  We study the classification of D-branes and Ramond-Ramond fields in Type I
string theory by developing a geometric description of KO-homology. We define
an analytic version of KO-homology using KK-theory of real C*-algebras, and
construct explicitly the isomorphism between geometric and analytic
KO-homology. The construction involves recasting the Cl(n)-index theorem and a
certain geometric invariant into a homological framework which is used, along
with a definition of the real Chern character in KO-homology, to derive
cohomological index formulas. We show that this invariant also naturally
assigns torsion charges to non-BPS states in Type I string theory, in the
construction of classes of D-branes in terms of topological KO-cycles. The
formalism naturally captures the coupling of Ramond-Ramond fields to background
D-branes which cancel global anomalies in the string theory path integral. We
show that this is related to a physical interpretation of bivariant KK-theory
in terms of decay processes on spacetime-filling branes. We also provide a
construction of the holonomies of Ramond-Ramond fields in Type II string theory
in terms of topological K-chains.
