"Not just an idle game":(examining some historical conceptual arguments in homotopy theory)

TL;DR

This paper analyzes the historical development of concepts in homotopy theory, focusing on initial reactions, key contributions, and the evolution of ideas like higher homotopy groups and fundamental group generalizations.

Contribution

It provides a detailed historical and conceptual analysis of the development of higher homotopy theory and related algebraic structures from the 1930s onward.

Findings

Reactions to Cech's seminar influenced homotopy theory development

Whitehead's use of free crossed modules advanced higher homotopy concepts

Evolution of fundamental group generalizations and Van Kampen theorem

Abstract

Part of the title of this article is taken from writings of Einstein, which argue that we need to exercise our ability to analyse familiar concepts, to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little $\ldots$. My aim is to do this for the initial negative reactions to the seminar by E. Cech on higher homotopy groups to the ICM meeting in Z\" urich in 1932; then the subsequent work of Hurewicz, the use of groupoids and so the use of many base points, and how J.H.C. Whitehead's use of free crossed modules gave rise to a successful search for higher dimensional versions of the fundamental group and of the theorem of Van Kampen.