An algorithmic discrete gradient field for non-colliding cell-like objects and the topology of pairs of points on skeleta of simplexes
Year: 2025
Type: article
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 $$\operatorname {DConf}(K,n)$$ DConf ( K , n ) . Computer experimentation suggests that the field is optimal for $$n=2$$ n = 2 and complexes K such as triangulations of surfaces. We study the field W(K, n) for $$n=2$$ n = 2 and $$K=\Delta ^{m,d}$$ K = Δ m , d , the d-dimensional skeleton of the m-dimensional simplex. In particular, we prove that $$\operatorname {DConf}(\Delta ^{m,d},2)$$ DConf ( Δ m , d , 2 ) is $$(\min \{d,m-1\}-1)$$ ( min { d , m - 1 } - 1 ) -connected, has torsion-free homology and admits a minimal cell structure. We compute the Betti numbers of $$\operatorname {DConf}(\Delta ^{m,d},2)$$ DConf ( Δ m , d , 2 ) and, for certain values of d, we prove that $$\operatorname {DConf}(\Delta ^{m,d},2)$$ DConf ( Δ m , d , 2 ) breaks, up to homotopy, as a wedge of (not necessarily equidimensional) spheres.
Source: Selecta Mathematica
Authors: Emilio J. González, Jesús González
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4411896562