An algorithmic discrete gradient field for non-colliding cell-like objects and the topology of pairs of points on skeleta of simplexes

TL;DR

This paper introduces an algorithmic discrete gradient field for discretized configuration spaces of non-colliding cell-like objects, analyzing their topological properties and computing Betti numbers for specific cases.

Contribution

It presents a new algorithmic procedure for constructing discrete gradient fields on configuration spaces and analyzes their topological features for certain simplicial complexes.

Findings

The discrete gradient field is generically optimal.

The space DConf(Δ^{m,d},2) is (min{d,m-1}-1)-connected.

DConf(Δ^{m,d},2) has torsion-free homology and admits a minimal cell structure.

Abstract

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 $\text{DConf}(K,n)$. Computer experimentation shows that the field is generically optimal. 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 $\text{DConf}(\Delta^{m,d},2)$ is $(\min\{d,m-1\}-1)$-connected, has torsion-free homology and admits a minimal cell structure. We compute the Betti numbers of $\text{DConf}(\Delta^{m,d},2)$ and, for certain values of $d$, we prove that $\text{DConf}(\Delta^{m,d},2)$ breaks, up to homotopy, as a wedge of (not necessarily equidimensional) spheres.