Characterization of f-vectors of families of convex sets in \({\mathbb{R}}^ d\). I: Necessity of Eckhoff's conditions
type: Item
instance of: scholarly article
title: Characterization of f-vectors of families of convex sets in \({\mathbb{R}}^ d\). I: Necessity of Eckhoff's conditions
zbMATH Open document ID: 0572.52006
DOI: 10.1007/BF02761163
author: Gil Kalai
published in: Israel Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: A finite simplicial complex C is called d-representable iff C is the nerve of a family of convex sets in \({\mathbb{R}}^ d\). J. Eckhoff conjectured in 1974 a characterization of the f-vectors \(f(C)=(f_ 0(C),f_ 1(C),...)\) [where \(f_ k(C)\) denotes the number of k- dimensional faces of C] of d-representable complexes. In this paper G. Kalai proves the necessity of Eckhoff's conditions (the hard part). For the sufficiency compare the forthcoming paper of the author in [J. Comb. Theory, Ser. A, to appear]. The challenging proof uses algebraic methods. The main idea is the introduction of some generalized homology groups for simplicial complexes, based on modules in the exterior algebra of a vector space associated with the simplicial complex. It turns out that the vanishing of some of these groups implies Eckhoff's conditions. The reader should be aware of some unnecessarily misleading notations, for instance \(N=[n]\), \(\bigwedge^ jF_{k+j}=\bigwedge^{j}F_{k+1}\), and the \(f_ i\) denote likewise the components of the f-vector and the members of a basis of a vector space.
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 3913147
zbMATH Keywords: necessity of the condition of Eckhoff
zbMATH Keywords: simplicial complex
zbMATH Keywords: nerve
zbMATH Keywords: family of convex sets
reviewed by: Gerd Wegner
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02761163
OpenAlex ID: W2062068855
description: scientific article; zbMATH DE number 3913147