Geometriae Dedicata

, Volume 73, Issue 2, pp 143–155 | Cite as

Pseudomanifolds with Complementarity

  • Basudeb Data


A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a simplex of the complex. In this article we show that if there exists a n-vertex d-dimensional pseudo-manifold M with complementarity and either n≤d+6 or d≤ 6 then d = 0, 2, 4 or 6 with n = 3d/2 + 3. We also show that if M is a d-dimensional pseudo-manifold with complementarity and the number of vertices in M is ≤ d+5 then M is either a set of three points or the unique 6-vertex real projective plane or the unique 9-vertex complex projective plane.

pseudomanifolds triangulation complementarity. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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
  2. 2.
    Bagchi, B. and Datta, B.: On Kühnel's 9-vertex complex projective plane, Geom. Dedicata 50 (1994), 1–13.Google Scholar
  3. 3.
    Bagchi, B. and Datta, B.: A structure theorem for pseudomanifolds, Discrete Math., 188 (1998), 41–60.Google Scholar
  4. 4.
    Barnette, D. and Gannon, D.:Manifolds with few vertices, Discrete Math. 16 (1976), 291–298.Google Scholar
  5. 5.
    Brehm, U. and Kühnel, W.: Combinatorial manifolds with few vertices, Topology 26 (1987), 465–473.Google Scholar
  6. 6.
    Brehm, U. and Kühnel, W.: 15-vertex triangulation of an 8-manifold, Math. Annal. 294 (1992), 167–193.Google Scholar
  7. 7.
    Datta, B.: Combinatorial manifolds with complementarity, Proc. Indian Acad. Sci. (Math. Sci.) 104 (1994), 385–388.Google Scholar
  8. 8.
    Datta, B.: Minimal triangulation, complementarity and projective planes, In: Geometry from the Pacific Rim, Gruyter, Berlin, 1997, pp. 77–84.Google Scholar
  9. 9.
    Eells, J. and Kuiper, N. H.: Manifolds which are like projective planes, Publ. Math. IHES 14 (1962), 181–222.Google Scholar
  10. 10.
    Hartsfield, N. and Ringel, G.: Clean triangulations, Combinatorica 11 (1991), 145–155.Google Scholar
  11. 11.
    Kühnel, W.: Triangulations of manifolds with few vertices, In: F. Tricerri (ed.). Advances in Differential Geometry and Topology, World Scientific, Singapore, 1990, pp. 59–114.Google Scholar
  12. 12.
    Kühnel, W. and Banchoff, T. F.: The 9-vertex complex projective plane, Math. Intell. 15(3) (1983), 11–22.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Basudeb Data
    • 1
  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia; e-mail

Personalised recommendations