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Cohomology Ring of Symplectic Quotients by Circle Actions

Published online by Cambridge University Press:  20 November 2018

Ramin Mohammadalikhani
Ramin Mohammadalikhani*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 3G3 e-mail: [email protected]
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Abstract

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In this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is isolated; however the circle action only needs to be “formally Hamiltonian”. By identifying the kernel, we obtain the cohomology ring of the symplectic quotient. Next we apply this result to some special cases and in particular to the case of products of two dimensional spheres. We show that the results of Kalkman and Hausmann-Knutson are special cases of our result.

Keywords

53D2053D3037J1037J1553D05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 3 , 01 June 2004 , pp. 553 - 565
Copyright
Copyright © Canadian Mathematical Society 2004

References

[Au] Audin, M., The topology of torus actions on symplectic manifolds. Birkhauser, Progress in Math. 93, 1991.Google Scholar
[BGV] Berline, N., Getzler, E. and Vergne, M., Heat Kernels and Dirac Operators. Springer-Verlag, 1992.Google Scholar
[G] Goldin, R., The cohomology of weight varieties. Ph.D. thesis, MIT, 1999.Google Scholar
[G2] Goldin, R., The cohomology of weight varieties and polygon spaces. Adv. in Math. 160(2001), 175–204.Google Scholar
[HK] Hausmann, J. and Knutson, A., The cohomology ring of polygon spaces. Ann. Inst. Fourier 48(1998), 281–321.Google Scholar
[JK1] Jeffrey, L. and Kirwan, F., Localization for nonabelian group actions. Topology 34(1995), 291–327.Google Scholar
[JK2] Jeffrey, L. and Kirwan, F., Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface. Ann. of Math. 148(1998), 109–191.Google Scholar
[JK3] Jeffrey, L. and Kirwan, F., Localization and the quantization conjecture. Topology 36(1997), 647–693.Google Scholar
[Ka] Kalkman, J., Cohomology rings of symplectic quotients. J. Reine Angew.Math. 458(1995), 37–52.Google Scholar
[Ki1] Kirwan, F., Cohomology of Quotients in Symplectic and Algebraic Geometry. Princeton University Press, 1984.Google Scholar
[TW1] Tolman, S. and Weitsman, J., The cohomology rings of symplectic quotients. Comm. in Anal. Geom., preprint.Google Scholar
[TW2] Tolman, S. and Weitsman, J., On semifree symplectic circle actions with isolated fixed points. Topology 39(2000), 299–309.Google Scholar