Homological optimality in Discrete Morse theory through chain homotopies

Morse theory is a fundamental tool for analyzing the geometry and topologyof smooth manifolds. This tool was translated by Forman to discretestructures such as cell complexes, by using discrete Morse functions or equivalentlygradient vector fields. Once a discrete gradient vector field has beendefined on a finite cell complex, information about its homology can be directlydeduced from it. In this paper we introduce the foundations of ahomology-based heuristic for finding optimal discrete gradient vector fieldson a general finite cell complex K. The method is based on a computationalhomological algebra representation (called homological spanning forest orHSF, for short) that is an useful framework to design fast and efficient algorithmsfor computing advanced algebraic-topological information (classificationof cycles, cohomology algebra, homology A(∞)-coalgebra, cohomologyoperations, homotopy groups,. . . ). Our approach is to consider the optimalityproblem as a homology computation process for a chain complex endowedwith an extra chain homotopy operator.