Effective homology techniques allow us to compute homology groups of a wide family of topological spaces. By the Whitehead tower method, this can also be used to compute higher homotopy groups. However, some of these techniques (in particular, the Whitehead tower) rely on the assumption that the starting space is simply connected. For some applications, this problem could be circumvented by replacing the space by its universal cover, which is a simply connected space that shares the higher homotopy groups of the initial space. In this paper, we formalize a simplicial construction for the universal cover, and represent it as a twisted Cartesian product. As we show with some examples, the universal cover of a space with effective homology does not necessarily have effective homology in general. We show two independent sufficient conditions that can ensure it: one is based on a nilpotency property of the fundamental group, and the other one on discrete vector fields. Some examples showing our implementation of these constructions in both SageMath and Kenzo are shown, together with an approach to compute the homology of the universal cover when the group is Abelian even in some cases where there is no effective homology, using the twisted homology of the space.

Topological image analysis is a powerful tool for understanding the structure and topology of images, being persistent homology one of its most popular methods. However, persistent homology requires a chain of inclusions of topological spaces, which can be challenging for digital images. In this article, we explore the use of zigzag persistence, a recent variant of traditional persistence, for digital image processing. To this end, new algorithms are developed to build a simplicial complex associated to a digital image and to compute the relationships between homology classes of a sequence of binary images via zigzag persistence. Additionally, we provide a simple software to use them. We demonstrate its effectiveness by applying it to a real-world problem of analyzing honey bee sperm videos.

Working in a simplicial and constructive context, a new spectral system is defined that relates Serre and Eilenberg–Moore spectral sequences associated to a principal simplicial fibration. The two Eilenberg–Moore spectral sequences (the one where the homology of the fiber is the output, and the other where the homology of the base is computed) are used in our construction. Explicit computer programs are developed, enhancing the Kenzo computer algebra tool to implement that spectral system.

In their original setting, both spectral sequences and persistent homology are algebraic topology tools defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set \Z of integer numbers. Recently, generalizations of both concepts have been proposed which originate from a different choice of the set of indices of the filtration, producing the new notions of multipersistence and spectral system. In this paper, we show that these notions are related, generalizing results valid in the case of filtrations over \Z. By using this relation and some previous programs for computing spectral systems, we have developed a new module for the Kenzo system computing multipersistence. We also present a new invariant providing information on multifiltrations and applications of our algorithms to spaces of infinite type.

In this paper, we present some algorithms and programs for computing generalized spectral sequences, a useful tool in Computational Algebraic Topology which provides topological information on spaces with generalized filtrations over a poset. Our programs have been implemented as a new module for the Kenzo system and solve the classical problems of spectral sequences which are differential maps and extensions. Moreover, combined with the use of effective homology and discrete vector fields, the programs make it possible to compute generalized spectral sequences of big spaces, sometimes of infinite type.

Formal verification of algorithms is traditionally taught in Computer Science studies in a theoretical way by means of the Hoare logic axioms and doing (by hand) exercises of verification of small programs. This work shows our experience with Krakatoa, an automatic theorem prover which allows students to interactively visualize the steps required to prove the correctness of a program.

This paper relates how a "simple" result in combinatorial homotopy eventually led to a totally new understanding of basic theorems in Algebraic Topology, namely the Eilenberg-Zilber theorem, the twisted Eilenberg-Zilber theorem, and finally the Eilenberg-MacLane correspondance between the Classifying Space and Bar constructions. In the last case, it was an amazing lucky consequence of computations based on conjectures not yet proved. The key new tool used in this context is Robin Forman's Discrete Vector Fields theory.

In this paper we report on an experience communicating between the GAP computional algebra system (in particular, its HAP package for homological algebra computations) and the Kenzo computer system for Algebraic Topology. Both systems were made to cooperate through an OpenMath link in order to perform computations in group cohomology. Furthermore, once HAP output is integrated into Kenzo, it can be used to compute more complicated algebraic invariants such as the homology groups of various 2-types.