On the topology of the free complexes of convex geometries
type: Item
instance of: scholarly article
title: On the topology of the free complexes of convex geometries
zbMATH Open document ID: 1113.52020
author: Masahiro Hachimori
author: Kenji Kashiwabara
published in: Discrete Mathematics
publication date: 2007-02-13T00:00:00Z
review text: For a finite point configuration \(A\subset\mathbb{R}^n\), a set \(K\) of vertices in \(A\) is free if \(\text{conv\,}K\cap A= K\). Edelman and Reiner (2000) confirmed the conjecture \[ |\text{int\,}A|= (-1)^{n-1} \sum_{K:\text{free}}(-1)^{|K|}|K|,\quad\text{int\,}A:= (\text{int\,conv\,}A)\cap A, \] of Ahrens et al. (1999), and they stated a conjecture about the topology of free complexes. In the present paper this topological conjecture is confirmed, and in view of the deletion of interior points a nice counterexample is presented, where also simplicial complexes play an essential role.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 55P99
zbMATH DE Number: 5124994
zbMATH Keywords: free complex
zbMATH Keywords: convex geometry
zbMATH Keywords: contractibility
zbMATH Keywords: simplicial complex
zbMATH Keywords: closure system
zbMATH Keywords: homotopy
MaRDI profile type: Publication
full work available at URL: j.disc.2006.06.020
OpenAlex ID: W2152440687
cites work: Convexity and the beta invariant
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cites work: A circuit set characterization of antimatroids
cites work: Counting the interior points of a point configuration
cites work: Convex, acyclic, and free sets of an oriented matroid
cites work: Greedoids
DOI: 10.1016/J.DISC.2006.06.020
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description: scientific article; zbMATH DE number 5124994