Skip to main content

On locally constructible spheres and balls

Abstract

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.)

This is a preview of subscription content, access via your institution.

References

  1. [1]

    Alon, N., The number of polytopes, configurations and real matroids. Mathematika, 33 (1986), 62–71.

    MATH  Article  MathSciNet  Google 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.

    Article  MathSciNet  Google 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.

    Article  MathSciNet  Google Scholar 

  5. [5]

    Aval, J. C., Multivariate Fuss–Catalan numbers. Discrete Math., 308 (2008), 4660–4669.

    MATH  Article  MathSciNet  Google 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.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    Benedetti, B., Locally Constructible Manifolds. Ph.D. Thesis, Technische Universität Berlin, Berlin, 2010. http://opus.kobv.de/tuberlin/volltexte/2010/2519/.

  8. [8]

    — Collapses, products and LC manifolds. J. Combin. Theory Ser. A, 118 (2011), 586–590.

    MATH  Article  MathSciNet  Google 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.

  10. [10]

    Björner, A., Topological methods, in Handbook of Combinatorics, Vol. 2, pp. 1819–1872. Elsevier, Amsterdam, 1995.

  11. [11]

    Catterall, S., Kogut, J. & Renken, R., Is there an exponential bound in fourdimensional simplicial gravity? Phys. Rev. Lett., 72 (1994), 4062–4065.

    MATH  Article  MathSciNet  Google 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.

  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.

  14. [14]

    Durhuus, B. & Jonsson, T., Remarks on the entropy of 3-manifolds. Nuclear Phys. B, 445 (1995), 182–192.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    Ehrenborg, R. & Hachimori, M., Non-constructible complexes and the bridge index. European J. Combin., 22 (2001), 475–489.

    MATH  Article  MathSciNet  Google Scholar 

  16. [16]

    Furch, R., Zur Grundlegung der kombinatorischen Topologie. Abh. Math. Sem. Univ. Hamburg, 3 (1923), 69–88.

    Article  Google 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.

    MATH  Article  MathSciNet  Google Scholar 

  18. [18]

    Goodrick, R. E., Non-simplicially collapsible triangulations of In. Math. Proc. Cambridge Philos. Soc., 64 (1968), 31–36.

    MATH  Article  MathSciNet  Google Scholar 

  19. [19]

    Gromov, M., Spaces and questions. Geom. Funct. Anal., 2000, Special Volume, Part I (2000), 118–161.

  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.

    MATH  Article  MathSciNet  Google Scholar 

  21. [21]

    Hachimori, M., Nonconstructible simplicial balls and a way of testing constructibility. Discrete Comput. Geom., 22 (1999), 223–230.

    MATH  Article  MathSciNet  Google Scholar 

  22. [22]

    Combinatorics of Constructible Complexes. Ph.D. Thesis, Tokyo University, Tokyo, 2000.

  23. [23]

    — Decompositions of two-dimensional simplicial complexes. Discrete Math., 308 (2008), 2307–2312.

    MATH  Article  MathSciNet  Google Scholar 

  24. [24]

    Simplicial complex library. Web archive, 2001. http://infoshako.sk.tsukuba.ac.jp/~HACHI/math/library/index eng.html.

  25. [25]

    Hachimori, M. & Shimokawa, K., Tangle sum and constructible spheres. J. Knot Theory Ramifications, 13 (2004), 373–383.

    MATH  Article  MathSciNet  Google 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.

    MATH  Article  MathSciNet  Google Scholar 

  27. [27]

    Hamstrom, M.-E. & Jerrard, R.P., Collapsing a triangulation of a “knotted” cell. Proc. Amer. Math. Soc., 21 (1969), 327–331.

    MATH  MathSciNet  Google 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.

    Book  Google Scholar 

  29. [29]

    Hudson, J. F. P., Piecewise Linear Topology. University of Chicago Lecture Notes. Benjamin, New York–Amsterdam, 1969.

