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Geometriae Dedicata

, Volume 50, Issue 1, pp 1–13 | Cite as

On Kühnel's 9-vertex complex projective plane

  • Bhaskar Bagchi
  • Basudeb Datta
Article

Abstract

We present an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of ℂP2. The original proof of existence, due to Kühnel, as well as the original proof of uniqueness, due to Kühnel and Lassmann, were based on extensive computer search. Recently Arnoux and Marin have used cohomology theory to present a computer-free proof. Our proof has the advantage of displaying a canonical copy of the affine plane over the three-element field inside this complex in terms of which the entire complex has a very neat and short description. This explicates the full automorphism group of the Kühnel complex as a subgroup of the automorphism group of this affine plane. Our method also brings out the rich combinatorial structure inside this complex.

Mathematics Subject Classification (1991)

Primary 57Q15 51E20 

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References

  1. 1.
    Altshuler, A.: Combinatorial 3-manifolds with few vertices,J. Combin. Theory Ser. A 16 (1974), 165–173.Google Scholar
  2. 2.
    Arnoux, P. and Marin, A.: The Kühnel triangulation of complex projective plane from the view-point of complex crystallography (part II),Mem. Fac. Sci. Kyushu Univ. Ser. A 45 (1991), 167–244.Google Scholar
  3. 3.
    Barnette, D.: The triangulations of the 3-sphere with up to 8 vertices,J. Combin. Theory Ser. A 14 (1973), 37–52.Google Scholar
  4. 4.
    Brehm, U. and Kühnel, W.: Combinatorial manifolds with few vertices,Topology 26 (1987), 465–473.Google Scholar
  5. 5.
    Brehm, U. and Kühnel, W.: 15-vertex triangulation of an 8-manifold,Math. Annal. 294 (1992), 167–193.Google Scholar
  6. 6.
    Datta, B.: Combinatorial manifolds with complementarityProc. (Math. Sci.) Indian Academy of Sciences, To appear.Google Scholar
  7. 7.
    Eells, J. Jr and Kuiper, N. H.: Manifolds which are like projective plane,Publ. Math. I.H.E.S. 14 (1962), 181–222.Google Scholar
  8. 8.
    Freedman, M.: The topology of four dimensional manifolds,J. Differential Geom. 17 (1983), 357–454.Google Scholar
  9. 9.
    Grünbaum, B. and Sreedharan, V. P.: An enumeration of simplicial 4-polytopes with 8 vertices,J. Combin. Theory 2 (1967), 437–465.Google Scholar
  10. 10.
    Hirsch, M. and Mazur, B.: Smoothing of piecewise linear manifolds,Ann. Math. Studies 80, Princeton Univ. Press, Princeton, 1974.Google Scholar
  11. 11.
    Kühnel, W. and Lassmann, G.: The unique 3-neighbourly 4-manifold with few vertices,J. Combin. Theory Ser. A 35 (1983), 173–184.Google Scholar
  12. 12.
    Kühnel, W. and Banchoff, T. F.: The 9-vertex complex projective plane,Math. Intelligencer 5, No. 3 (1983), 11–22.Google Scholar
  13. 13.
    Morin, B. and Yoshida, M.: The Kühnel triangulation of complex projective plane from the view-point of complex crystallography (part I),Mem. Fac. Sci. Kyushu Univ. Ser. A 45 (1991), 55–142.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Bhaskar Bagchi
    • 1
  • Basudeb Datta
    • 2
  1. 1.Stat-Math UnitIndian Statistical Institute, R. V. College PostBangaloreIndia
  2. 2.Mathematics DepartmentIndian Institute of ScienceBangaloreIndia

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