Arrangements of Hyperplanes, Higher Braid Groups and Higher Bruhat Orders

Triangulations of cyclic polytopes and higher Bruhat orders

“…In this paper we examine the structure of the first higher Stasheff-Tamari order y x (n, d) on the set of all triangulations of the cyclic polytope C(n, d) (definitions below), introduced by Edelman and Reiner [7]. It turns out that it is similarly structured as the higher Bruhat order i -2, d-1) of Manin and Schechtman [15]; in particular it is bounded.Given a triangulation of the convex hull of a finite point configuration in Euclidean (/-space that is not satisfying a certain quality measure, can one find a better, or even the best triangulation (with respect to this measure) by performing a finite sequence of (computationally cheap) local transformations? A necessary condition for the latter case is that any possible triangulation is accessible by this kind of transformation.…”

“…In this paper we examine the structure of the first higher Stasheff-Tamari order y x (n, d) on the set of all triangulations of the cyclic polytope C(n, d) (definitions below), introduced by Edelman and Reiner [7]. It turns out that it is similarly structured as the higher Bruhat order i -2, d-1) of Manin and Schechtman [15]; in particular it is bounded.…”

Kohno Systems on Manin–Schechtman Configuration Spaces

“…The Manin-Schechtman configuration space U (k + 2, k) has a simple geometric structure [10]. It is the complement in C n to the union of (k + 2) hyperplanes D J , |J| = k + 1, J ⊂ K, |K| = k + 2, which contains the only plane D K , |K| = k + 2, of codimension 2.…”

“…gives a description of generators of homology groups H 2 (U (n, k), Z) and generating relations of the higher braid groups B n (k) [3,6,10].…”

“…Thus, U (n, k) = C n \ ∪ J∈C(n, k+1) D J . By definition, the fundamental group B n (k) = π 1 (U (n, k), c 0 ) is called a Manin-Schechtman higher braid group [10]. Now let C(n, m), 1 ≤ m ≤ n, be the set of subset K ⊂ {1, .…”