## A point set whose space of triangulations is disconnected (2000)

Venue: | J. AMER. MATH. SOC |

Citations: | 18 - 5 self |

### BibTeX

@ARTICLE{Santos00apoint,

author = {Francisco Santos},

title = {A point set whose space of triangulations is disconnected},

journal = {J. AMER. MATH. SOC},

year = {2000},

volume = {13},

pages = {611--637}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

358 |
Lectures on polytopes
- Ziegler
- 1995
(Show Context)
Citation Context ...he secondary polytope of A introduced by Gel’fand et al. [20]. This is a polytope of dimension |A| − dim(A) − 1 whose vertices are in bijection with the regular (or coherent) triangulations of A (see =-=[6, 21, 27, 47]-=- and also Section 4.2). Triangulations with less than |A| − dim(A) − 1 geometric bistellar operations are called flip-deficient. Flip-deficiency cannot occur either in dimension two or in convex posit... |

319 | Homotopy associativity of H-spaces - Stasheff - 1963 |

269 |
Gröbner Bases and Convex Polytopes
- Sturmfels
- 1996
(Show Context)
Citation Context ... vertices whose graph of triangulations hasanisolatedelement. Our construction is likely to have an impact in algebraic geometry too, via the connections between lattice polytopes and toric varieties =-=[21, 23, 31, 43]-=-. For example, in [2, Section 2] and [24, Section 4] the different authors study algebraic schemes based on the poset of subdivisions of an integer point configuration. The connectivity of these schem... |

197 |
Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 15
- Oda
- 1988
(Show Context)
Citation Context ... vertices whose graph of triangulations hasanisolatedelement. Our construction is likely to have an impact in algebraic geometry too, via the connections between lattice polytopes and toric varieties =-=[21, 23, 31, 43]-=-. For example, in [2, Section 2] and [24, Section 4] the different authors study algebraic schemes based on the poset of subdivisions of an integer point configuration. The connectivity of these schem... |

153 |
Incremental topological flipping works for regular triangulations, Algorithmica 15
- Edelsbrunner, Shah
- 1996
(Show Context)
Citation Context ...n 234 with 552 vertices. Connectivity of the graph of triangulations is an important question in computational geometry, where flips are used to enumerate triangulations or to search for optimal ones =-=[12, 17]-=-. But it is also relevant theoretically: if the graph is connected, properties which hold for a particular triangulation and are preserved under flips must hold for any other one. Our negative result ... |

149 |
Topological methods, in Handbook of Combinatorics
- Björner
- 1995
(Show Context)
Citation Context ...and the π-induced subdivisions are its zonotopal tilings [47]. See [37] for a very complete account of the different contexts in which Baues posets appear. The Baues complex of A is the order complex =-=[9]-=- of the strict Baues poset, that is to say, the abstract simplicial complex whose simplices are the chains in the poset. Billera et al. [7] and Rambau and Ziegler [36] proved, respectively, that if di... |

149 | Foundations of algebraic topology - Eilenberg, Steenrod - 1952 |

108 | Rotation distance, triangulations, and hyperbolic geometry - Sleator, Tarjan, et al. |

85 |
Constructions and complexity of secondary polytopes
- Billera, Filliman, et al.
- 1990
(Show Context)
Citation Context ...gulation. More combinatorial definitions are convenient if A is not in convex position, i.e. if some element of A is not a vertex of the convex hull. See Definitions 4.1 and 1.1 for details, and also =-=[6]-=-, [21, Chapter 7], [36], [47, Chapter 9], or the monograph in preparation [14]. There are at least the following three ways to give a structure to the collection of all triangulations of a point confi... |

68 |
The associahedron and triangulations of the n-gon
- Lee
- 1989
(Show Context)
Citation Context ...5]. For the vertex set of a convex polygon, the graph is a classical object in combinatorics, first studied by Stasheff and Tamari [46, 42] and related to associativity structures and to binary trees =-=[26, 41]-=-. It is disturbing that in dimension three, and even assuming convex position, we do not know whether the graph is always connected or not. The graph of triangulations of A contains as an induced subg... |

