Ascending and descending regions of a discrete Morse function

We present an algorithm which produces a decomposition of a regular cellular complex with a discrete Morse function analogous to the Morse-Smale decomposition of a smooth manifold with respect to a smooth Morse function. The advantage of our algorithm compared to similar existing results is that it works, at least theoretically, in any dimension. Practically, there are dimensional restrictions due to the size of cellular complexes of higher dimensions, though. We prove that the algorithm is correct in the sense that it always produces a decomposition into descending and ascending regions of the critical cells in a finite number of steps, and that, after a finite number of subdivisions, all the regions are topological discs. The efficiency of the algorithm is discussed and its performance on several examples is demonstrated.Comment: 23 pages, 12 figure

Coincidence points of maps on Zpα_p{\alpha}-spaces

Sia X uno spazio con una azione libera del gruppo ciclico Z$_{p^{\alpha}}$ ed f : X $\rightarrow$ M una mappa continua. Lo scopo di questo articolo è stimare per mezzo dell'indice Z$_{p^{\alpha}}$ $$A_{f}=\left\{ x\epsilon X\mid f\left(gx\right)=f\left(x\right)\: for\: all\: g\:\epsilon Z_{p^{\alpha}}\right\}$$ quando l'indice dello spazio X è noto ed M verifica opportune proprietà.Let X be a space with a free action of the cyclic group Z$_{p^{\alpha}}$ and f : X $\rightarrow$ M a continuous map. The purpose of this paper is to estimate by means of the Z$_{p^{\alpha}}$- index the size of the set $$A_{f}=\left\{ x\epsilon X\mid f\left(gx\right)=f\left(x\right)\: for\: all\: g\:\epsilon Z_{p^{\alpha}}\right\}$$ when the index of the space X is known, and the space M satisfies certain conditions