Cancellation of a critical pair in discrete Morse theory and its effect on (co)boundary operators

Abstract: Discrete Morse theory helps us compute the homology groups of simplicial complexes in an efficient manner. A "good" gradient vector field reduces the number of critical simplices, simplifying the homology calculations by reducing them to the computation of homology groups of a simpler chain complex. This homology computation hinges on an efficient enumeration of gradient trajectories. The technique of cancelling pairs of critical simplices reduces the number of critical simplices, though it also perturbs the gradient trajectories. In this article, we demonstrate that (the matrix of) a certain modified boundary operator of interest can be derived from the corresponding original boundary operator through a sequence of elementary row operations. Thus, it eliminates the need of enumeration of the new gradient trajectories. We also obtain a similar result for coboundary operators.