Discrete gradient fields on infinite complexes
Let \(M\) be a simplicial complex, and let us say that a pair of simplices \((\sigma,\tau)\) forms a face-pair if \(\sigma\) is a face of \(\tau\) and the difference in their dimensions is exactly one. A discrete vector field \(V\) is a collection of face-pairs such that each simplex in \(M\) belongs to at most one pair in \(V\). To depict this graphically, we draw an arrow from the center of \(\sigma\) to the center of \(\tau\). One way to construct a discrete vector field is from a discrete Morse function: a real-valued function on the set of all simplices of \(M\) with the property that for any simplex \(\sigma\), there is at most one face-pair \((\sigma,\tau)\) such that (1) \(f(\sigma)\geq f(\tau)\), and at most one face-pair \((\nu,\sigma)\) with (2) \(f(\nu)\geq f(\sigma)\). We obtain a discrete vector field on \(M\) from the set of all face-pairs that satisfy (1) or (2). A discrete vector field obtained in this way is said to be a discrete gradient vector field. It is a standard result that if \(M\) is a finite simplicial complex, a discrete vector field \(V\) is gradient if and only if there are no closed \(V\)-paths. Here, a \(V\)-path is a sequence of simplices \(\sigma_0,\tau_0,\sigma_1,\tau_1,\dots\) with \((\sigma_i,\tau_i)\) a face-pair from \(V\) and \((\sigma_{i+1},\tau_i)\) a face-pair not from \(V\). The authors of the paper extend this result to infinite complexes that are locally finite. In the special case when the complex is one-dimensional (i.e., a graph), it is shown that \(V\) having no closed \(V\)-paths is necessary and sufficient to guarantee that \(V\) is gradient. However in the general case, additional criteria must be placed on the \(V\)-paths, in particular on the incidence of two paths, in order to ensure that \(V\) is a discrete gradient field.
scientific article; zbMATH DE number 5919129