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Equivariant Cohomology of S1-Actions on 4-Manifolds

Published online by Cambridge University Press:  20 November 2018

Leonor Godinho
Leonor Godinho*
Affiliation:
Centro de Análise Matemática, Geometria et Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal e-mail: [email protected]
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Abstract

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Let $M$ be a symplectic 4-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations.

Keywords

53D2055N9157S15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 3 , 01 September 2007 , pp. 365 - 376
Copyright
Copyright © Canadian Mathematical Society 2007

References

[A] Audin, M., The topology of torus actions on symplectic manifolds. Progress in Mathematics 93, Birkhäuser Verlag, Basel, 1991.Google Scholar
[AH] Ahara, K. and Hattori, A., 4-dimensional symplectic S 1 -manifolds admitting moment map. J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 38(1991), no. 2, 251–298.Google Scholar
[CS] Chang, T. and Skjelbred, T., The topological Schur lemma and related results. Ann. of Math. 100(1974), 307–321.Google Scholar
[G] Goldin, R. F., An effective algorithm for the cohomology ring of symplectic reductions. Geom. Funct. Anal. 12(2002), no. 3, 567–583.Google Scholar
[GH] Goldin, R. F. and Holm, T., The equivariant cohomology of Hamiltonian G-spaces from residual S 1 actions. Math. Res. Lett. 8(2001), no. 1–2, 67–77.Google Scholar
[GKM] Goresky, M., Kottwitz, R. and MacPherson, R., Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131(1998), no. 1, 25–83.Google Scholar
[GZ] Guillemin, V. and Zara, C., One skeleta, Betti numbers and equivariant cohomology. Duke Math. J. 107(2001), no. 2, 283–349.Google Scholar
[H] Huck, W., Circle actions on 4-manifolds. I. Manuscripta Math. 87(1995), no. 1, 51–70.Google Scholar
[K] Karshon, Y., Periodic Hamiltonian flows on four dimensional manifolds, Mem. Amer. Math Soc. 141(1999), no. 672.Google Scholar
[Ki] Kirwan, F. C., The cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes 31, Princeton University Press, 1984.Google Scholar
[TW1] Tolman, S. and Weitsman, J., On semifree symplectic circle actions with isolated fixed points. Topology 39(2000), no. 2, 299–309.Google Scholar
[TW2] Tolman, S. and Weitsman, J., On the cohomology rings of Hamiltonian T-spaces, In: Northern California Symplectic Geometry Seminar, Amer.Math. Soc. Transl. Ser. 2 196, American Mathematical Society, Providence, RI, 1999, pp. 251–258.Google Scholar
[TW3] Tolman, S. and Weitsman, J., The cohomology rings of symplectic quotients. Comm. Anal. Geom. 11(2003), no. 4, 751–773.Google Scholar