Lifting relations in right orderable groups

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.