2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/196173We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of T-duality as well as establishing a very general formula for D-brane charges. This formula is closely related to a noncommutative Grothendieck-Riemann-Roch theorem that is proved here. Our approach relies on a very general form of Poincare duality, which is studied here in detail. Among the technical tools employed are calculations with iterated products in bivariant K-theory and cyclic theory, which are simplified using a novel diagram calculus reminiscent of Feynman diagrams.56 pages; v3: final version, to appear in CMPHigh Energy Physics - TheoryK-Theory and Homologytext