Not just an idle game" (the story of higher dimensional versions of the Poincar{\'e} fundamental group)

TL;DR

This paper reviews the development of higher-dimensional, nonabelian versions of the fundamental group using groupoids and multiple base points, building on Whitehead's work and addressing historical debates on homotopy groups.

Contribution

It introduces a construction of higher homotopy group analogues that are nonabelian, extending classical ideas with new algebraic structures like groupoids.

Findings

Construction of higher-dimensional nonabelian fundamental groups

Use of groupoids and multiple base points in homotopy theory

Addressing historical limitations of abelian higher homotopy groups

Abstract

The title of this article is partially taken from writings of A. Einstein. In the 1932 ICM at Z\"urich, when E. \vCech gave a seminar on higher homotopy groups of a pointed space and proved they were abelian for n > 1. On these grounds, H. Hopf and P.S. Aleksandrov persuaded \^Cech to withdraw his paper, so that only a small paragraph appeared in the Proceedings. This article reviews the eventual construction by the author and P.J. Higgins, of reasonably nonabelian higher dimensional versions of the fundamental group, using groupoids and many base points, and stimulated by long term work of J.H.C. Whitehead on crossed modules.