The second homotopy group in terms of colorings of locally finite models and new results on asphericity

TL;DR

This paper introduces a novel approach to describe the second homotopy group of CW-complexes using G-colorings of posets, leading to new insights on asphericity and a generalized Hurewicz theorem.

Contribution

It provides a new method to analyze the second homotopy group via locally finite models and G-colorings, extending classical theorems to broader classes of complexes.

Findings

Generalizes the Hurewicz theorem for non-simply-connected complexes

Derives new criteria for asphericity of 2D complexes

Provides a framework linking homotopy and homology through G-colorings

Abstract

We describe the second homotopy group of any CW-complex $K$ by analyzing the universal cover of a locally finite model of $K$ using the notion of $G$-coloring of a partially ordered set. As applications we prove a generalization of the Hurewicz theorem, which relates the homotopy and homology of non-necessarily simply-connected complexes, and derive new results on asphericity for two-dimensional complexes and group presentations.