Acta Mathematica

, Volume 206, Issue 2, pp 205–243 | Cite as

On locally constructible spheres and balls

  • Bruno Benedetti
  • Günter M. ZieglerEmail author


Durhuus and Jonsson (1995) introduced the class of “locally constructible” (LC) 3-spheres and showed that there are only exponentially many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity.

We characterize the LC property for d-spheres (“the sphere minus a facet collapses to a (d−2)-complex”) and for d-balls. In particular, we link it to the classical notions of collapsibility, shellability and constructibility, and obtain hierarchies of such properties for simplicial balls and spheres. The main corollaries from this study are:

– Not all simplicial 3-spheres are locally constructible. (This solves a problem by Durhuus and Jonsson.)

There are only exponentially many shellable simplicial 3-spheres with given number of facets. (This answers a question by Kalai.)

– All simplicial constructible 3-balls are collapsible. (This answers a question by Hachimori.)

– Not every collapsible 3-ball collapses onto its boundary minus a facet. (This property appears in papers by Chillingworth and Lickorish.)


Span Tree Simplicial Complex Dual Graph Interior Vertex Free Face 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Alon, N., The number of polytopes, configurations and real matroids. Mathematika, 33 (1986), 62–71.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Ambjørn, J., Boulatov, D. V., Kawamoto, N. & Watabiki, Y., Recursive sampling simulations of 3D gravity coupled to scalar fermions. Phys. Lett. B, 480 (2000), 319–330.CrossRefMathSciNetGoogle Scholar
  3. [3]
    Ambjørn, J., Durhuus, B. & Jonsson, T., Quantum Geometry. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1997.Google Scholar
  4. [4]
    Ambjørn, J. & Varsted, S., Three-dimensional simplicial quantum gravity. Nuclear Phys. B, 373 (1992), 557–577.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Aval, J. C., Multivariate Fuss–Catalan numbers. Discrete Math., 308 (2008), 4660–4669.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Bartocci, C., Bruzzo, U., Carfora, M. & Marzuoli, A., Entropy of random coverings and 4D quantum gravity. J. Geom. Phys., 18 (1996), 247–294.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Benedetti, B., Locally Constructible Manifolds. Ph.D. Thesis, Technische Universität Berlin, Berlin, 2010.
  8. [8]
    — Collapses, products and LC manifolds. J. Combin. Theory Ser. A, 118 (2011), 586–590.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Bing, R. H., Some aspects of the topology of 3-manifolds related to the Poincaré conjecture, in Lectures on Modern Mathematics, Vol. II, pp. 93–128. Wiley, New York, 1964.Google Scholar
  10. [10]
    Björner, A., Topological methods, in Handbook of Combinatorics, Vol. 2, pp. 1819–1872. Elsevier, Amsterdam, 1995.Google Scholar
  11. [11]
    Catterall, S., Kogut, J. & Renken, R., Is there an exponential bound in fourdimensional simplicial gravity? Phys. Rev. Lett., 72 (1994), 4062–4065.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Cheeger, J., Critical points of distance functions and applications to geometry, in Geometric Topology: Recent Developments (Montecatini Terme, 1990), Lecture Notes in Math., 1504, pp. 1–38. Springer, Berlin–Heidelberg, 1991.Google Scholar
  13. [13]
    Chillingworth, D. R. J., Collapsing three-dimensional convex polyhedra. Math. Proc. Cambridge Philos. Soc., 63 (1967), 353–357. Correction in Math. Proc. Cambridge Philos. Soc., 88 (1980), 307–310.Google Scholar
  14. [14]
    Durhuus, B. & Jonsson, T., Remarks on the entropy of 3-manifolds. Nuclear Phys. B, 445 (1995), 182–192.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Ehrenborg, R. & Hachimori, M., Non-constructible complexes and the bridge index. European J. Combin., 22 (2001), 475–489.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Furch, R., Zur Grundlegung der kombinatorischen Topologie. Abh. Math. Sem. Univ. Hamburg, 3 (1923), 69–88.CrossRefGoogle Scholar
  17. [17]
    Goodman, J.E. & Pollack, R., There are asymptotically far fewer polytopes than we thought. Bull. Amer. Math. Soc., 14 (1986), 127–129.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Goodrick, R. E., Non-simplicially collapsible triangulations of In. Math. Proc. Cambridge Philos. Soc., 64 (1968), 31–36.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Gromov, M., Spaces and questions. Geom. Funct. Anal., 2000, Special Volume, Part I (2000), 118–161.Google Scholar
  20. [20]
