  RR fields in string backgrounds including orientifold planes and branes on
top of them are classified by K-theory. Following the idea introduced in
hep-th/0103183, we also classify such fluxes by cohomology. Both of them are
compared through the Atiyah-Hirzebruch Spectral Sequence. Some new correlations
between branes on orientifold planes $Op^\pm$ and obstructions to the existence
of some branes are found. Finally, we find a topological condition that avoid
the presence of global gauge anomalies in lower dimensional systems.
