A point set whose space of triangulations is disconnected

Santos, Francisco¹

¹ Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, E-39005, Santander, Spain

Journal of the American Mathematical Society, Tome 13 (2000) no. 3, pp. 611-637

Résumé

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.

Détail

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

Bibliographie

[1] Baues, H. J. Geometry of loop spaces and the cobar construction Mem. Amer. Math. Soc. 1980

[2] Billera, Louis J., Filliman, Paul, Sturmfels, Bernd Constructions and complexity of secondary polytopes Adv. Math. 1990 155 179

[3] Billera, L. J., Kapranov, M. M., Sturmfels, B. Cellular strings on polytopes Proc. Amer. Math. Soc. 1994 549 555

[4] Billera, Louis J., Sturmfels, Bernd Fiber polytopes Ann. of Math. (2) 1992 527 549

[5] Bjã¶Rner, A. Topological methods 1995 1819 1872

[6] Bjã¶Rner, Anders, Las Vergnas, Michel, Sturmfels, Bernd, White, Neil, Ziegler, Gã¼Nter M. Oriented matroids 1993

[7] De Loera, Jesãºs A., HoåŸTen, Serkan, Santos, Francisco, Sturmfels, Bernd The polytope of all triangulations of a point configuration Doc. Math. 1996

[8] De Loera, J. A., Santos, F., Urrutia, J. The number of geometric bistellar neighbors of a triangulation Discrete Comput. Geom. 1999 131 142

[9] Edelman, P. H., Reiner, V. Visibility complexes and the Baues problem for triangulations in the plane Discrete Comput. Geom. 1998 35 59

[10] Edelsbrunner, H., Shah, N. R. Incremental topological flipping works for regular triangulations Algorithmica 1996 223 241

[11] Hebroni, P. Sur les inverses des éléments dérivables dans un anneau abstrait C. R. Acad. Sci. Paris 1939 285 287

[12] Ewald, Gã¼Nter Über stellare Äquivalenz konvexer Polytope Results Math. 1978 54 60

[13] Gel′Fand, I. M., Zelevinskiä­, A. V., Kapranov, M. M. Discriminants of polynomials in several variables and triangulations of Newton polyhedra Algebra i Analiz 1990 1 62

[14] Gel′Fand, I. M., Kapranov, M. M., Zelevinsky, A. V. Discriminants, resultants, and multidimensional determinants 1994

[15] Itenberg, Ilia, Roy, Marie-Franã§Oise Interactions between real algebraic geometry and discrete and computational geometry 1999 217 236

[16] Kapranov, M. M., Sturmfels, B., Zelevinsky, A. V. Quotients of toric varieties Math. Ann. 1991 643 655

[17] Mathematical software. III 1977

[18] Lee, Carl W. The associahedron and triangulations of the 𝑛-gon European J. Combin. 1989 551 560

[19] Macpherson, Robert Combinatorial differential manifolds 1993 203 221

[20] Morelli, Robert The birational geometry of toric varieties J. Algebraic Geom. 1996 751 782

[21] Oda, Tadao Convex bodies and algebraic geometry 1988

[22] Pachner, Udo P.L. homeomorphic manifolds are equivalent by elementary shellings European J. Combin. 1991 129 145

[23] Rambau, J., Ziegler, G. M. Projections of polytopes and the generalized Baues conjecture Discrete Comput. Geom. 1996 215 237

[24] Sleator, Daniel D., Tarjan, Robert E., Thurston, William P. Rotation distance, triangulations, and hyperbolic geometry J. Amer. Math. Soc. 1988 647 681

[25] Stasheff, James Dillon Homotopy associativity of 𝐻-spaces. I, II Trans. Amer. Math. Soc. 1963 293 312

[26] Sturmfels, Bernd Gröbner bases and convex polytopes 1996

[27] Sturmfels, Bernd, Ziegler, Gã¼Nter M. Extension spaces of oriented matroids Discrete Comput. Geom. 1993 23 45

[28] Tamari, Dov The algebra of bracketings and their enumeration Nieuw Arch. Wisk. (3) 1962 131 146

[29] Ziegler, Gã¼Nter M. Lectures on polytopes 1995

[30] Ziegler, Gã¼Nter M. Recent progress on polytopes 1999 395 406