    Grove, K., Petersen, P. V & Wu, J.Y., Geometric finiteness theorems via controlled topology. Invent. Math., 99 (1990), 205–213. Correction in Invent. Math., 104 (1991), 221–222.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Hachimori, M., Nonconstructible simplicial balls and a way of testing constructibility. Discrete Comput. Geom., 22 (1999), 223–230.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Combinatorics of Constructible Complexes. Ph.D. Thesis, Tokyo University, Tokyo, 2000.Google Scholar
  23. [23]
    — Decompositions of two-dimensional simplicial complexes. Discrete Math., 308 (2008), 2307–2312.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Simplicial complex library. Web archive, 2001. eng.html.
  25. [25]
    Hachimori, M. & Shimokawa, K., Tangle sum and constructible spheres. J. Knot Theory Ramifications, 13 (2004), 373–383.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    Hachimori, M. & Ziegler, G. M., Decompositons of simplicial balls and spheres with knots consisting of few edges. Math. Z., 235 (2000), 159–171.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    Hamstrom, M.-E. & Jerrard, R.P., Collapsing a triangulation of a “knotted” cell. Proc. Amer. Math. Soc., 21 (1969), 327–331.zbMATHMathSciNetGoogle Scholar
  28. [28]
    Hog-Angeloni, C. & Metzler, W., Geometric aspects of two-dimensional complexes, in Two-Dimensional Homotopy and Combinatorial Group Theory, London Math. Soc. Lecture Note Ser., 197, pp. 1–50. Cambridge Univ. Press, Cambridge, 1993.CrossRefGoogle Scholar
  29. [29]
    Hudson, J. F. P., Piecewise Linear Topology. University of Chicago Lecture Notes. Benjamin, New York–Amsterdam, 1969.Google Scholar
  30. [30]
    Kalai, G., Many triangulated spheres. Discrete Comput. Geom., 3 (1988), 1–14.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    Kamei, S., Cones over the boundaries of nonshellable but constructible 3-balls. Osaka J. Math., 41 (2004), 357–370.zbMATHMathSciNetGoogle Scholar
  32. [32]
    Kawauchi, A., A Survey of Knot Theory. Birkhäuser, Basel, 1996.zbMATHGoogle Scholar
  33. [33]
    Klee, V. & Kleinschmidt, P., The d-step conjecture and its relatives. Math. Oper. Res., 12 (1987), 718–755.zbMATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    Lee, C. W., Kalai’s squeezed spheres are shellable. Discrete Comput. Geom., 24 (2000), 391–396.zbMATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    Lickorish, W.B. R., An unsplittable triangulation. Michigan Math. J., 18 (1971), 203–204.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    — Unshellable triangulations of spheres. European J. Combin., 12 (1991), 527–530.zbMATHMathSciNetGoogle Scholar
  37. [37]
    Lickorish, W. B. R. & Martin, J. M., Triangulations of the 3-ball with knotted spanning 1-simplexes and collapsible rth derived subdivisions. Trans. Amer. Math. Soc., 137 (1969), 451–458.zbMATHMathSciNetGoogle Scholar
  38. [38]
    Lutz, F. H., Small examples of nonconstructible simplicial balls and spheres. SIAM J. Discrete Math., 18 (2004), 103–109.zbMATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    Matoušek, J. & Nešetřil, J., Invitation to Discrete Mathematics. Oxford University Press, Oxford, 2009.zbMATHGoogle Scholar
  40. [40]
    Pfeifle, J. & Ziegler, G. M., Many triangulated 3-spheres. Math. Ann., 330 (2004), 829–837.zbMATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    Provan, J. S. & Billera, L. J., Decompositions of simplicial complexes related to diameters of convex polyhedra. Math. Oper. Res., 5 (1980), 576–594.zbMATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    Regge, T., General relativity without coordinates. Nuovo Cimento, 19 (1961), 558–571.CrossRefMathSciNetGoogle Scholar
  43. [43]
    Regge, T. & Williams, R. M., Discrete structures in gravity. J. Math. Phys., 41 (2000), 3964–3984.zbMATHCrossRefMathSciNetGoogle Scholar
  44. [44]
    Stanley, R. P., Enumerative Combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999.Google Scholar
  45. [45]
    Tutte, W. T., A census of planar triangulations. Canad. J. Math., 14 (1962), 21–38.zbMATHCrossRefMathSciNetGoogle Scholar
  46. [46]
    — On the enumeration of convex polyhedra. J. Combin. Theory Ser. B, 28 (1980), 105–126.zbMATHCrossRefMathSciNetGoogle Scholar
  47. [47]
    Weingarten, D., Euclidean quantum gravity on a lattice. Nuclear Phys. B, 210 (1982), 229–245.CrossRefGoogle Scholar
  48. [48]
    Zeeman, E. C., Seminar on Combinatorial Topology. Institut des Hautes Études Scientifiques and University of Warwick, Paris–Coventry, 1966.Google Scholar
  49. [49]
    Ziegler, G. M., Lectures on Polytopes. Graduate Texts in Mathematics, 152. Springer, New York, 1995.Google Scholar
  50. [50]
    — Shelling polyhedral 3-balls and 4-polytopes. Discrete Comput. Geom., 19 (1998), 159–174.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2011

Authors and Affiliations

  1. 1.Institute of MathematicsFreie Universität BerlinBerlinGermany

Personalised recommendations