Unions of Cockroft two-complexes

TL;DR

This paper introduces a combinatorial group-theoretic hypothesis that characterizes when unions of connected Cockcroft two-complexes are Cockcroft, linking topological properties to group homology.

Contribution

It presents a new necessary and sufficient condition for unions of Cockcroft two-complexes to be Cockcroft, involving the second homology of groups.

Findings

Established a homological criterion for Cockcroft unions

Applied the hypothesis to analyze third homology of groups

Connected topological and algebraic properties in group complexes

Abstract

A combinatorial group-theoretic hypothesis is presented that serves as a necessary and sufficient condition for a union of connected Cockcroft two-complexes to be Cockcroft. This hypothesis has a component that can be expressed in terms of the second homology of groups. The hypothesis is applied to the study of the third homology of groups given by generators and relators.