Characterization of f-vectors of families of convex sets in \({\mathbb{R}}^ d\). I: Necessity of Eckhoff's conditions (Q1062277)

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.