  This is an introduction to work on the generalisation to quantum groups of
Mackey's approach to quantisation on homogeneous spaces. We recall the
bicrossproduct models of the author, which generalise the quantum double. We
describe the general extension theory of Hopf algebras and the nonAbelian
cohomology spaces $\CH^2(H,A)$ which classify them. They form a new kind of
topological quantum number in physics which is visible only in the quantum
world. These same cross product quantisations can also be viewed as trivial
quantum principal bundles in quantum group gauge theory. We also relate this
nonAbelian cohomology $\CH^2(H,\C )$ to Drinfeld's theory of twisting.
