For a positive integer n and a finite simplicial complex K, we describe an algorithmic procedure constructing a maximal discrete gradient field W(K, n) on Abrams’ discretized configuration space \(\operatorname {DConf}(K,n)\) . Computer experimentation suggests that the field is optimal for \(n=2\) and complexes K such as triangulations of surfaces. We study the field W(K, n) for \(n=2\) and \(K=\Delta ^{m,d}\) , the d-dimensional skeleton of the m-dimensional simplex. In particular, we prove that \(\operatorname {DConf}(\Delta ^{m,d},2)\) is \((\min \{d,m-1\}-1)\) -connected, has torsion-free homology and admits a minimal cell structure. We compute the Betti numbers of \(\operatorname {DConf}(\Delta ^{m,d},2)\) and, for certain values of d, we prove that \(\operatorname {DConf}(\Delta ^{m,d},2)\) breaks, up to homotopy, as a wedge of (not necessarily equidimensional) spheres.
