Paper detail

Infinitary commutativity and fundamental groups of topological monoids

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-$Δ$-monoids. In particular, we show that every pre-$Δ$-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 $e\in M$ 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.