Acta Mathematica

On locally constructible spheres and balls

Bruno Benedetti and Günter M. Ziegler

Full-text: Open access

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.)

Note

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”.

Article information

Source
Acta Math. Volume 206, Number 2 (2011), 205-243.

Dates
Received: 22 June 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document
http://projecteuclid.org/euclid.acta/1485892546

Digital Object Identifier
doi:10.1007/s11511-011-0062-2

Mathematical Reviews number (MathSciNet)
MR2810852

Zentralblatt MATH identifier
1237.57025

Rights
2011 © Institut Mittag-Leffler

Citation

Benedetti, Bruno; Ziegler, Günter M. On locally constructible spheres and balls. Acta Math. 206 (2011), no. 2, 205--243. doi:10.1007/s11511-011-0062-2. http://projecteuclid.org/euclid.acta/1485892546.


Export citation

References

  • A lon, N., The number of polytopes, configurations and real matroids. Mathematika, 33 (1986), 62–71.
  • A mbjørn, J., B oulatov, D. V., K awamoto, N. & W atabiki, Y., Recursive sampling simulations of 3D gravity coupled to scalar fermions. Phys. Lett. B, 480 (2000), 319–330.
  • A mbjørn, J., D urhuus, B. & J onsson, T., Quantum Geometry. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1997.
  • A mbjørn, J. & V arsted, S., Three-dimensional simplicial quantum gravity. Nuclear Phys. B, 373 (1992), 557–577.
  • A val, J. C., Multivariate Fuss–Catalan numbers. Discrete Math., 308 (2008), 4660–4669.
  • B artocci, C., Bruzzo, U., C arfora, M. & M arzuoli, A., Entropy of random coverings and 4D quantum gravity. J. Geom. Phys., 18 (1996), 247–294.
  • B enedetti, B., Locally Constructible Manifolds. Ph.D. Thesis, Technische Universität Berlin, Berlin, 2010.
  • — Collapses, products and LC manifolds. J. Combin. Theory Ser. A, 118 (2011), 586–590.
  • B ing, 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.
  • B jörner, A., Topological methods, in Handbook of Combinatorics, Vol. 2, pp. 1819–1872. Elsevier, Amsterdam, 1995.
  • C atterall, S., K ogut, J. & R enken, R., Is there an exponential bound in fourdimensional simplicial gravity? Phys. Rev. Lett., 72 (1994), 4062–4065.
  • C heeger, 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.
  • C hillingworth, 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.
  • D urhuus, B. & J onsson, T., Remarks on the entropy of 3-manifolds. Nuclear Phys. B, 445 (1995), 182–192.
  • E hrenborg, R. & H achimori, M., Non-constructible complexes and the bridge index. European J. Combin., 22 (2001), 475–489.
  • F urch, R., Zur Grundlegung der kombinatorischen Topologie. Abh. Math. Sem. Univ. Hamburg, 3 (1923), 69–88.
  • G oodman, J.E. & P ollack, R., There are asymptotically far fewer polytopes than we thought. Bull. Amer. Math. Soc., 14 (1986), 127–129.
  • G oodrick, R. E., Non-simplicially collapsible triangulations of In. Math. Proc. Cambridge Philos. Soc., 64 (1968), 31–36.
  • G romov, M., Spaces and questions. Geom. Funct. Anal., 2000, Special Volume, Part I (2000), 118–161.
  • G rove, K., P etersen, P. V & W u, J.Y., Geometric finiteness theorems via controlled topology. Invent. Math., 99 (1990), 205–213. Correction in Invent. Math., 104 (1991), 221–222.
  • H achimori, M., Nonconstructible simplicial balls and a way of testing constructibility. Discrete Comput. Geom., 22 (1999), 223–230.
  • Combinatorics of Constructible Complexes. Ph.D. Thesis, Tokyo University, Tokyo, 2000.
  • — Decompositions of two-dimensional simplicial complexes. Discrete Math., 308 (2008), 2307–2312.
  • Simplicial complex library. Web archive, 2001.
  • H achimori, M. & S himokawa, K., Tangle sum and constructible spheres. J. Knot Theory Ramifications, 13 (2004), 373–383.
  • H achimori, M. & Z iegler, G. M., Decompositons of simplicial balls and spheres with knots consisting of few edges. Math. Z., 235 (2000), 159–171.
  • H amstrom, M.-E. & J errard, R.P., Collapsing a triangulation of a “knotted” cell. Proc. Amer. Math. Soc., 21 (1969), 327–331.
  • H og-A ngeloni, C. & M etzler, 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.
  • H udson, J. F. P., Piecewise Linear Topology. University of Chicago Lecture Notes. Benjamin, New York–Amsterdam, 1969.
  • K alai, G., Many triangulated spheres. Discrete Comput. Geom., 3 (1988), 1–14.
  • K amei, S., Cones over the boundaries of nonshellable but constructible 3-balls. Osaka J. Math., 41 (2004), 357–370.
  • K awauchi, A., A Survey of Knot Theory. Birkhäuser, Basel, 1996.
  • K lee, V. & K leinschmidt, P., The d-step conjecture and its relatives. Math. Oper. Res., 12 (1987), 718–755.
  • L ee, C. W., Kalai’s squeezed spheres are shellable. Discrete Comput. Geom., 24 (2000), 391–396.
  • L ickorish, W.B. R., An unsplittable triangulation. Michigan Math. J., 18 (1971), 203–204.
  • — Unshellable triangulations of spheres. European J. Combin., 12 (1991), 527–530.
  • L ickorish, W. B. R. & M artin, 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.
  • L utz, F. H., Small examples of nonconstructible simplicial balls and spheres. SIAM J. Discrete Math., 18 (2004), 103–109.
  • M atoušek, J. & N ešetřil, J., Invitation to Discrete Mathematics. Oxford University Press, Oxford, 2009.
  • P feifle, J. & Z iegler, G. M., Many triangulated 3-spheres. Math. Ann., 330 (2004), 829–837.
  • P rovan, J. S. & B illera, L. J., Decompositions of simplicial complexes related to diameters of convex polyhedra. Math. Oper. Res., 5 (1980), 576–594.
  • R egge, T., General relativity without coordinates. Nuovo Cimento, 19 (1961), 558–571.
  • R egge, T. & Williams, R. M., Discrete structures in gravity. J. Math. Phys., 41 (2000), 3964–3984.
  • S tanley, R. P., Enumerative Combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999.
  • T utte, W. T., A census of planar triangulations. Canad. J. Math., 14 (1962), 21–38.
  • — On the enumeration of convex polyhedra. J. Combin. Theory Ser. B, 28 (1980), 105–126.
  • W eingarten, D., Euclidean quantum gravity on a lattice. Nuclear Phys. B, 210 (1982), 229–245.
  • Z eeman, E. C., Seminar on Combinatorial Topology. Institut des Hautes Études Scientifiques and University of Warwick, Paris–Coventry, 1966.
  • Z iegler, G. M., Lectures on Polytopes. Graduate Texts in Mathematics, 152. Springer, New York, 1995.
  • — Shelling polyhedral 3-balls and 4-polytopes. Discrete Comput. Geom., 19 (1998), 159–174.