  30. [30]

    Kalai, G., Many triangulated spheres. Discrete Comput. Geom., 3 (1988), 1–14.

    MATH  Article  MathSciNet  Google Scholar 

  31. [31]

    Kamei, S., Cones over the boundaries of nonshellable but constructible 3-balls. Osaka J. Math., 41 (2004), 357–370.

    MATH  MathSciNet  Google Scholar 

  32. [32]

    Kawauchi, A., A Survey of Knot Theory. Birkhäuser, Basel, 1996.

    MATH  Google Scholar 

  33. [33]

    Klee, V. & Kleinschmidt, P., The d-step conjecture and its relatives. Math. Oper. Res., 12 (1987), 718–755.

    MATH  Article  MathSciNet  Google Scholar 

  34. [34]

    Lee, C. W., Kalai’s squeezed spheres are shellable. Discrete Comput. Geom., 24 (2000), 391–396.

    MATH  Article  MathSciNet  Google Scholar 

  35. [35]

    Lickorish, W.B. R., An unsplittable triangulation. Michigan Math. J., 18 (1971), 203–204.

    MATH  Article  MathSciNet  Google Scholar 

  36. [36]

    — Unshellable triangulations of spheres. European J. Combin., 12 (1991), 527–530.

    MATH  MathSciNet  Google 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.

    MATH  MathSciNet  Google Scholar 

  38. [38]

    Lutz, F. H., Small examples of nonconstructible simplicial balls and spheres. SIAM J. Discrete Math., 18 (2004), 103–109.

    MATH  Article  MathSciNet  Google Scholar 

  39. [39]

    Matoušek, J. & Nešetřil, J., Invitation to Discrete Mathematics. Oxford University Press, Oxford, 2009.

    MATH  Google Scholar 

  40. [40]

    Pfeifle, J. & Ziegler, G. M., Many triangulated 3-spheres. Math. Ann., 330 (2004), 829–837.

    MATH  Article  MathSciNet  Google 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.

    MATH  Article  MathSciNet  Google Scholar 

  42. [42]

    Regge, T., General relativity without coordinates. Nuovo Cimento, 19 (1961), 558–571.

    Article  MathSciNet  Google Scholar 

  43. [43]

    Regge, T. & Williams, R. M., Discrete structures in gravity. J. Math. Phys., 41 (2000), 3964–3984.

    MATH  Article  MathSciNet  Google Scholar 

  44. [44]

    Stanley, R. P., Enumerative Combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999.

  45. [45]

    Tutte, W. T., A census of planar triangulations. Canad. J. Math., 14 (1962), 21–38.

    MATH  Article  MathSciNet  Google Scholar 

  46. [46]

    — On the enumeration of convex polyhedra. J. Combin. Theory Ser. B, 28 (1980), 105–126.

    MATH  Article  MathSciNet  Google Scholar 

  47. [47]

    Weingarten, D., Euclidean quantum gravity on a lattice. Nuclear Phys. B, 210 (1982), 229–245.

    Article  Google Scholar 

  48. [48]

    Zeeman, E. C., Seminar on Combinatorial Topology. Institut des Hautes Études Scientifiques and University of Warwick, Paris–Coventry, 1966.

  49. [49]

    Ziegler, G. M., Lectures on Polytopes. Graduate Texts in Mathematics, 152. Springer, New York, 1995.

  50. [50]

    — Shelling polyhedral 3-balls and 4-polytopes. Discrete Comput. Geom., 19 (1998), 159–174.

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Günter M. Ziegler.

Additional information

B. B. was supported by DFG via the Berlin Mathematical School, G. M. Z. was partially supported by DFG. Both authors are supported by ERC Advanced Grant No. 247029 “SDModels”.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Benedetti, B., Ziegler, G.M. On locally constructible spheres and balls. Acta Math 206, 205–243 (2011). https://doi.org/10.1007/s11511-011-0062-2

Download citation

Keywords

  • Span Tree
  • Simplicial Complex
  • Dual Graph
  • Interior Vertex
  • Free Face