Polyhedral representation of discrete Morse functions (Q389487)

Two common discrete analogs of Morse theory are due to \textit{T. F. Banchoff} [J. Differ. Geom. 1, 245--256 (1967; Zbl 0164.22903)] and \textit{R. Forman} [Adv. Math. 134, No. 1, 90--145, Art. No. AI971650 (1998; Zbl 0896.57023)]. This article establishes a connection between these two approaches. To make this connection, the proof relies on Forman's discrete Morse functions on posets. Specifically, it is shown that for every discrete Morse-Forman function on a finite regular CW complex, the cell is Forman-critical if and only if its barycenter is Banchoff-critical with respect to projection onto a line spanned by a predetermined unit vector in \(S^{m-1}\).