Skip to main content
SpringerLink
Log in

The topology of certain 3-Sasakian 7-manifolds

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We calculate the integer cohomology ring and stable tangent bundle of a family of compact, 3-Sasakian 7-manifolds constructed by Boyer, Galicki, Mann and Rees. The most important part of the cohomology ring is a previously undetermined torsion group that we describe explicitly and whose order we compute. There is a surprising connection with the combinatorics of trees.

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

Access this article

Log in via an institution

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. Boyer C.P., Galicki K. and Mann B.M. (1994). The geometry and topology of 3-Sasakian manifolds. J. Reine Angew. Math. 455: 183–220

    MATH  MathSciNet  Google Scholar 

  2. Boyer C.P., Galicki K. and Mann B.M. (1998). Hypercomplex structures from 3-Sasakian structures. J. Reine Angew. Math. 501: 115–141

    MATH  MathSciNet  Google Scholar 

  3. Boyer C.P., Galicki K., Mann B.M. and Rees E.G. (1998). Compact 3-Sasakian 7-manifolds with arbitrary second Betti number. Invent. Math. 131(2): 321–344

    Article  MATH  MathSciNet  Google Scholar 

  4. Chinburg, T., Escher, C., Ziller, W.: Topological properties of Eschenburg spaces and 3-Sasakian manifolds. Preprint, Math. Ann. (2007) (to appear)

  5. Hepworth, R.A.: Generalized Kreck-Stolz invariants and the topology of certain 3-Sasakian 7-Manifolds. PhD Thesis, Edinburgh University (2006)

  6. Hodge, W.V.D., Pedoe, D.: Methods of algebraic geometry. In: Cambridge Mathematical Library, vol. I. Cambridge University Press, Cambridge (1994). Book I: Algebraic preliminaries, Book II: Projective space, Reprint of the 1947 original

  7. Kreck M. and Stolz S. (1988). A diffeomorphism classification of 7-dimensional homogeneous Einstein manifolds with SU(3) × (2) × U(1)-symmetry. Ann. Math. (2) 127(2): 373–388

    Article  MathSciNet  Google Scholar 

  8. Kreck M. and Stolz S. (1991). Some nondiffeomorphic homeomorphic homogeneous 7-manifolds with positive sectional curvature. J. Differ. Geom. 33(2): 465–486

    MathSciNet  Google Scholar 

  9. Stanley, R.P.: Enumerative combinatorics, vol. 2. In: Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999). With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin

  10. van Lint J.H. and Wilson R.M. (2001). A Course in Combinatorics, 2nd edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, EH8 9XD, UK

    Richard A. Hepworth

Authors
  1. Richard A. Hepworth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Richard A. Hepworth.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hepworth, R.A. The topology of certain 3-Sasakian 7-manifolds. Math. Ann. 339, 733–755 (2007). https://doi.org/10.1007/s00208-007-0100-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-007-0100-8

Keywords

Search

Navigation