Relative Chern Character, Boundaries and Index Formulæ

For three classes of elliptic pseudodifferential operators on a compact manifold with boundary which have ‘geometric K-theory’, namely the ‘transmission algebra’ introduced by Boutet de Monvel [5], the ‘zero algebra’ introduced by Mazzeo in [9, 10] and the ‘scattering algebra’ from [16] we give explicit formulæ for the Chern character of the index bundle in terms of the symbols (including normal operators at the boundary) of a Fredholm family of fibre operators. This involves appropriate descriptions, in each case, of the cohomology with compact supports in the interior of the total space of a vector bundle over a manifold with boundary in which the Chern character, mapping from the corresponding realization of K-theory, naturally takes values.