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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 P0, 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 P0 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 integral

Year: 2009

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