63 |
PL homeomorphic manifolds are equivalent by elementary shellings
- Pachner
- 1991
(Show Context)
Citation Context ...s. (Geometric bistellar flips in triangulations with rational vertices correspond to pairs blow-up/blow-down in toric varieties.)sA POINT SET WHOSE SPACE OF TRIANGULATIONS IS DISCONNECTED 613 Pachner =-=[32]-=- has shown that any two PL-homeomorphic combinatorial manifolds can be connected by a sequence of the (non-geometric) bistellar operations used in combinatorial topology [10, 19]. (B) The Baues poset.... |

58 | Complete moduli in the presence of semiabelian group action
- Alexeev
(Show Context)
Citation Context ...geometric bistellar flips. (iv) Let Π ′ be a coherent refinement of Π. Then every coherent refinement of Π which refines Π ′ is also a coherent refinement of Π ′ . Proof. (i) This is Lemma 2.12.11 in =-=[2]-=-, where Σc(Π, A) is constructed in a way similar to the construction of fiber polytopes in [8]. For our purposes here it would be sufficient to prove that the poset of coherent refinements of Π is ant... |

46 | The polytope of all triangulations of a point configuration, Documenta Mathematica 1
- Loera, Hoşten, et al.
- 1996
(Show Context)
Citation Context ...erior if its convex hull intersects the interior of conv(A) andisaboundary facet otherwise. It has some combinatorial advantages to consider as elements of T only the maximal simplices, as is done in =-=[6, 13, 21]-=-. Here (but not in Section 4) we prefer to use the convention that lower dimensional ones are also elements, to work more easily with the links and stars of simplices. If S ∈T,thenstarT (S) ={S ′ ∈T :... |

35 | Quotients of toric varieties - Kapranov, Sturmfels, et al. - 1991 |

32 | TOPCOM: Triangulations of point configurations and oriented matroids
- Rambau
(Show Context)
Citation Context ...two point configurations A1 ⊂ R4 and A2 ⊂ R2 with 81 and 4 points respectively. The triangulation has 9 × 64 × 3 × � � 6 2 maximal simplices. Jörg Rambau has checked, with his computer program TOPCOM =-=[34]-=-, that our six-dimensional triangulation with the integer coordinates described in Section 3.4 is in fact a triangulation and has no flips. The current release of TOPCOM includes files which generate ... |

31 |
Subdivisions and triangulations of polytopes
- Lee
- 1997
(Show Context)
Citation Context ...he secondary polytope of A introduced by Gel’fand et al. [20]. This is a polytope of dimension |A| − dim(A) − 1 whose vertices are in bijection with the regular (or coherent) triangulations of A (see =-=[6, 21, 27, 47]-=- and also Section 4.2). Triangulations with less than |A| − dim(A) − 1 geometric bistellar operations are called flip-deficient. Flip-deficiency cannot occur either in dimension two or in convex posit... |

29 |
Geometry of loop spaces and the cobar construction
- Baues
- 1980
(Show Context)
Citation Context ...at if dim(A) =1ordim(P ) − dim(A) = 2, then the Baues complex of any projection π : vert(P ) → A is homotopy equivalent to a (dim(P ) − dim(A) − 1)-sphere. The first case solved a conjecture of Baues =-=[5]-=-. The conjecture that this holds for arbitrary P and A was since called the generalized Baues conjecture, orGBC.Itwas inspired by the fact that the fiber polytope of the projection [8] (a generalizati... |

29 |
The birational geometry of toric varieties
- Morelli
- 1996
(Show Context)
Citation Context ...hold for any other one. Our negative result contrasts the following two positive results in more algebraic-geometric and topological settings: Morelli’s factorization theorem of toric birational maps =-=[1, 30]-=- has as an implication that any two triangulations of a rational point configuration are connected by geometric bistellar operations if we allow the use of arbitrarily many additional rational vertice... |

