A point set whose space of triangulations is disconnected
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- by Francisco Santos;
- J. Amer. Math. Soc. 13 (2000), 611-637
- DOI: https://doi.org/10.1090/S0894-0347-00-00330-1
- Published electronically: March 29, 2000
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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.References
- 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 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 10.1016/0001-8708(90)90077-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 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 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 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 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 10.1007/s004539900013
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- 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, DOI 10.1007/978-0-8176-4771-1 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 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 10.1007/BF01459264
- Mathematical software. III, Publication of the Mathematics Research Center, University of Wisconsin, vol. 39, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. MR 474682
- Carl W. Lee, The associahedron and triangulations of the $n$-gon, European J. Combin. 10 (1989), no. 6, 551–560. MR 1022776, DOI 10.1016/S0195-6698(89)80072-1 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 10.1016/S0195-6698(13)80080-7 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 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 10.1090/S0894-0347-1988-0928904-4
- James Dillon Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 293–312. 108 (1963), 275-292; ibid. MR 158400, DOI 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, DOI 10.1090/ulect/008 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 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, DOI 10.1007/978-1-4613-8431-1
- 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 10.1090/conm/223/03151
Bibliographic Information
- Francisco Santos
- Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, E-39005, Santander, Spain
- MR Author ID: 360182
- ORCID: 0000-0003-2120-9068
- Email: santos@matesco.unican.es
- 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.
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1758756