac operators due to Bismut and Cheeger and to Piazza and the first author. In showing that any elliptic family of symbols has a realization as an invertible family of pseudodifferential operators, which is a form of the cobordism invariance of the index, a crucial role is played by the weak contractibility of the group of cusp smoothing operators on a compact manifold with non-trivial boundary and the associated exact sequence of classifying spaces of odd and even K-theory.