Covering morphisms of groupoids, derived modules and a 1-dimensional Relative Hurewicz Theorem

TL;DR

This paper establishes a 1-dimensional version of the Relative Hurewicz Theorem, connecting it to abelianizations of groupoids, covering spaces, and derived modules, filling a gap in the mathematical literature.

Contribution

It introduces a dimension 1 Relative Hurewicz Theorem and relates it to abelianizations, covering morphisms, and derived modules of groupoids, providing new insights in algebraic topology.

Findings

A 1-dimensional Relative Hurewicz Theorem is formulated.

The theorem relates to abelianizations of groupoids and covering morphisms.

Connections to Crowell's derived modules are established.

Abstract

We fill a lacuna in the literature by giving a version in dimension 1 of the Relative Hurewicz Theorem, and relate this to abelianisations of groupoids, covering spaces and covering morphisms of groupoids, and Crowell's notion of derived modules.