P.H. Edelman, J. Rambau and V. Reiner, On subdivision posets of cyclic polytopes, in Combinatorics of Polytopes (K. Fukuda and G.M. Ziegler, eds.), European J. Combin. 21 (2000), 85-101.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....on n and d. Our main object of study will be the set of zonotopal tilings [25, x7.5] or zonotopal subdivisions, of Z(n; d) The significance of this set comes from two directions. First, it provides an analogue to the much studied set of polyhedral subdivisions of the classical cyclic polytope [1, 2, 10, 11, 17, 18, 19]. Second, by the BohneDress Theorem on zonotopal tilings [8] it bijects to the set of single element extensions of the dual oriented matroid of C n;d , a set which is important in the study of the higher Bruhat orders of Manin and Schechtman [16] see [15, 24] Our first result is motivated by ....

P.H. Edelman, J. Rambau and V. Reiner, On subdivision posets of cyclic polytopes, in "Combinatorics of Convex Polytopes" (K. Fukuda and G.M. Ziegler, eds.), European J. Combin. 21 (2000), xx--xx.

....apply the Suspension Lemma to show that the higher Bruhat orders by MANIN SCHECHTMAN [5] a certain generalization of the weak Bruhat order on the symmetric group) are spherical, no matter whether we order by inclusion or by single step inclusion. The Suspension Lemma has been applied again in [2] to uniformly prove the sphericity of the two (possibly different) higher Stasheff Tamari orders [3] on the set of triangulations of a cyclic polytope. The author would like to thank Anders Bj#orner, Victor Reiner, and G#unter M. Ziegler for helpful discussions. 2. THE LEMMA In this section we ....

Paul Edelman, Victor Reiner, and J#org Rambau, On subdivision posets of cyclic polytopes, Manuscript, 1997.

....[14] various special cases have remained of interest in the literature; see [15, Section 4] One such relates to subdivisions of cyclic polytopes. Another is the case where P is a simplex, in which #(P Q) is the poset of all proper polyhedral subdivisions of Q and is simply denoted #(Q) In [9] an affirmative answer to the problem was given in the case of the poset of all subdivisions of cyclic polytopes of dimension at most 3. This was recently improved in [13] to all dimensions, as follows. Theorem 1.1. 13, Theorem 1.1] For all 1 # d n, the Baues poset #(C(n, d) of all proper ....

P.H. EDELMAN, J. RAMBAU, AND V. REINER, On subdivision posets of cyclic polytopes, MSRI Preprint 1997-030.

....surjection : C(n; d 0 ) C(n; d) and one can show that the set of subdivisions of C(n; d) and also the subset of induced subdivisions are independent of the choice of the t i . Subdivisions and the GBP for these maps between cyclic polytopes have been studied a great deal in the recent past [1, 7, 12, 8, 13], and in [15, Conjecture 19] we conjectured that the GBP always has a positive answer for these maps : C(n; d 0 ) C(n; d) This is known to be true in the following cases: ffl d = 1 [3] ffl d 0 Gamma d 2 [14] ffl n = d 0 1 [13] ffl n = d 0 2 and d = 2 [1] We will consider ....

P. H. Edelman, J. Rambau, and V. Reiner, On subdivision posets of cyclic polytopes, MSRI preprint 1997-030, available at http://www.msri.org/MSRI-preprints/online/1997-030.html.

....[15] and the recent survey [18] for an overview. The Generalized Baues Problem as it is investigated in this paper 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 [8] it is shown that the Generalized Baues Problem has an affirmative answer for cyclic polytopes in dimensions not exceeding three. We show that this is actually true in all dimensions. Theorem 1.1. For all d 0 and n d the Baues poset w(C(n;d) of all proper polyhedral subdivisions of the ....

Paul Edelman, Victor Reiner, and Jörg Rambau, On subdivision posets of cyclic polytopes, Preprint 1997-030, MSRI, April 1997.

....[14] various special cases have remained of interest in the literature; see [15, Section 4] One such relates to subdivisions of cyclic polytopes. Another is the case where P is a simplex, in which (P Q) is the poset of all proper polyhedral subdivisions of Q and is simply denoted (Q) In [9] an affirmative answer to the problem was given in the case of the poset of all subdivisions of cyclic polytopes of dimension at most 3. This was recently improved in [13] to all dimensions, as follows. Theorem 1.1. 13, Theorem 1.1] For all 1 d n, the Baues poset (C(n; d) of all proper ....

P.H. EDELMAN, J. RAMBAU, AND V. REINER, On subdivision posets of cyclic polytopes, MSRI Preprint 1997-030.

No context found.

Paul Edelman, Victor Reiner, and Jörg Rambau, On subdivision posets of cyclic polytopes, Preprint 1997-030, MSRI, April 1997.

....proper part of the Baues poset (suitably topologized [9] is homotopy equivalent to a (dim(P ) Gamma dim(Q) Gamma 1) dimensional sphere. This is known to be true when dim(Q) 1 [7] and when dim(P ) Gamma dim(Q) 2, but false in general [25, 27] In previous work on cyclic polytopes ( 26] and [15] for d 3) it was shown to be true for C(n; n Gamma 1) C(n; d) We prove the following result, which in particular answers the question positively for C(n; n Gamma 2) C(n; 2) Further progress on this question for C(n; d) C(n; 2) is contained in [29] Theorem 1.2. Let : P ....

P.H. Edelman, J. Rambau and V. Reiner, On subdivision posets of cyclic polytopes, MSRI preprint 1997-030, available at http://www.msri.org/MSRI-preprints/online/1997-030.html.

No context found.

P.H. Edelman, J. Rambau and V. Reiner, On subdivision posets of cyclic polytopes, in Combinatorics of Polytopes (K. Fukuda and G.M. Ziegler, eds.), European J. Combin. 21 (2000), 85-101.

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