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The generalized Baues problem
, 1998
"... Abstract. We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex polytope. Along with a discussion of most of the known results, we survey the motivat ..."
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Abstract. We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex polytope. Along with a discussion of most of the known results, we survey the motivation for the problem and its relation to triangulations, zonotopal tilings, monotone paths in linear programming, oriented matroid Grassmannians, singularities, and homotopy theory. Included are several open questions and problems. 1.
The Generalized Baues Problem For Cyclic Polytopes
, 1998
"... The Generalized Baues Problem asks whether for a given point configuration the order complex of all its proper polyhedral subdivisions, partially ordered by refinement, is homotopy equivalent to a sphere. In this paper, an affirmative answer is given for the vertex sets of cyclic polytopes in all di ..."
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The Generalized Baues Problem asks whether for a given point configuration the order complex of all its proper polyhedral subdivisions, partially ordered by refinement, is homotopy equivalent to a sphere. In this paper, an affirmative answer is given for the vertex sets of cyclic polytopes in all dimensions. This yields the first nontrivial class of point configurations with neither a bound on the dimension, the codimension, nor the number of vertices for which this is known to be true. Moreover, it is shown that all triangulations of cyclic polytopes are lifting triangulations. This contrasts the fact that in general there are many nonregular triangulations of cyclic polytopes. Beyond this, we find triangulations of C 11 5 with flip deficiency. This proves—among other things—that there are triangulations of cyclic polytopes that are nonregular for every choice of points on the moment curve.
Zonotopal Subdivisions of Cyclic Zonotopes
, 2001
"... The cyclic zonotope Z…n; d † is the zonotope in R d generated by any ndistinct vectors of the form …1; t; t2;...; td 1 †. It is proved that the refinement poset of all proper zonotopal subdivisions of Z…n; d † which are induced by the canonical projection p: Z…n; d0 †!Z…n; d†, in the sense of Biller ..."
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The cyclic zonotope Z…n; d † is the zonotope in R d generated by any ndistinct vectors of the form …1; t; t2;...; td 1 †. It is proved that the refinement poset of all proper zonotopal subdivisions of Z…n; d † which are induced by the canonical projection p: Z…n; d0 †!Z…n; d†, in the sense of Billera and Sturmfels, is homotopy equivalent to a sphere and that any zonotopal subdivision of Z…n; d † is shellable. The first statement gives an affirmative answer to the generalized Baues problem in a new special case and refines a theorem of Sturmfels and Ziegler on the extension space of an alternating oriented matroid. An important ingredient in the proofs is the fact that all zonotopal subdivisions of Z…n; d † are stackable in a suitable direction. It is shown that, in general, a zonotopal subdivision is stackable in a given direction if and only if a certain associated oriented matroid program is Euclidean, in the sense of Edmonds and Mandel.
The generalized Baues problem for cyclic polytopes II
, 1999
"... Given an affine surjection of polytopes : P! Q, the Generalized Baues Problem asks whether the poset of all proper polyhedral subdivisions of Q which are induced by the map has the homotopy type of a sphere. We extend earlier work of the last two authors on subdivisions of cyclic polytopes to give ..."
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Given an affine surjection of polytopes : P! Q, the Generalized Baues Problem asks whether the poset of all proper polyhedral subdivisions of Q which are induced by the map has the homotopy type of a sphere. We extend earlier work of the last two authors on subdivisions of cyclic polytopes to give an affirmative answer to the problem for the natural surjections between cyclic polytopes: C(n; d # ) ! C(n; d) for all 1 d! d # ! n.
On Some Instances Of The Generalized Baues Problem
"... . We present an approach applicable to certain instances of the generalized Baues problem of Billera, Kapranov, and Sturmfels. This approach involves two applications of Alexander/SpanierWhitehead duality. We use this to show that the generalized Baues problem has a positive answer for the surj ..."
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. We present an approach applicable to certain instances of the generalized Baues problem of Billera, Kapranov, and Sturmfels. This approach involves two applications of Alexander/SpanierWhitehead duality. We use this to show that the generalized Baues problem has a positive answer for the surjective map of cyclic polytopes C(n; d) ! C(n; 2) if n ! 2d + 2 and d 9. Draft version April 1998. Not to be submitted for publication, just for the fun of it! 1. Introduction The generalized Baues problem (GBP) of Billera, Kapranov, and Sturmfels [3, x3] asks whether a certain poset associated to an affine surjection : P ! Q of polytopes has the homotopy type of a sphere, when the poset is endowed with a standard topology. Although it is known that this question has a negative answer in general, there are many interesting special cases for which the answer is known or conjectured to be positive. For motivation and a survey of general results on the GBP, see [15]. The purpose of this ...
Projections Of Polytopes On The Plane And The Generalized Baues Problem
 Proc. Amer. Math. Soc
, 1999
"... . Given an affine projection : P ! Q of a dpolytope P onto a polygon Q, it is proved that the poset of proper polytopal subdivisions of Q which are induced by has the homotopy type of a sphere of dimension d \Gamma 3 if maps all vertices of P into the boundary of Q. This result, originally c ..."
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. Given an affine projection : P ! Q of a dpolytope P onto a polygon Q, it is proved that the poset of proper polytopal subdivisions of Q which are induced by has the homotopy type of a sphere of dimension d \Gamma 3 if maps all vertices of P into the boundary of Q. This result, originally conjectured by Reiner, is an analogue of a result of Billera, Kapranov and Sturmfels on cellular strings on polytopes and explains the significance of the interior point of Q present in the counterexample to their generalized Baues conjecture, constructed by Rambau and Ziegler. 1. Introduction Motivated by their theory of fiber polytopes [6] [18, Lecture 9], Billera and Sturmfels have associated to any affine projection of convex polytopes : P ! Q the Baues poset !(P ! Q) of proper polytopal subdivisions of Q which are induced by . This poset reduces to the poset of proper cellular strings [7] on P with respect to , if dim(Q) = 1, and can be described in general as the poset of proper ...
The number of triangulations of the cyclic polytope C(n,n4)
"... We show that the exact number of triangulations of the cyclic polytope C(n; n \Gamma 4) is (n + 4)2 n\Gamma4 2 \Gamma n if n is even and i 3n+11 2 p 2 j 2 n\Gamma4 2 \Gamma n if n is odd. These formulas were previously conjectured by the second author. Our techniques are based on Gal ..."
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We show that the exact number of triangulations of the cyclic polytope C(n; n \Gamma 4) is (n + 4)2 n\Gamma4 2 \Gamma n if n is even and i 3n+11 2 p 2 j 2 n\Gamma4 2 \Gamma n if n is odd. These formulas were previously conjectured by the second author. Our techniques are based on Gale duality and the concept of virtual chambers. They further provide formulas for the number of triangulations which use a specific simplex. We also compute a tight upper bound for the number of regular triangulations of C(n; n \Gamma 4) in terms of n. Introduction By a triangulation of a finite point set A ae R d we mean a simplicial complex geometrically realized in R d with vertex set contained in A and which covers the convex hull of A. If A is the vertex set of a polytope P this definition agrees with the standard definition of triangulation of P . The collection of all triangulations of a fixed point set has attracted attention in recent years for its connections to algebraic ...