28 | Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere
- Björner, Lutz
(Show Context)
Citation Context ... DISCONNECTED 613 Pachner [32] has shown that any two PL-homeomorphic combinatorial manifolds can be connected by a sequence of the (non-geometric) bistellar operations used in combinatorial topology =-=[10, 19]-=-. (B) The Baues poset. The polyhedral subdivisions of A form a partially ordered set (poset) with the refinement relation. Its minimal elements are the triangulations and its unique maximal element is... |

25 | A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension
- Abramovich, Matsuki, et al.
- 1999
(Show Context)
Citation Context ...hold for any other one. Our negative result contrasts the following two positive results in more algebraic-geometric and topological settings: Morelli’s factorization theorem of toric birational maps =-=[1, 30]-=- has as an implication that any two triangulations of a rational point configuration are connected by geometric bistellar operations if we allow the use of arbitrarily many additional rational vertice... |

25 |
Discriminants of polynomials in several variables and triangulations of Newton polyhedra
- Gelfand, Zelevinski, et al.
- 1991
(Show Context)
Citation Context ... do not know whether the graph is always connected or not. The graph of triangulations of A contains as an induced subgraph the 1-skeleton of the secondary polytope of A introduced by Gel'fand et al. =-=[20]-=-. This is a polytope of dimension jAj \Gamma dim(A) \Gamma 1 whose vertices are in bijection with the regular (or coherent) triangulations of A; see [6, 21, 27, 47] and also Section 4.2. Triangulation... |

25 | Triangulations of Oriented Matroids - Santos |

22 | Cellular strings on polytopes - Billera, Kapranov, et al. |

22 |
Extension Spaces of Oriented Matroids
- Sturmfels, Ziegler
- 1993
(Show Context)
Citation Context ...g cases of P being a simplex or a cube remain open. They have been solved (positively) only if dim(A) =2,ifdim(P ) − dim(A) =3orifπ is the natural projection onto a cyclic polytope or cyclic zonotope =-=[4, 16, 35, 45]-=-. Thecubecasewithdim(P )=n and dim(A) =d is a special case of the simplex case with dim(P )=n − 1 and dim(A) =d − 1(seebelow). Triangulations of A and bistellar flips between them are precisely the mi... |

20 | Projections of polytopes and the generalized baues conjecture
- Rambau, Ziegler
- 1996
(Show Context)
Citation Context ...torial definitions are convenient if A is not in convex position, i.e. if some element of A is not a vertex of the convex hull. See Definitions 4.1 and 1.1 for details, and also [6], [21, Chapter 7], =-=[36]-=-, [47, Chapter 9], or the monograph in preparation [14]. There are at least the following three ways to give a structure to the collection of all triangulations of a point configuration A: (A) Flips. ... |

18 | The number of geometric bistellar neighbors of a triangulation, Discrete and Computational Geometry 21
- Loera, Urrutia
- 1999
(Show Context)
Citation Context ... Triangulations with less than |A| − dim(A) − 1 geometric bistellar operations are called flip-deficient. Flip-deficiency cannot occur either in dimension two or in convex position in dimension three =-=[15]-=-. In non-convex position in dimension three there are triangulations with more than n 2 vertices and less than 4n flips for arbitrarily large n [38, Section 2]. In dimension four there are triangulati... |

17 | The generalized Baues problem - Reiner - 1999 |

12 |
Visibility complexes and the Baues problem for triangulations in the plane, Discrete Compt. geom
- Edelman, Reiner
(Show Context)
Citation Context ...g cases of P being a simplex or a cube remain open. They have been solved (positively) only if dim(A) =2,ifdim(P ) − dim(A) =3orifπ is the natural projection onto a cyclic polytope or cyclic zonotope =-=[4, 16, 35, 45]-=-. Thecubecasewithdim(P )=n and dim(A) =d is a special case of the simplex case with dim(P )=n − 1 and dim(A) =d − 1(seebelow). Triangulations of A and bistellar flips between them are precisely the mi... |

