Urgent: Betti numbers, basic cycles
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 \intk p(alpha) dq(alpha) = lk(mod 4) + O(h), 1<=k<=k0
k0 - Betti number of a sphere, lk - index of a cycle, \intk - 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.