Dualizing complex of the incidence algebra of a finite regular cell complex

We introduce the visibility complex (a 2-dimensional regular cell complex) of a collection of n pairwise disjoint convex obstacles in the plane. It can be considered as a subdivision of the set of free rays (i.e., rays whose origins lie in free space, the complement of the obstacles). Its cells correspond to collections of rays with the same backward and forward views. The combinatorial complexity of the visibility complex is proportional to the number k of free bitangents of the collection of obstacles. We give an O(n log n+k) time and O(k) working space algorithm for its construction. Furthermore we show how the visibility complex can be used to compute the visibility polygon from a point in O(m log n) time, where m is the size of the visibility polygon. Our method is based on the notions of pseudotriangle and pseudo-triangulation, introduced in this paper.

Download Full-text

Strong Euler well-composedness

Journal of Combinatorial Optimization ◽

10.1007/s10878-021-00837-8 ◽

2021 ◽

Author(s):

Nicolas Boutry ◽

Rocio Gonzalez-Diaz ◽

Maria-Jose Jimenez ◽

Eduardo Paluzo-Hildago

Keyword(s):

Euler Characteristic ◽

General Setting ◽

Cell Complex ◽

Cubical Complexes ◽

Cell Complexes ◽

Boundary Cell ◽

Dimensional Ball ◽

Regular Cell Complex

AbstractIn this paper, we define a new flavour of well-composedness, called strong Euler well-composedness. In the general setting of regular cell complexes, a regular cell complex of dimension n is strongly Euler well-composed if the Euler characteristic of the link of each boundary cell is 1, which is the Euler characteristic of an $$(n-1)$$ ( n - 1 ) -dimensional ball. Working in the particular setting of cubical complexes canonically associated with $$n$$ n D pictures, we formally prove in this paper that strong Euler well-composedness implies digital well-composedness in any dimension $$n\ge 2$$ n ≥ 2 and that the converse is not true when $$n\ge 4$$ n ≥ 4 .

Download Full-text

Coxeter-like complexes

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.312 ◽

2004 ◽

Vol Vol. 6 no. 2 ◽

Author(s):

Eric Babson ◽

Victor Reiner

Keyword(s):

Symmetric Group ◽

Type A ◽

Cell Complex ◽

Coxeter System ◽

Coxeter Complex ◽

Homology Groups ◽

Finite Set ◽

International Audience ◽

Regular Cell Complex ◽

Chessboard Complexes

International audience Motivated by the Coxeter complex associated to a Coxeter system (W,S), we introduce a simplicial regular cell complex Δ (G,S) with a G-action associated to any pair (G,S) where G is a group and S is a finite set of generators for G which is minimal with respect to inclusion. We examine the topology of Δ (G,S), and in particular the representations of G on its homology groups. We look closely at the case of the symmetric group S_n minimally generated by (not necessarily adjacent) transpositions, and their type-selected subcomplexes. These include not only the Coxeter complexes of type A, but also the well-studied chessboard complexes.

Download Full-text

Dualizing Complex of a Toric Face Ring

Nagoya Mathematical Journal ◽

10.1017/s0027763000009806 ◽

2009 ◽

Vol 196 ◽

pp. 87-116 ◽

Cited By ~ 5

Author(s):

Ryota Okazaki ◽

Kohji Yanagawa

Keyword(s):

Free Module ◽

Cell Complex ◽

Topological Properties ◽

Semigroup Rings ◽

Module Theory ◽

Graded Ring ◽

Face Ring ◽

Dualizing Complex ◽

Normality Assumption ◽

Affine Semigroup

A toric face ring, which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Römer and their coauthors recently. In this paper, under the “normality” assumption, we describe a dualizing complex of a toric face ring R in a very concise way. Since R is not a graded ring in general, the proof is not straightforward. We also develop the square-free module theory over R, and show that the Cohen-Macaulay, Buchsbaum, and Gorenstein* properties of R are topological properties of its associated cell complex.

Download Full-text