Lifting relations in right orderable groups

TL;DR

This paper investigates when chain complexes over group rings can be realized as cellular chain complexes of simply connected G-CW complexes, providing solutions for right orderable groups with specific conditions on the second differential.

Contribution

It establishes that for right orderable groups, certain chain complexes are realizable, and solves the relation lifting problem in cases with cyclic relation modules.

Findings

Realizability of chain complexes for right orderable groups under specific conditions.

Solution to the relation lifting problem for groups with cyclic relation modules.

Identification of conditions on the second differential for realizability.

Abstract

In this article we study the following problem: given a chain complex $A_*$ of free $\mathbb{Z}G$-modules, when is $A_*$ isomorphic to the cellular chain complex of some simply connected $G$-CW-complex? Such a chain complex is called realisable. Wall studied this problem in the 60's and reduced it to a problem involving only the second differential $d_2$, now known as the relation lifting problem. We show that if $G$ is right orderable and $d_2$ is given by a matrix of a certain form, then $A_*$ is realisable. As a special case, we solve the relation lifting problem for right orderable groups with cyclic relation module.