On the Baues Conjecture in Corank 3
نویسنده
چکیده
A special case of the Generalized Baues Conjecture states that the order complex of the Baues poset of an acyclic vector connguration A (the Baues complex of A) is homotopy equivalent to a sphere of dimension equal to the corank of A minus one. The Baues poset of A is the set of proper polyhedral subdivisions of A ordered by reenement. Recently, Santos has found a counterexample in corank 317 to the Baues conjecture. Here, we study the case of corank 3. The techniques we use also allow us to show that if a corank 3 vector connguration is not acyclic, then its Baues complex is contractible.
منابع مشابه
The Graph of Triangulations of a Point Configuration with d +4 Vertices Is 3-Connected
We study the graph of bistellar ips between triangulations of a vector connguration A with d + 4 elements in rank d + 1 (i.e. with corank 3), as a step in the Baues problem. We prove that the graph is connected in general and 3-connected for acyclic vector conngurations, which include all point conngurations of dimension d with d + 4 elements. Hence, every pair of triangulations can be joined b...
متن کاملh-Vectors of Small Matroid Complexes
Stanley conjectured in 1977 that the h-vector of a matroid simplicial complex is a pure O-sequence. We give simple constructive proofs that the conjecture is true for matroids of rank less than or equal to 3, and corank 2. We used computers to verify that Stanley’s conjecture holds for all matroids on at most nine elements.
متن کاملRealizable but not Strongly Euclidean Oriented Matroids
The extension space conjecture of oriented matroid theory claims that the space of all (non-zero, non-trivial, single-element) extensions of a re-alizable oriented matroid of rank r is homotopy equivalent to an (r ? 1)-sphere. In 1993, Sturmfels and Ziegler proved the conjecture for the class of strongly Euclidean oriented matroids, which includes those of rank at most 3 or corank at most 2. Th...
متن کاملProjections of Polytopes on the Plane and Thegeneralized Baues
Given an aane projection : P ! Q of a d-polytope 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 ? 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 polyto...
متن کاملProjections of Polytopes on the Plane and the Generalized Baues Problem
Given an affine projection π : P → Q of a d-polytope P onto a polygon Q, it is proved that the poset of proper polytopal subdivisions of Q which are induced by π has the homotopy type of a sphere of dimension d− 3 if π 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...
متن کامل