Computing the homology of universal covers via effective homology and discrete vector fields

TL;DR

This paper develops methods to compute the homology of universal covers of topological spaces using effective homology and discrete vector fields, addressing challenges when the universal cover lacks effective homology.

Contribution

It formalizes a simplicial construction for universal covers as twisted cartesian products and provides conditions ensuring effective homology of these covers.

Findings

Universal cover construction as twisted cartesian product.

Sufficient conditions for effective homology of universal covers.

Implementation in SageMath and Kenzo for examples.

Abstract

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.