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

TL;DR

This paper derives a combinatorial formula for updating boundary operators in discrete Morse theory after cancelling a critical pair, avoiding the enumeration of gradient trajectories.

Contribution

It provides an explicit, combinatorial method to compute modified boundary operators post-cancellation, simplifying homology calculations.

Findings

Derived an explicit formula for boundary operator modification.

The formula can be obtained via elementary row operations.

Eliminates the need to enumerate gradient trajectories.

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, in a purely combinatorial manner, we derive an explicit formula for computing the modified boundary operators after cancelling a critical pair, in terms of the original boundary operators. The same formula can be obtained through a sequence of elementary row operations on the original boundary operators. Thus, it eliminates the need of enumeration of the new gradient trajectories. We also obtain a similar result for coboundary operators.