Infinitary commutativity and fundamental groups of topological monoids

Abstract

The well-known Eckmann-Hilton Principle may be applied to prove that fundamental groups of H-spaces are commutative. In this paper, we identify an infinitary analogue of the Eckmann-Hilton Principle that applies to fundamental groups of all topological monoids and slightly more general objects called pre-∆-monoids. In particular, we show that every pre-∆-monoid M is "transfinitely π 1 -commutative" in the sense that permutation of the factors of any infinite loop-concatenation indexed by a countably infinite order and based at the identity e P M is a homotopy invariant action. We also give a detailed account of fundamental groups of James reduced products and apply transfinite π 1 -commutativity to make several computations.