2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102144A families index theorem in K-theory is given for the setting of Atiyah, Patodi and Singer of a family of Dirac operators with spectral boundary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred cusp, pseudodifferential operators on the fibres (with boundary) of a fibration; a version of Poincare duality is also shown in this setting, identifying the stable Fredholm families with elements of a bivariant K-group.64 pages, corrected typosDifferential GeometryK-Theory and Homology58J40 (Primary) 19K56, 19K35 (Secondary)text