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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