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

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.