  The various descent and duality relations among BPS and non-BPS D-branes are
classified using topological K-theory. It is shown how the descent procedures
for producing type-II D-branes from brane-antibrane bound states by tachyon
condensation and $\klein$ projections arise as natural homomorphisms of
K-groups generating the brane charges. The transformations are generalized to
type-I theories and type-II orientifolds, from which the complete set of vacuum
manifolds and field contents for tachyon condensation is deduced. A new set of
internal descent relations is found which describes branes over orientifold
planes as topological defects in the worldvolumes of brane-antibrane pairs on
top of planes of higher dimension. The periodicity properties of these
relations are shown to be a consequence of the fact that all fundamental bound
state constructions and hence the complete spectrum of brane charges are
associated with the topological solitons which classify the four Hopf
fibrations.
