Free crossed resolutions from simplicial resolutions with given CW-basis

TL;DR

This paper explores the connection between CW-bases in free simplicial groups and the generation of crossed complexes, focusing on resolutions to compare simplicial and crossed resolutions of groups.

Contribution

It introduces a method to derive crossed resolutions from simplicial resolutions with a specified CW-basis, enhancing understanding of their relationship.

Findings

Established a correspondence between free simplicial resolutions and crossed resolutions.

Provided a systematic approach to generate crossed complexes from simplicial resolutions.

Clarified the role of CW-bases in the construction of resolutions.

Abstract

In this paper, we examine the relationship between a CW-basis for a free simplicial group and methods of freely generating the corresponding crossed complex. Attention is concentrated on the case of resolutions, thus comparing free simplicial resolutions with crossed resolutions of a group.