Computational Cohomology Project

Introduction

The purpose of this website is to disseminate the results of work conducted in the framework of the research project Computational Cohomology and Applications, no. FCOMP-01-0124-FEDER-010645 (FCT reference PTDC/MAT/098871/2008).

The team working for this project consists of three people:

Dr. Paweł Pilarczyk (Centre of Mathematics, University of Minho, Braga, Portugal), the Principal Investigator of the project,
Dr. Pedro Real (Departamento de Matemática Aplicada I, E.T.S. de Ingeniería Informática, Universidad de Sevilla, Spain).
Dr. Jacek Szybowski (Faculty of Applied Mathematics, AGH University of Science and Technology, Krakow, Poland).

The project is aimed at the development of algorithms and software for effective computation of comprehensive homological information (homology, cohomology, and related operations) on cubical complexes, using the cubical chain complex structure directly (as opposed to going through its simplicial decomposition). An extract from the project summary follows.

The recent rapid progress in the computing hardware and software incurs high demand for new technologies with advanced mathematical background. On the other hand, the increasing power of computers opens new possibilities in applying computational methods to solving mathematical problems. In this project we address both demands by contributing to the development of new methods for the effective computation of specific algebraic-topological invariants.

The purpose of this research is to develop efficient cohomology computation methods in the context of cubical complexes, that is, cellular complexes with respect to a fixed rectangular grid in Rⁿ. The results will be published in scientific journals and at this website, and publicised through talks delivered at international conferences and workshops.

Although algebraic topology is in principle an abstract branch of pure mathematics, it has been recently proved to provide a theoretical basis for powerful computational methods. Many results, however, focus on the computation of the homology of topological spaces. Unfortunately, in some cases the homology groups alone provide insufficient amount of topological information, and thus it is necessary to study finer invariants. Therefore, in this project we plan to study the cohomology, whose additional multiplicative structure provides considerable advantage over plain homology.

In the classical approach, compact polyhedra are represented by means of simplicial complexes. However, in many applications, including digital imagery and numerical methods, it is more natural to represent subsets of Rⁿ by means of n-dimensional hypercubes with respect to a fixed rectangular grid. Subdividing these hypercubes into simplices yields a considerable overhead and detrimentally impacts the speed of computations. Using the cubical cellular structure directly is substantially more efficient, and thus we are going to focus on this approach in our project.

The homology theory based on cubical cells with respect to a rectangular grid has been recently undergoing intensive development, and algorithms have been proposed for the computation of homology of cubical sets (polyhedra built upon cubical cells), and homomorphisms induced in homology by continuous maps between such sets. Moreover, effective geometric reduction techniques have been recently developed in the context of cubical sets, which substantially increases the speed of homology computation. In our project we are going to benefit from these reduction techniques, and adapt them for the purpose of cohomology computation.

Algorithms for the computation of cohomological operations have been recently studied in the context of simplicial complexes. The crucial idea which allows to go beyond the computation of cohomology generators alone, and to address the issue of computing cohomological operations, is to construct a cochain contraction operator, which provides means to determine the cohomology class of every cochain. Although very powerful, this method is also memory-consuming, and thus needs substantial improvements. We plan to address this issue and combine the purely algebraic construction with geometric approach in order to achieve a solution feasible for practical applications.

The definition of cohomological operations is originally based on simplicial complexes. The case of cubical complexes is a non-trivial generalization and will require a considerable amount of investigation. Using the ideas of simplicial cohomology and singular cubical cohomology, we are going to investigate the meaning of the Hom functor from the combinatorial point of view in order to develop an approach to cohomology computation based directly on finite cubical complexes.

Although the potential scale of applications of the methods developed within the framework of this project is immense, we are going to limit our attention to two specific applications: the computation of the Conley index in dynamical systems, and the analysis of multi-dimensional digital images. The advantage of using cohomology for the computation of the Conley index is undeniable, as it will allow to distinguish various cases currently identified by homology, especially important if the Conley index over a base is considered. In the case of digital images, the approach based directly on the cubical structure of pixels, voxels or higher-dimensional boxes will result in new quality of efficient algorithms, especially if they are combined with geometric reduction techniques.