Computing the homology of groups: the geometric way

TL;DR

This paper introduces algorithms for computing the homology of groups using geometric methods, implemented as modules in the Kenzo system, enabling calculations for various classes of groups and extensions.

Contribution

The paper develops new algorithms for homology computation of groups via geometric techniques, integrated into the Kenzo system, including methods for effective homology and free resolutions.

Findings

Algorithms for homology of Eilenberg-MacLane spaces implemented in Kenzo

Effective homology constructed from finite free resolutions

Methods for computing homology of 2-types and central extensions

Abstract

In this paper we present several algorithms related with the computation of the homology of groups, from a geometric perspective (that is to say, carrying out the calculations by means of simplicial sets and using techniques of Algebraic Topology). More concretely, we have developed some algorithms which, making use of the effective homology method, construct the homology groups of Eilenberg-MacLane spaces K(G,1) for different groups G, allowing one in particular to determine the homology groups of G. Our algorithms have been programmed as new modules for the Kenzo system, enhancing it with the following new functionalities: - construction of the effective homology of K(G,1) from a given finite free resolution of the group G; - construction of the effective homology of K(A,1) for every finitely generated Abelian group A (as a consequence, the effective homology of K(A,n) is also available in Kenzo, for all n); - computation of homology groups of some 2-types; - construction of the effective homology for central extensions. In addition, an inverse problem is also approached in this work: given a group G such that K(G,1) has effective homology, can a finite free resolution of the group G be obtained? We provide some algorithms to solve this problem, based on a notion of norm of a group, allowing us to control the convergence of the process when building such a resolution.