12 |
The generalized Baues problem. New perspectives in algebraic combinatorics
- Reiner
(Show Context)
Citation Context ...n |A| − 1, all subdivisions of A are π-induced and this is the case of interest to us. When P is a cube its projection is a zonotope and the π-induced subdivisions are its zonotopal tilings [47]. See =-=[37]-=- for a very complete account of the different contexts in which Baues posets appear. The Baues complex of A is the order complex [9] of the strict Baues poset, that is to say, the abstract simplicial ... |

11 |
Homotopy associativity of H-spaces, Trans
- Stasheff
- 1963
(Show Context)
Citation Context ...own to be connected since the early days of computational geometry [25]. For the vertex set of a convex polygon, the graph is a classical object in combinatorics, first studied by Stasheff and Tamari =-=[46, 42]-=- and related to associativity structures and to binary trees [26, 41]. It is disturbing that in dimension three, and even assuming convex position, we do not know whether the graph is always connected... |

9 | F.: The generalized Baues problem for cyclic polytopes
- Rambau, Santos
(Show Context)
Citation Context ...g cases of P being a simplex or a cube remain open. They have been solved (positively) only if dim(A) =2,ifdim(P ) − dim(A) =3orifπ is the natural projection onto a cyclic polytope or cyclic zonotope =-=[4, 16, 35, 45]-=-. Thecubecasewithdim(P )=n and dim(A) =d is a special case of the simplex case with dim(P )=n − 1 and dim(A) =d − 1(seebelow). Triangulations of A and bistellar flips between them are precisely the mi... |

9 | Triangulations with very few geometric bistellar neighbors - Santos |

9 | Oriented Matroids - orner, Vergnas, et al. - 1993 |

7 |
Discriminants of polynomials in several variables and triangulations of Newton polyhedra, Algebra i Analiz 2
- fand, Kapranov
- 1990
(Show Context)
Citation Context ... do not know whether the graph is always connected or not. The graph of triangulations of A contains as an induced subgraph the 1-skeleton of the secondary polytope of A introduced by Gel’fand et al. =-=[20]-=-. This is a polytope of dimension |A| − dim(A) − 1 whose vertices are in bijection with the regular (or coherent) triangulations of A (see [6, 21, 27, 47] and also Section 4.2). Triangulations with le... |

7 |
Software for C 1 -interpolation
- Lawson
- 1977
(Show Context)
Citation Context ...ns of A has the triangulations of A as vertices and the flips between them as edges. Graphs of triangulations in dimension two are known to be connected since the early days of computational geometry =-=[25]-=-. For the vertex set of a convex polygon, the graph is a classical object in combinatorics, first studied by Stasheff and Tamari [46, 42] and related to associativity structures and to binary trees [2... |

6 |
The algebra of bracketings and their enumeration, Nieuw Archief voor Wiskunde 3-10
- Tamari
- 1962
(Show Context)
Citation Context ...own to be connected since the early days of computational geometry [25]. For the vertex set of a convex polygon, the graph is a classical object in combinatorics, first studied by Stasheff and Tamari =-=[46, 42]-=- and related to associativity structures and to binary trees [26, 41]. It is disturbing that in dimension three, and even assuming convex position, we do not know whether the graph is always connected... |

6 |
Topological methods, in: Handbook of combinatorics, pp.1819--1872
- orner
- 1995
(Show Context)
Citation Context ... and the -induced subdivisions are its zonotopal tilings [47]. See [37] for a very complete account of the different contexts in which Baues posets appear. The Baues complex of A is the order complex =-=[9]-=- of the strict Baues poset. That is to say, the abstract simplicial complex whose simplices are the chains in the poset. Billera et al. [7] and Rambau and Ziegler [36] proved, respectively, that if di... |

6 |
Software for C -interpolation
- Lawson
- 1977
(Show Context)
Citation Context ...ns of A has the triangulations of A as vertices and the flips between them as edges. Graphs of triangulations in dimension two are known to be connected since the early days of computational geometry =-=[25]-=-. For the vertex set of a convex polygon, the graph is a classical object in combinatorics, first studied by Stasheff and Tamari [46, 42] and related to associativity structures and to binary trees [2... |

