An analytic index is defined for a family of cusp pseudodifferential
operators, Pb​, on a fibration with fibres which are compact manifolds with
boundaries, provided the family is elliptic and has invertible indicial family
at the boundary. In fact there is always a perturbation Qb​ by a family of
cusp operators of order −∞ such that each Pb​+Qb​ is invertible. Thus
any elliptic family of symbols has a realization as an invertible family of
cusp 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.Comment: 21 pages; corrected typos, changed the abstract, added a paragraph in
the introductio