H. Molina-Abril, P. Real:

"Homological optimality in Discrete Morse theory through chain homotopies";

Pattern Recognition Letters,33(2012), 11; 1501 - 1506.

Morse theory is a fundamental tool for analyzing the geometry and topology

of smooth manifolds. This tool was translated by Forman to discrete

structures such as cell complexes, by using discrete Morse functions or equivalently

gradient vector fields. Once a discrete gradient vector field has been

defined on a finite cell complex, information about its homology can be directly

deduced from it. In this paper we introduce the foundations of a

homology-based heuristic for finding optimal discrete gradient vector fields

on a general finite cell complex K. The method is based on a computational

homological algebra representation (called homological spanning forest or

HSF, for short) that is an useful framework to design fast and efficient algorithms

for computing advanced algebraic-topological information (classification

of cycles, cohomology algebra, homology A(∞)-coalgebra, cohomology

operations, homotopy groups,. . . ). Our approach is to consider the optimality

problem as a homology computation process for a chain complex endowed

with an extra chain homotopy operator.

Discrete Morse Theory, cell complex, integral-chain complex, chain homotopy, graph, homology, gradient vector field

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