A colimit of classifying spaces

TL;DR

This paper revisits a group-theoretic approach to understanding the homotopy groups of certain spaces, providing new formulas and an alternative proof for the homotopy groups of a 2-sphere.

Contribution

It introduces a colimit construction of classifying spaces that offers new insights and formulas for homotopy and homology groups, including an alternative proof for the 2-sphere case.

Findings

Derived new formulas for homotopy and homology groups

Provided an alternative proof for the homotopy groups of a 2-sphere

Recalled a group-theoretic description of the first non-vanishing homotopy group

Abstract

We recall a group-theoretic description of the first non-vanishing homotopy group of a certain (n+1)-ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu's group-theoretic description of the homotopy groups of a 2-sphere.