Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



A point set whose space of triangulations is disconnected

Author: Francisco Santos
Journal: J. Amer. Math. Soc. 13 (2000), 611-637
MSC (2000): Primary 52B11; Secondary 52B20
Published electronically: March 29, 2000
MathSciNet review: 1758756
Full-text PDF

Abstract | References | Similar Articles | Additional Information


By the ``space of triangulations" of a finite point configuration $\mathcal{A}$ we mean either of the following two objects: the graph of triangulations of $\mathcal{A}$, whose vertices are the triangulations of $\mathcal{A}$ and whose edges are the geometric bistellar operations between them or the partially ordered set (poset) of all polyhedral subdivisions of $\mathcal{A}$ ordered by coherent refinement. The latter is a modification of the more usual Baues poset of $\mathcal{A}$. It is explicitly introduced here for the first time and is of special interest in the theory of toric varieties.

We construct an integer point configuration in dimension 6 and a triangulation of it which admits no geometric bistellar operations. This triangulation is an isolated point in both the graph and the poset, which proves for the first time that these two objects cannot be connected.

References [Enhancements On Off] (What's this?)

  • 1. D. ABRAMOVICH, K. MATSUKI AND S. RASHID, A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension, Tohoku Math. J. (2) 51 (1999), no. 4, 489-537. Available at Los Alamos e-print archive, CMP 2000:05
  • 2. V. ALEXEEV, Complete moduli in the presence of semiabelian group action, preprint 1999. Available at Los Alamos e-print archive http//
  • 3. L. ANDERSON, Matroid bundles and sphere bundles, in: New Perspectives in Algebraic Combinatorics (L. J. Billera, A. Björner, C. Greene, R. E. Simion and R. P. Stanley, eds.), MSRI publications 38 (1999), Cambridge University Press, pp. 1-21. CMP 2000:07
  • 4. M. AZAOLA, The Baues conjecture in corank 3, Topology, to appear.
  • 5. J. BAUES, Geometry of loop spaces and the cobar construction, Mem. Am. Math. Soc. 25 (1980), 99-124. MR 81m:55010
  • 6. L. BILLERA, P. FILLIMAN AND B. STURMFELS, Constructions and complexity of secondary polytopes, Adv. Math. 83 (1990), 155-179. MR 92d:52028
  • 7. L. BILLERA, M.M. KAPRANOV AND B. STURMFELS, Cellular strings on polytopes, Proc. Am. Math. Soc. 122(2) (1994), 549-555. MR 95a:52020
  • 8. L. BILLERA AND B. STURMFELS, Fiber polytopes, Ann. Math. 135 (1992), 527-549. MR 93e:52019
  • 9. A. BJÖRNER, Topological methods, in: Handbook of Combinatorics (R.L. Graham, M. Grötschel and L. Lovász, eds.), Elsevier, Amsterdam, 1995, pp. 1819-1872. MR 96m:52012
  • 10. A. BJÖRNER AND F. H. LUTZ, Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere, Experiment. Math., to appear.
  • 11. A. BJÖRNER, M. LAS VERGNAS, B. STURMFELS, N. WHITE AND G. M. ZIEGLER, Oriented Matroids, Cambridge University Press, Cambridge, 1992. MR 95e:52023
  • 12. J.A. DE LOERA, Triangulations of Polytopes and Computational Algebra, Ph.D. thesis, Cornell University, 1995.
  • 13. J.A. DE LOERA, S. HO¸STEN, F. SANTOS AND B. STURMFELS, The polytope of all triangulations of a point configuration, Doc. Math. 1 (1996), 103-119. MR 97e:52017
  • 14. J.A. DE LOERA, J. RAMBAU AND F. SANTOS, Triangulations of polyhedra and point sets, in preparation.
  • 15. J.A. DE LOERA, F. SANTOS AND J. URRUTIA, The number of geometric bistellar neighbors of a triangulation, Discrete Comput. Geom. 21 (1999) 1, 131-142. MR 99k:52025
  • 16. P. EDELMAN AND V. REINER, Visibility complexes and the Baues problem for triangulations in the plane, Discrete Comput. Geom. 20 (1998), 35-59. MR 99h:52013
  • 17. H. EDELSBRUNNER AND N.R. SHAH, Incremental topological flipping works for regular triangulations, Algorithmica 15 (1996), 223-241. MR 96j:65161
  • 18. S. EILENBERG AND N. E. STEENROD, Foundations of algebraic topology, Princeton University Press, 1952. MR 14:398b
  • 19. G. EWALD, Über stellare Äquivalenz konvexer Polytope, Resultate Math. 1 (1978), 54-60. MR 80b:52012
  • 20. I.M. GEL'FAND, M.M. KAPRANOV AND A.V. ZELEVINSKY, Discriminants of polynomials in several variables and triangulations of Newton polyhedra, Leningrad Math. J. 2 (1990), 449-505. (English translation of Algebra i Analiz 2 (1990), 1-62, in Russian). MR 91m:14080
  • 21. I.M. GEL'FAND, M.M. KAPRANOV AND A.V. ZELEVINSKY, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994. MR 95e:14045
  • 22. B. HUBER, J. RAMBAU AND F. SANTOS, The Cayley Trick, Lawrence polytopes, and the Bohne-Dress theorem on zonotopal tilings, preprint 1999. To appear in J. Eur. Math. Soc. (JEMS).
