Resumen de An algorithmic discrete gradient field for non-colliding cell-like objects and the topology of pairs of points on skeleta of simplexes
J. Emilio González, Jesús González
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 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 = m,d , the d-dimensional skeleton of the m-dimensional simplex. In particular, we prove that DConf(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 DConf(m,d , 2) and, for certain values of d, we prove that DConf(m,d , 2) breaks, up to homotopy, as a wedge of (not necessarily equidimensional) spheres.
