  D-branes are classified by twisted K-theory. Yet twisted K-theory is often
hard to calculate. We argue that, in the case of a compactification on a
simply-connected six manifold, twisted K-theory is isomorphic to a much simpler
object, twisted homology. Unlike K-theory, homology can be twisted by a class
of any degree and so it classifies not only D-branes but also M-branes. Twisted
homology classes correspond to cycles in a certain bundle over spacetime, and
branes may decay via Kachru-Pearson-Verlinde transitions only if this cycle is
trivial. We provide a spectral sequence which calculates twisted homology, the
kth step treats D(p-2k)-branes ending on Dp-branes.
