Characterizing equivalent discrete Morse functions (Q732766)

\textit{R. Forman} in [Adv. Math. 134, No.~1, 90--145 (1998; Zbl 0896.57023)] and [Math. Z. 228, No.~4, 629--681 (1998; Zbl 0922.58063)] introduced the notion of discrete Morse function defined on a finite regular \(cw-\)complex and developed a discrete Morse theory as a purely combinatorial tool for studying the topology of the considered complex. In the present paper \(M\) is a (finite or infinite) simplicial complex. We denote by \(\sigma^{(p)}\in M\) a simplex of dimension \(p\), where \((p)\), when needed, is indicating. A discrete Morse function on \(M\) is a proper function \(f:M\rightarrow\mathbb{R}\) such that: \(\text{card}\{\tau^{(p+1)}>\sigma:f(\tau)\leq f(\sigma)\}\leq1\) and \(\text{card}\{\upsilon^{(p-1)}<\sigma:f(\upsilon)\geq f(\sigma)\}\leq1\), where \(\sigma\), \(\tau\), \(\upsilon\) are simplices of \(M\). Associated to the concept of discrete Morse function we have the notion of critical simplex. A \(p\)-simplex \(\sigma\) is said to be critical with respect to \(f\) if \(\text{card}\{\tau^{(p+1)}>\sigma:f(\tau)\leq f(\sigma)\}=0\) and \(\text{card}\{\upsilon^{(p-1)}<\sigma:f(\upsilon)\geq f(\sigma)\}=0\). A discrete vector field \(V\) defined on \(M\) is a collection of pairs \((\alpha^{(p)}<\beta^{(p+1)})\) of simplices of \(M\) such that every simplex is in, at most, one pair \(V\). Using a characterization of non-critical simplices one can prove that a Morse function \(f:M\rightarrow\mathbb{R}\) induces a discrete vector field on \(M,\) called gradient vector field induced by \(f\), as follows: a pair \((\tau^{(p-1)}<\sigma^{(p)})\) is in the gradient vector field induced by \(f\) if and only if \(f(\sigma^{(p)})\leq f(\tau^{(p-1)}).\) Now two discrete Morse functions \(f\) and \(g\) defined on \(M\) are said to be equivalent if for every pair of simplices \(\alpha^{(p)}\) and \(\beta^{(p+1)} \) in \(M\) such that \(\alpha^{(p)}<\beta^{(p+1)}\) it holds that, \(f(\alpha)<f(\beta)\) if and only if \(g(\alpha)<g(\beta).\) The main theorem of the paper is the following: Two discrete Morse functions \(f\) and \(g\) defined on a simplicial complex \(M\) are equivalent if and only if \(f\) and \(g\) induce the same gradient vector field.