The Cubical Cohomology Ring: An Algorithmic Approach

Abstract

A cohomology ring algorithm in a dimension-independent framework of combinatorial cubical complexes is developed with the aim of applying it to the topological analysis of high-dimensional data. This approach is convenient in the cupproduct computation and motivated, among others, by interpreting pixels or voxels in digital images as cubes. The S-complex theory and so called co-reductions are adopted to build a cohomology ring algorithm speeding up the algebraic computations.

Distributed computation of low-dimensional cup products

“…To implement (2) one can use the Alexander-Whitney diagonal approximation formula in the case of simplicial spaces, and Serre's analogue of this for cubical spaces. Details of the cubical analogue can be found in [26], and details on practical implementations of these two formulae can be found in [16,18,14,15,28,17,24]. In this section we assume that X is an arbitrary connected regular finite CW-space and observe that for k = 2 the homomorphism (2), and hence the cup product (1), can be read directly from a group presentation P = x | r for the fundamental group π 1 X.…”

“…In Section 3 we illustrate the method on the integral cohomology ring of a 3-dimensional digital image. Previous papers [16,18,14,15] have described different approaches to computing the cohomology ring, over Z/2Z, of cubical and simplicial spaces arising from 3-dimensional digital images; these papers are based on techniques in [28,17,24]. The fundamental group algorithm in [1] involves the construction of an admissible discrete vector field on X, and this construction can consume significant memory and time for large CW-spaces X.…”

Computation of cubical homology, cohomology, and (co)homological operations via chain contraction

“…An equivalent recursive formula for the cubical cup product (1) was also derived in [35] (see also [32]) in an explicit way in the context of cubical sets as in [34], with [44] as a theoretical basis. The key idea of that construction was to see the cup product in cohomology as a map induced by the composition of two chain maps (see [27,Chapter 3] and [39, Chapter XIII, §3]):…”

Computing Cup Products in $$\mathbb {Z}_2$$ Z 2 -Cohomology of 3D Polyhedral Complexes

“…R. Gonzalez-Diaz, Jimenez and B. Medrano [8] introduced a method for computing cup products directly from a cubical complex associated with a given 3D digital image (no additional subdivisions are necessary). More recently, motivated by problems in high-dimensional data analysis, T. Kaczynski and M. Mrozek [17] gave an algorithm for computing the cohomology algebra of a cubical complex of arbitrary dimension; they are currently in the process of implementing their algorithm. In the context of persistent cohomology, A. Yarmola [34] discussed the computation of cup products in the cohomology of a finite simplicial complex over a field.…”