On the topology of the free complexes of convex geometries (Q864169)

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.