On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras

Abstract

The aim here is to sketch the development of ideas related to brackets and similar concepts: Some purely group theoretical combinatorics due to Ph. Hall led to a proof of the Jacobi identity for the Whitehead product in homotopy theory. Whitehead introduced crossed modules to characterize a second relative homotopy group; guided by combinatorial group theory considerations, Reidemeister and Peiffer explored this kind of structure to develop normal forms for the decomposition of a 3-manifold; but crossed modules are also lurking behind a forgotten approach of Turing to the extension problem for groups: Turing concocted the obstruction 3-cocycle isolated later by Eilenberg-Mac Lane and already proved the Eilenberg-Mac Lane theorem to the effect that the vanishing of the class of that cocycle is equivalent to the existence of a solution for the corresponding extension problem. This Turing cocycle is related to what has come to be known as Teichmueller cocycle. There was a parallel development for Lie algebras including a forgotten paper by Goldberg and, likewise, for Lie-Rinehart algebras and Lie algebroids. Versions of Turing's theorem were discovered several times under such circumstances, and there is rarely a hint at the mutual relationship. Also, Lie-Rinehart algebras have for long occurred in the literature on differential algebra, at least implicitly.