  • 23. I. ITENBERG AND M.-F. ROY, Interactions between real algebraic geometry and discrete and computational geometry, in: Advances in discrete and computational geometry. Proceedings of the 1996 AMS-IMS-SIAM joint summer research conference on discrete and computational geometry: ten years later (Bernard Chazelle et al. eds.), pp. 217-236. Contemp. Math. 223, Am. Math. Soc., Providence, 1999. MR 99j:14052
  • 24. M.M. KAPRANOV, B. STURMFELS AND A.V. ZELEVINSKY, Quotients of toric varieties, Math. Ann. 290 (1991), 643-655. MR 92g:14050
  • 25. C.L. LAWSON, Software for $C^1$-interpolation, in Mathematical Software III (John Rice ed.), Academic Press, New York, 1977. MR 57:14316
  • 26. C.W. LEE, The associahedron and triangulations of the $n$-gon, European J. Combin. 10 (1990), 551-560. MR 90i:52010
  • 27. C.W. LEE, Subdivisions and triangulations of polytopes, in Handbook of Discrete and Computational Geometry (J.E. Goodman and J. O'Rourke eds.), CRC Press, New York, 1997, pp. 271-290. CMP 2000:06
  • 28. D. MACLAGAN AND R. THOMAS, Combinatorics of the Toric Hilbert Scheme, preprint 1999. Available at Los Alamos e-print archive,
  • 29. R. D. MACPHERSON, Combinatorial Differential Manifolds, in Topological Methods in Modern Mathematics: a Symposium in Honor of John Milnor's Sixtieth Birthday (L. R. Goldberg and A. Phillips, eds.), Publish or Perish, Houston, 1993, pp. 203-221. MR 94d:57047
  • 30. R. MORELLI, The birational geometry of toric varieties, J. Alg. Geom. 5 (1996), 751-782. MR 99b:14056
  • 31. T. ODA, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Springer-Verlag, 1988. MR 88m:14038
  • 32. U. PACHNER, PL-homeomorphic manifolds are equivalent by elementary shellings, European J. Combin. 12 (1991), 129-145. MR 92d:52040
  • 33. I. PEEVA AND N. STILLMAN, Toric Hilbert schemes, preprint 1999. Available at
  • 34. J. RAMBAU, TOPCOM: Triangulations Of Point Configurations and Oriented Matroids, software available at
  • 35. J. RAMBAU AND F. SANTOS, The generalized Baues problem for cyclic polytopes, European J. Combin. 21 (2000), 65-83.
  • 36. J. RAMBAU AND G.M. ZIEGLER, Projections of polytopes and the generalized Baues conjecture, Discrete Comput. Geom. (3) 16 (1996), 215-237. MR 97i:52011
  • 37. V. REINER, The generalized Baues problem, in: New Perspectives in Algebraic Combinatorics (L. J. Billera, A. Björner, C. Greene, R. E. Simion and R. P. Stanley, eds.), MSRI publications 38 (1999), Cambridge University Press, pp. 293-336. CMP 2000:07
  • 38. F. SANTOS, Triangulations with very few geometric bistellar neighbors, Discrete Comput. Geom. 23 (2000), 15-33. CMP 2000:05
  • 39. F. SANTOS, Triangulations of Oriented Matroids, Mem. Am. Math. Soc., to appear. Available at
  • 40. F. SANTOS, On the refinements of a polyhedral subdivision, preprint, 1999, 26 pages. Available at
  • 41. D. SLEATOR, R. TARJAN AND W. THURSTON, Rotation distance, triangulations, and hyperbolic geometry, J. Am. Math. Soc. 1 (1988), 647-681. MR 90h:68026
  • 42. J.D. STASHEFF, Homotopy associativity of H-spaces, Trans. Am. Math. Soc. 108 (1963), 275-292. MR 28:1623
  • 43. B. STURMFELS, Gröbner bases and convex polytopes, University Series Lectures 8, American Mathematical Society, Providence, 1995. MR 97b:13034
  • 44. B. STURMFELS, The geometry of $A$-graded algebras, preprint 1994. Available at Los Alamos e-print
  • 45. B. STURMFELS AND G. M. ZIEGLER, Extension spaces of oriented matroids, Discrete Comput. Geom. 10 (1993), 23-45. MR 94i:52015
  • 46. D. TAMARI, The algebra of bracketings and their enumeration, Nieuw Arch. Wisk. 10 (1962), 131-146. MR 26:3749
  • 47. G. M. ZIEGLER, Lectures on Polytopes, Springer-Verlag, New York, 1994. MR 96a:52011
  • 48. G. M. ZIEGLER, Recent progress on polytopes, in ``Advances in Discrete and Computational Geometry" (B. Chazelle, J.E. Goodman, R. Pollack, eds.), Contemporary Mathematics 223 (1998), Am. Math. Soc., Providence, pp. 395-406. MR 99g:52011

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 52B11, 52B20

Retrieve articles in all journals with MSC (2000): 52B11, 52B20

Additional Information

Francisco Santos
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, E-39005, Santander, Spain

Keywords: Triangulation, point configuration, bistellar flip, polyhedral subdivision, Baues problem
Received by editor(s): August 3, 1999
Received by editor(s) in revised form: March 6, 2000
Published electronically: March 29, 2000
Additional Notes: This research was partially supported by grant PB97–0358 of the Spanish Dirección General de Enseñanza Superior e Investigación Científica.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society