Homological optimality in Discrete Morse Theory through chain homotopies

Abstract

a b s t r a c tMorse 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(1)-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.

Searching combinatorial optimality using graph-based homology information

“…First steps towards the simplification of this operator are taken in [1], where HSFs are introduced. More precisely, HSF is a special set of direct graphs on the connectivity graph G(K ), that generalizes the notion of V -path in DMT [7].…”

“…Working with coefficients in Z/2Z, an algebraic topological representation related to a finite cell complex K = (K n ) n≥0 is introduced in [1]. This representation, called The maps ∂ and φ are nilpotent of degree two, that is, ∂∂ = 0 and φφ = 0.…”

“…This is a high computationally complex problem [11] that will be referred here as combinatorial optimality. On the other hand, the global combinatorial structure that underlies in some homology integral operators has recently been discovered under the umbrella of the Homological Spanning Forest (HSF) theory [1,12,13].…”

“…In fact, this kind of labelling process of high dimensional homology classes is performed using directed subgraphs (in general, they are not acyclic) on the connectivity graph of the cell complex. In [1], a homologybased heuristic for finding near optimal discrete gradient vector fields based on iterated Morse reductions is given.…”

“…Informally, an HSF for a cell complex can be seen as a kind of combinatorial compressed representation of a homology integral operator. As main result, we refine the heuristic for computing DGVFs based on the iterative Morse complex reduction technique of [1], reducing the search space for an optimal DGVF to an HSF associated to a homology integral operator. …”