Urgent: Betti numbers, basic cycles

Hello!

I'm a physics student and I need to know how to use multi-dimensional WKB method(quasiclassical quantization)

I've read one book on that and came to a theorem below and I experience troubles applying it to my case.

I need to know the following

- how to calculate an integral via basic cycle of the manifold

- how to calculate an index of that cycle

- how to get one-dimensional Betti number of the manifold

My case is a sphere(so I have 3 coordinates - one r and 2 angles, and I have to compute 3 integrals as I understand) - how to apply the following formula to a sphere

2/pi h \int_{k} p(alpha) dq(alpha) = l_{k}(mod 4) + O(h), 1<=k<=k_{0}

k_{0} - Betti number of a sphere, l_{k} - index of a cycle, \int_{k} - is an integral via k-th basic cycle of a manifold.

Any help please, it's urgent.

Re: Urgent: Betti numbers, basic cycles

There are lots of good texts around describing how to integrate on manifolds. Do you have access to any? I recommend Bott and Tu: Differential Forms in Algebraic Topology of Warner: Foundations of Differential Manifolds and Lie Groups.