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manuscripta mathematica

, Volume 95, Issue 1, pp 149–158 | Cite as

A note on smooth toral reductions of spheres

  • Charles P. Boyer
  • Krzysztof Galicki
  • Benjamin M. Mann
Article

Abstract

In this paper we show that there exist mod 2 obstructions to the smoothness of 3-Sasakian reductions of spheres. Specifically, ifS is a smooth 3-Sasakian manifold obtained by reduction of the 3-Sasakian sphereS 4n−1 by a torus, and if the second Betti numberb 2(S)≥2 then dimS=7, 11, 15, whereas, ifb 2 (S)≥5 then dimS=7. We also show that the above bounds are sharp, in that we construct explicit examples of 3-Sasakian manifolds in the cases not excluded by these bounds.

Mathematics Subject Classification (1991)

53C25 57S25 

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References

  1. [1]
    Bielawski, R.: Betti Numbers of 3-Sasakian Quotients of Spheres by Tori. Bull. London Math. Soc. (to appear)Google Scholar
  2. [2]
    Bielawski, R.: Reconstruction of HamiltonianT n Spaces in Quaternionic Riemannian Geometry. MPI preprint (1997)Google Scholar
  3. [3]
    Bielawski, R., Dancer, A.S.: The Geometry and Topology of Toric Hyperkähler Manifolds. McMaster Univ. reprint (1996)Google Scholar
  4. [4]
    Boyer, C.P., Galicki, K.: The Twistor Space of a 3-Sasakian Manifold. Int. J. of Math.8(1), 31–60 (1997)zbMATHCrossRefGoogle Scholar
  5. [5]
    Boyer, C.P., Galicki, K., Mann, B.M.: The Geometry and Topology of 3-Sasakian Manifolds. J. reine angew. Math.455, 183–220 (1994)zbMATHGoogle Scholar
  6. [6]
    Boyer, C.P., Galicki, K., Mann, B.M.: Hypercomplex Structures on Stiefel Manifolds. Ann. Global Anal. and Geom.14, 81–105 (1996)zbMATHCrossRefGoogle Scholar
  7. [7]
    Boyer, C.P., Galicki, K., Mann, B.M., Rees, E.G.: Compact 3-Sasakian 7-Manifolds with Arbitrary Second Betti Number. Inventiones Math. (to appear)Google Scholar
  8. [8]
    Boyer, C.P., Galicki, K., Mann, B.M., Rees, E.G.: Positive Einstein Manifolds with Arbitrary Second Betti Number. Balkan Jour. of Geom. and Its Appl.1(2), 1–8 (1996)zbMATHGoogle Scholar
  9. [9]
    Galicki, K., Salamon, S.: On Betti Numbers of 3-Sasakian Manifolds. Geom. Ded.63, 45–68 (1996)zbMATHGoogle Scholar
  10. [10]
    Guillemin, V.: Moment Maps and Combinatorial Invariants of HamiltonianT n-spaces. Boston: Birkhäuser 1994Google Scholar
  11. [11]
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, Vol. 1. New York: Interscience 1963Google Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Charles P. Boyer
    • 1
  • Krzysztof Galicki
    • 1
  • Benjamin M. Mann
    • 1
  1. 1.Department of MathematicsUniversity of New MexicoAlbuquerqueUSA

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