A point set whose space of triangulations is disconnected

Santos, Francisco

doi:10.1090/S0894-0347-00-00330-1

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
DOI: https://doi.org/10.1090/S0894-0347-00-00330-1
Published electronically: March 29, 2000
MathSciNet review: 1758756
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

AbMaRaD. 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, http://xxx.lanl.gov/e-print/math.AG/9803126.

AlexeevV. Alexeev, Complete moduli in the presence of semiabelian group action, preprint 1999. Available at Los Alamos e-print archive http//xxx.lanl.gov/e-print/math.AG/9905103.

AndersonL. 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.

Azaola M. Azaola, The Baues conjecture in corank 3, Topology, to appear.

H. J. Baues, Geometry of loop spaces and the cobar construction, Mem. Amer. Math. Soc. 25 (1980), no. 230, ix+171. MR 567799, DOI https://doi.org/10.1090/memo/0230
Louis J. Billera, Paul Filliman, and Bernd Sturmfels, Constructions and complexity of secondary polytopes, Adv. Math. 83 (1990), no. 2, 155–179. MR 1074022, DOI https://doi.org/10.1016/0001-8708%2890%2990077-Z
L. J. Billera, M. M. Kapranov, and B. Sturmfels, Cellular strings on polytopes, Proc. Amer. Math. Soc. 122 (1994), no. 2, 549–555. MR 1205482, DOI https://doi.org/10.1090/S0002-9939-1994-1205482-0
Louis J. Billera and Bernd Sturmfels, Fiber polytopes, Ann. of Math. (2) 135 (1992), no. 3, 527–549. MR 1166643, DOI https://doi.org/10.2307/2946575
A. Björner, Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819–1872. MR 1373690

BjoLut 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.

Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler, Oriented matroids, Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1993. MR 1226888

deLoera J.A. de Loera, Triangulations of Polytopes and Computational Algebra, Ph.D. thesis, Cornell University, 1995.

Jesús A. de Loera, Serkan Hoşten, Francisco Santos, and Bernd Sturmfels, The polytope of all triangulations of a point configuration, Doc. Math. 1 (1996), No. 04, 103–119. MR 1386049

TheBook J.A. de Loera, J. Rambau and F. Santos, Triangulations of polyhedra and point sets, in preparation.

J. A. de Loera, F. Santos, and J. Urrutia, The number of geometric bistellar neighbors of a triangulation, Discrete Comput. Geom. 21 (1999), no. 1, 131–142. MR 1661291, DOI https://doi.org/10.1007/PL00009405
P. H. Edelman and V. Reiner, Visibility complexes and the Baues problem for triangulations in the plane, Discrete Comput. Geom. 20 (1998), no. 1, 35–59. MR 1626683, DOI https://doi.org/10.1007/PL00009377
H. Edelsbrunner and N. R. Shah, Incremental topological flipping works for regular triangulations, Algorithmica 15 (1996), no. 3, 223–241. MR 1368251, DOI https://doi.org/10.1007/s004539900013

EilSte S. Eilenberg and N. E. Steenrod, Foundations of algebraic topology, Princeton University Press, 1952.

Günter Ewald, Über stellare Äquivalenz konvexer Polytope, Results Math. 1 (1978), no. 1, 54–60 (German). MR 510150
I. M. Gel′fand, A. V. Zelevinskiĭ, and M. M. Kapranov, Discriminants of polynomials in several variables and triangulations of Newton polyhedra, Algebra i Analiz 2 (1990), no. 3, 1–62 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 3, 449–505. MR 1073208
I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417

HuRaSaB. 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).

Ilia Itenberg and Marie-Françoise Roy, Interactions between real algebraic geometry and discrete and computational geometry, Advances in discrete and computational geometry (South Hadley, MA, 1996) Contemp. Math., vol. 223, Amer. Math. Soc., Providence, RI, 1999, pp. 217–236. MR 1661384, DOI https://doi.org/10.1090/conm/223/03139
M. M. Kapranov, B. Sturmfels, and A. V. Zelevinsky, Quotients of toric varieties, Math. Ann. 290 (1991), no. 4, 643–655. MR 1119943, DOI https://doi.org/10.1007/BF01459264
Mathematical software. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Publication of the Mathematics Research Center, No. 39. MR 0474682
Carl W. Lee, The associahedron and triangulations of the $n$-gon, European J. Combin. 10 (1989), no. 6, 551–560. MR 1022776, DOI https://doi.org/10.1016/S0195-6698%2889%2980072-1

Leesurvey 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.

MacTho D. Maclagan and R. Thomas, Combinatorics of the Toric Hilbert Scheme, preprint 1999. Available at Los Alamos e-print archive, http://xxx.lanl.gov/e-print/math.AG/9912014.

Robert MacPherson, Combinatorial differential manifolds, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 203–221. MR 1215966
Robert Morelli, The birational geometry of toric varieties, J. Algebraic Geom. 5 (1996), no. 4, 751–782. MR 1486987
Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
Udo Pachner, P.L. homeomorphic manifolds are equivalent by elementary shellings, European J. Combin. 12 (1991), no. 2, 129–145. MR 1095161, DOI https://doi.org/10.1016/S0195-6698%2813%2980080-7

PeeStiI. Peeva and N. Stillman, Toric Hilbert schemes, preprint 1999. Available at http://www.math.cornell.edu/~irena/publications.html.

topcom J. Rambau, TOPCOM: Triangulations Of Point Configurations and Oriented Matroids, software available at http://www.zib.de/rambau/TOPCOM.html.

RamSan J. Rambau and F. Santos, The generalized Baues problem for cyclic polytopes, European J. Combin. 21 (2000), 65–83.

J. Rambau and G. M. Ziegler, Projections of polytopes and the generalized Baues conjecture, Discrete Comput. Geom. 16 (1996), no. 3, 215–237. MR 1410159, DOI https://doi.org/10.1007/BF02711510

ReinersurveyV. 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.

SantosfewflipsF. Santos, Triangulations with very few geometric bistellar neighbors, Discrete Comput. Geom. 23 (2000), 15–33.

SantosOMtriF. Santos, Triangulations of Oriented Matroids, Mem. Am. Math. Soc., to appear. Available at http://www.matesco.unican.es/~santos/Articulos/OMtri.ps.gz

Santosrefine F. Santos, On the refinements of a polyhedral subdivision, preprint, 1999, 26 pages. Available at http://www.matesco.unican.es/~santos/Articulos/refine.ps.gz

Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc. 1 (1988), no. 3, 647–681. MR 928904, DOI https://doi.org/10.1090/S0894-0347-1988-0928904-4
James Dillon Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293–312. MR 0158400, DOI https://doi.org/10.1090/S0002-9947-1963-0158400-5
Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949

Sturmfelsagraded B. Sturmfels, The geometry of $A$-graded algebras, preprint 1994. Available at Los Alamos e-print http://xxx.lanl.gov/e-print/math.AG/9410032.

Bernd Sturmfels and Günter M. Ziegler, Extension spaces of oriented matroids, Discrete Comput. Geom. 10 (1993), no. 1, 23–45. MR 1215321, DOI https://doi.org/10.1007/BF02573961
Dov Tamari, The algebra of bracketings and their enumeration, Nieuw Arch. Wisk. (3) 10 (1962), 131–146. MR 146227
Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028
Günter M. Ziegler, Recent progress on polytopes, Advances in discrete and computational geometry (South Hadley, MA, 1996) Contemp. Math., vol. 223, Amer. Math. Soc., Providence, RI, 1999, pp. 395–406. MR 1661396, DOI https://doi.org/10.1090/conm/223/03151