Subdivisions, shellability, and collapsibility of products


We prove that the second derived subdivision of any rectilinear triangulation of any convex polytope is shellable. Also, we prove that the first derived subdivision of every rectilinear triangulation of any convex 3-dimensional polytope is shellable. This complements Mary Ellen Rudin's classical example of a non-shellable rectilinear triangulation of the tetrahedron. Our main tool is a new relative notion of shellability that characterizes the behavior of shellable complexes under gluing.

As a corollary, we obtain a new characterization of the PL property in terms of shellability: A triangulation of a sphere or of a ball is PL if and only if it becomes shellable after sufficiently many derived subdivisions. This improves on PL approximation theorems by Whitehead, Zeeman and Glaser, and answers a question by Billera and Swartz.

We also show that any contractible complex can be made collapsible by repeatedly taking products with an interval. This strengthens results by Dierker and Lickorish, and resolves a conjecture of Oliver. Finally, we give an example that this behavior extends to non-evasiveness, thereby answering a question of Welker.

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


  1. [1]

    K. A. Adiprasito: Methods from Differential Geometry in Polytope Theory, Berlin, DE, May 2013, published on

    Google Scholar 

  2. [2]

    K. A. Adiprasito and B. Benedetti: Metric geometry, convexity and collapsibility, preprint, available at arXiv:1107.5789.

  3. [3]

    K. A. Adiprasito and B. Benedetti: Tight complexes in 3-space admit perfect discrete Morse functions, European J. Combinatorics 45 (2015), 71–84.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    K. A. Adiprasito and I. Izmestiev: Derived subdivisions make every PL sphere polytopal, Isr. J. Math. (2014), to appear.

    Google Scholar 

  5. [5]

    K. A. Adiprasito and R. Sanyal: Relative Stanley-Reisner Theory and Upper Bound Theorems for Minkowski sums, preprint, arXiv:1405.7368.

  6. [6]

    J. W. Alexander, The combinatorial theory of complexes, Ann. of Math. 31 (2) (1930), 292–320.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    J. A. Barmak and E. G. Minian: Strong homotopy types, nerves and collapses, Discrete Comput. Geom. 47 (2012), 301–328.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    B. Benedetti: Discrete Morse theory for manifolds with boundary, Trans. Amer. Math. Soc. 364 (2012), 6631–6670.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    R. H. Bing: Some aspects of the topology of 3-manifolds related to the Poincaré conjecture, Lectures on modern mathematics, Vol. II, Wiley, New York, 1964, 93–128.

    Google Scholar 

  10. [10]

    A. Björner: Topological methods, Handbook of Combinatorics, Vol. 1, 2, Elsevier, Amsterdam, 1995, 1819–1872.

    Google Scholar 

  11. [11]

    A. Björner and M. L. Wachs: Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc. 348 (1996), 1299–1327.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    H. Bruggesser and P. Mani: Shellable decompositions of cells and spheres, Math. Scand. 29 (1971), 197–205 (1972).

    MathSciNet  MATH  Google Scholar 

  13. [13]

    V. M. Buchstaber and T. E Panov: Torus actions and their applications in topology and combinatorics, vol. 24, American Mathematical Society, 2002.

    Google Scholar 

  14. [14]

    D. R. J. Chillingworth: Collapsing three-dimensional convex polyhedra, Proc. Cambridge Philos. Soc. 63 (1967), 353–357, Correction in Math. Proc. Cambridge Philos. Soc. 88 (2) (1980), 307–310.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    M. M. Cohen: A course in Simple-Homotopy Theory, Springer-Verlag, New York, 1973, Graduate Texts in Mathematics, Vol. 10.

    Google Scholar 

  16. [16]

    M. M. Cohen: Dimension estimates in collapsing X×I q, Topology 14 (1975), 253–256.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    M. Courdurier: On stars and links of shellable polytopal complexes, Journal of Combinatorial Theory, Series A 113 (2006), 692–697.

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    S. L. Čukić and E. Delucchi: Simplicial shellable spheres via combinatorial blowups, Proc. Amer. Math. Soc. 135 (2007), 2403–2414 (electronic).

    MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    G. Danaraj and V. Klee: Shellings of spheres and polytopes, Duke Mathematical Journal 41 (1974), 443–451.

    MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    M. W. Davis and G. Moussong: Notes on nonpositively curved polyhedra, in: Low-dimensional topology (Eger 1996/Budapest 1998), Bolyai Soc. Math. Stud., vol. 8, János Bolyai Math. Soc., Budapest, 1999, 11-94.

    Google Scholar 

  21. [21]

    P. Dierker: Note on collapsing K × I where K is a contractible polyhedron, Proc. Amer. Math. Soc. 19 (1968), 425–428.

    MathSciNet  MATH  Google Scholar 

  22. [22]

    R. Edwards and D. Gillman: Any spine of the cube is 2-collapsible, Canad. J. Math. 35 (1983), 43–48.

    MathSciNet  Article  MATH  Google Scholar 

  23. [23]

    D. Gillman and D. Rolfsen: The Zeeman conjecture for standard spines is equivalent to the Poincaré conjecture, Topology 22 (1983), 315–323.

    MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    L. C. Glaser: Geometrical combinatorial topology, i., Van Nostrand Reinhold Mathematical Studies Vol. 27. New York, 1970.

    Google Scholar 

  25. [25]

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

    MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    B. Grünbaum: Convex polytopes, second ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003.

    Google Scholar 

  27. [27]

    M. Hachimori and G. M. Ziegler: Decompositons of simplicial balls and spheres with knots consisting of few edges, Math. Z. 235 (2000), 159–171.

    MathSciNet  Article  MATH  Google Scholar 

  28. [28]

    G. Hetyei: Invariants des complexes cubiques, Ann. Sci. Math. Qué. 20 (1996), 35–52 (French).

    MathSciNet  MATH  Google Scholar 

  29. [29]

    J. F. P. Hudson: Piecewise Linear Topology, University of Chicago Lecture Notes, W. A. Benjamin, Inc., New York-Amsterdam, 1969.

    Google Scholar 

  30. [30]

    J. Kahn, M. Saks and D. Sturtevant: A topological approach to evasiveness, Combinatorica 4 (1984), 297–306.

    MathSciNet  Article  MATH  Google Scholar 

  31. [31]

    W. B. R. Lickorish: On collapsing X 2×I, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, 157–160.

    Google Scholar 

  32. [32]

    W. B. R. Lickorish: Unshellable triangulations of spheres, European J. Combin. 12 (1991), 527–530.

    MathSciNet  Article  MATH  Google Scholar 

  33. [33]

    W. B. R. Lickorish: Simplicial moves on complexes and manifolds, Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, 299–320 (electronic).

    Google Scholar 

  34. [34]

    S. Matveev and D. Rolfsen: Zeeman's collapsing conjecture, Two-dimensional Homotopy and Combinatorial Group Theory, London Math. Soc. Lecture Note Ser., vol. 197, Cambridge Univ. Press, Cambridge, 1993, 335–364.

    Google Scholar 

  35. [35]

    M. H. A. Newman: On the foundations of combinatorial analysis situs, Proc. Royal Acad. Amsterdam, vol. 29, 1926, 610–641.

    Google Scholar 

  36. [36]

    U. Pachner: Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten, Abh. Math. Sem. Univ. Hamburg 57 (1987), 69–86.

    MathSciNet  Article  MATH  Google Scholar 

  37. [37]

    J. S. Provan and L. J. Billera: Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), 576–594.

    MathSciNet  Article  MATH  Google Scholar 

  38. [38]

    C. P. Rourke and B. J. Sanderson: Introduction to Piecewise-Linear Topology, Springer, New York, 1972, Ergebnisse Series vol. 69.

    Google Scholar 

  39. [39]

    M. E. Rudin: An unshellable triangulation of a tetrahedron, Bull. Amer. Math. Soc. 64 (1958), 90–91.

    MathSciNet  Article  MATH  Google Scholar 

  40. [40]

    P. Schenzel: On the number of faces of simplicial complexes and the purity of Frobe-nius, Math. Z. 178 (1) (1981), 125–142.

    MathSciNet  Article  MATH  Google Scholar 

  41. [41]

    R. P. Stanley: Combinatorics and Commutative Algebra, Progress in Mathematics, vol. 41, Birkhäuser Boston Inc., Boston, MA, 1983.

    Google Scholar 

  42. [42]

    G. C. Verchota and A. L. Vogel: The multidirectional Neumann problem in R4, Math. Ann. 335 (2006), 571–644.

    MathSciNet  Article  MATH  Google Scholar 

  43. [43]

    I. A. Volodin, V. E. Kuznecov and A. T. Fomenko: The problem of the algorithmic discrimination of the standard three-dimensional sphere, Uspehi Mat. Nauk 29 (1974), 71–168, Appendix by S. P. Novikov.

    MathSciNet  Google Scholar 

  44. [44]

    V. Welker: Constructions preserving evasiveness and collapsibility, Discrete Math. 207 (1999), 243–255.

    MathSciNet  Article  MATH  Google Scholar 

  45. [45]

    J. H. C. Whitehead: Simplicial Spaces, Nuclei and m-Groups, Proc. London Math. Soc. S2-45 (1939), 243.

    MathSciNet  Article  MATH  Google Scholar 

  46. [46]

    E. C. Zeeman Seminar on Combinatorial Topology, Institut des Hautes Etudes Scientifiques, Paris, 1966 (English).

    Google Scholar 

  47. [47]

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

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Karim A. Adiprasito.

Additional information

This work was supported by the DFG within the research training group “Methods for Discrete Structures” (GRK1408) and by the Romanian NASR, CNCS — UEFISCDI, project PN-II-ID-PCE-2011-3-0533.

Supported by the Swedish Research Council, grant “Triangulerade Mångfalder, Knutteori i diskrete Morseteori”, and by the DFG grant “Discretization in Geometry and Dynamics”.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Adiprasito, K.A., Benedetti, B. Subdivisions, shellability, and collapsibility of products. Combinatorica 37, 1–30 (2017).

Download citation

Mathematics Subject Classification (2000)

  • 52B22
  • 05E45
  • 57Q10