I think that you have to investigate in the direction of fibrations. And then, maybe, spectral sequences.
edit: Concernig your comment -- I am interested in that as well.
Right now my investigation got me there -- check the book "Modern Geometry - Methods and Applications: Part 3: Introduction to Homology Theory" by: B.A. Dubrovin, A.T. Fomenko, S.P. Novikov.
Chapter 20.
It is quite difficult for me to read the book (Yeah, you got me, I'm a physicist.) so I cannot give you anything but the direction.
