Relations among Fixed Points

Let M be a smooth manifold with a circle action, and {P } be the fixed point sets. The problem I want to discuss in this paper is how to get the topological information of one relatively complicated fixed point set, say P 0 , from the other much simpler fixed points. Such problems are interesting in symplectic geometry and geometric invariant theory, especially in the study of moduli spaces. In this paper I derive several very simple integral formulas which express integrals over P 0 in terms of integrals over the other fixed point sets P 's. As applications, I use these formulas to give an explicit expression for integrations of cohomology classes on the moduli space of higher rank stable bundles over a Riemann surface in terms of integrals over lower rank moduli spaces. In rank 2 case, these formulas express the integrals over the moduli spaces in terms of integrals over symmetric products of the Riemann surface. These formulas are also useful in computing the changes of integrals on the quotient manifolds when the polarization is altered in geometric invariant theory [DH], [Th], or when the level of moment map is changed in sympletic geometry [GS]. On the other hand, recently Pid-strigach and Tyurin [PT] have constructed a circle action on the mod-uli space of solutions of a rank 2 Seiberg-Witten equation whose fixed point sets are respectively given by the moduli spaces of self-dual connections and a rank 1 Seiberg-Witten equation. This indicates that our formulas may be useful in relating the Donaldson invariants to the Seiberg-Witten invariants. In §5 we also derive a similar formula in equivariant K-theory which relates the theorem in [GS1] about geometric quantization commuting with symplectic reduction to the Verlinde formula. Note that for rank 2 case, the formulas we derived can be used to recover many reasults about moduli spaces of vector bundles on a Riemann surface, such as the Verlinde formula and the Newstead conjectures about the vanishing of Chern classes and Pontryagin classes. By using the same idea we can derive similar formulas on noncompact manifolds, such as the Hitchin moduli spaces of vector bundles with Higgs fields on a compact Riemann surface. We note that, when the