3 |
Matroid bundles and sphere bundles, in: New Perspectives in Algebraic Combinatorics
- Anderson
- 1999
(Show Context)
Citation Context ... k of any realizable oriented matroid M of rank d (the OM-Grassmannian of rank k of M) is homotopy equivalent to the real Grassmannian G k (R d ). This conjecture is relevant in matroid bundle theory =-=[3]-=- and the combinatorial differential geometry introduced by MacPherson [29]. 1. Triangulations and flips Let A be a finite subset of the real affine space Rk and let d denote the dimension of the affin... |

3 |
Toric Hilbert schemes, preprint
- Peeva, Stillman
- 1999
(Show Context)
Citation Context ... and every mono-A-GI is radical [43, Lemma 10.14]. Our point configuration is not unimodular. The A-graded ideals are the closed points of the toric Hilbert scheme of A, defined by Peeva and Stillman =-=[33]-=- (see also [44, Section 5]). Using a notion of flip between mono-A-GI’s, Maclagan and Thomas [28] have shown that a triangulation without flips whose Stanley ideal equals the radical of some mono-A-GI... |

3 | Recent progress on polytopes - Ziegler - 1998 |

3 |
The Baues conjecture
- Azaola
(Show Context)
Citation Context ... P being a simplex or a cube remain open. They have been solved (positively) only if dim(A) = 2, if dim(P ) \Gamma dim(A) = 3 or ifsis the natural projection onto a cyclic polytope or cyclic zonotope =-=[4, 16, 35, 45]-=-. The cube case with dim(P ) = n and dim(A) = d is a special case of the simplex case with dim(P ) = n \Gamma 1 and dim(A) = d \Gamma 1. See below. Triangulations of A and bistellar flips between them... |

3 | Topological methods - orner - 1995 |

3 | The generalized Baues problem for cyclic polytopes I, in "Combinatorics of Convex Polytopes - Rambau, Santos |

2 | The Baues conjecture in corank 3
- Azaola
(Show Context)
Citation Context |

2 |
Triangulations of polyhedra and point sets, in preparation
- Loera, Rambau, et al.
(Show Context)
Citation Context ... position, i.e. if some element of A is not a vertex of the convex hull. See Definitions 4.1 and 1.1 for details, and also [6], [21, Chapter 7], [36], [47, Chapter 9], or the monograph in preparation =-=[14]-=-. There are at least the following three ways to give a structure to the collection of all triangulations of a point configuration A: (A) Flips. Geometric bistellar operations, or flips, are the minim... |

2 |
Über stellare Äquivalenz konvexer Polytope
- Ewald
- 1978
(Show Context)
Citation Context ... DISCONNECTED 613 Pachner [32] has shown that any two PL-homeomorphic combinatorial manifolds can be connected by a sequence of the (non-geometric) bistellar operations used in combinatorial topology =-=[10, 19]-=-. (B) The Baues poset. The polyhedral subdivisions of A form a partially ordered set (poset) with the refinement relation. Its minimal elements are the triangulations and its unique maximal element is... |

2 |
Interactions between real algebraic geometry and discrete and computational geometry, in: Advances in discrete and computational geometry
- Itenberg, Roy
- 1999
(Show Context)
Citation Context ... vertices whose graph of triangulations hasanisolatedelement. Our construction is likely to have an impact in algebraic geometry too, via the connections between lattice polytopes and toric varieties =-=[21, 23, 31, 43]-=-. For example, in [2, Section 2] and [24, Section 4] the different authors study algebraic schemes based on the poset of subdivisions of an integer point configuration. The connectivity of these schem... |

2 | Triangulations of Oriented Matroids, Mem - Santos - 2002 |

2 |
On the refinements of a polyhedral subdivision”, preprint
- Santos
- 1999
(Show Context)
Citation Context ... of the negative part is “flipped out” and the star of the positive part is “flipped in”. This convention is not made by other authors (see [21, page 231]) and will be relevant in our exposition. See =-=[40]-=- for a generalization of the concept of flip in the framework of Baues posets and fiber polytopes. It is sometimes more convenient to focus on flippable facets, which we now introduce, instead of flip... |