Skip to main content
Log in

Basic Kirwan surjectivity for K-contact manifolds

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We prove an analogue of Kirwan surjectivity in the setting of equivariant basic cohomology of K-contact manifolds. If the Reeb vector field induces a free \(S^1\)-action, the \(S^1\)-quotient is a symplectic manifold, and our result reproduces Kirwan’s surjectivity for these symplectic manifolds. We further prove a Tolman–Weitsman type description of the kernel of the basic Kirwan map for \(S^1\)-actions and show that torus actions on a K-contact manifold that preserve the contact form and admit 0 as a regular value of the contact moment map are equivariantly formal in the basic setting.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atiyah, M.F., Bott, R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. Ser. A 308, 523–615 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boyer, C.P., Galicki, K.: Sasakian Geometry. Oxford University Press, Oxford (2008)

    MATH  Google Scholar 

  3. Baird, T., Lin, Y.: Topology of generalized complex quotients. J. Geom. Phys. 60(10), 1539–1557 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blair, D.E.: Contact Manifolds in Riemannian Geometry, Vol. 509 of Lecture Notes in Mathematics. Springer, Berlin (1976)

    Book  Google Scholar 

  5. Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Springer, Berlin (2013)

    MATH  Google Scholar 

  6. Cartan, H.: La transgression dans un groupe de lie et dans un espace fibré principal, pp. 57–71. Colloque de Topologie Bruxelles, Bruxelles (1950)

    Google Scholar 

  7. Cannas da Silva, A.: Lectures on Symplectic Geometry, Vol. 1764 of Lecture Notes in Mathematics. Springer, Berlin (2001)

    MATH  Google Scholar 

  8. Casselmann, L., Fisher, J.M.: Localization for \(K\)-contact manifolds. arXiv:1703.00333 (2017)

  9. Duflot, J.: Smooth toral actions. Topology 22(3), 253–265 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ginzburg, V.L., Guillemin, V., Karshon, Y.: Moment maps, Cobordisms, and Hamiltonian group actions, Vol. 98 of Math. Surveys and Monographs. American Mathematical Society, Providence (2002)

  11. Goertsches, O., Nozawa, H., Töben, D.: Equivariant cohomology of \(K\)-contact manifolds. Math. Ann. 354, 1555–1582 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Goertsches, O., Nozawa, H., Töben, D.: Localization of Chern-Simons type invariants of Riemannian foliations. to appear in Israel. J. Math. arXiv: 1508.07973 (2015)

  13. Goertsches, O., Töben, D.: Torus actions whose equivariant cohomology is Cohen–Macaulay. J. Topol. 3(4), 819–846 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Goertsches, O., Töben, D.: Equivariant basic cohomology of Riemannian foliations. Journal für die reine und angewandte Mathematik (Crelles Journal) (2016)

  15. Goldin, R.F.: An effective algorithm for the cohomology ring of symplectic reductions. Geom. Funct. Anal. 12(3), 567–583 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guillemin, V.W., Sternberg, S.: Supersymmetry and Equivariant de Rham Theory. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  17. Kirwan, F.: Cohomology of Quotients in Symplectic and Algebraic Geometry. Princeton University Press, Princeton (1984)

    MATH  Google Scholar 

  18. Kobayashi, S.: Fixed points of isometries. Nagoya Math. J. 13(1), 63–68 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lerman, E.: Question 3.1. In: AIM Workshop on Moment Maps and Surjectivity in Various Geometries 2004: Conjectures and Further Questions. http://aimath.org/WWN/momentmaps/momentmaps (2013). Accessed 21 June 2013

  20. Molino, P.: Riemannian Foliations. Birkhauser Boston Inc., Cambridge (1988)

    Book  MATH  Google Scholar 

  21. Paradan, P.-E., Vergne, M.: Equivariant relative Thom forms and Chern characters. arXiv:0711.3898 (2007)

  22. Reinhart, B.L.: Harmonic integrals on foliated manifolds. Am. J. Math. 81(2), 529–536 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rukimbira, P.: Topology and closed characteristics of K-contact manifolds. Bull. Belg. Math. Soc. 2(3), 349–356 (1995)

    MathSciNet  MATH  Google Scholar 

  24. Rukimbira, P.: On K-contact manifolds with minimal number of closed characteristics. Proc. Am. Math. Soc. 127, 3345–3352 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tolman, S., Weitsman, J.: The cohomology rings of symplectic quotients. Commun. Anal. Geom. 11(4), 751–774 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wasserman, A.G.: Equivariant differential topology. Topology 8(2), 127–150 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  27. Weibel, C.A.: An Introduction to Homological Algebra, Vol. 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997)

    Google Scholar 

Download references

Acknowledgements

The author would like to thank Oliver Goertsches for suggesting the topic of this paper, numerous helpful discussions, his careful guidance and critical reading of the manuscript, as well as Jonathan Fisher for helpful conversations. The author would further like to thank the anonymous reviewers for their careful reading and constructive comments on the paper. This research was supported by the RTG 1670 “Mathematics inspired by String Theory and Quantum Field Theory”, funded by the Deutsche Forschungsgemeinschaft (DFG).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lana Casselmann.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Casselmann, L. Basic Kirwan surjectivity for K-contact manifolds. Ann Glob Anal Geom 52, 157–185 (2017). https://doi.org/10.1007/s10455-017-9552-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-017-9552-6

Keywords

Mathematics Subject Classification

Navigation